Structure of -Lie Algebras with Involutive...
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Research Article Structure of -Lie Algebras with Involutive Derivations Ruipu Bai , 1 Shuai Hou, 2 and Yansha Gao 2 1 College of Mathematics and Information Science, Hebei University, Key Laboratory of Machine Learning and Computational Intelligence of Hebei Province, Baoding 071002, China 2 College of Mathematics and Information Science, Hebei University, Baoding 071002, China Correspondence should be addressed to Ruipu Bai; [email protected] Received 30 January 2018; Revised 28 June 2018; Accepted 11 July 2018; Published 2 September 2018 Academic Editor: Kaiming Zhao Copyright © 2018 Ruipu Bai et al. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We study the structure of -Lie algebras with involutive derivations for ≥2. We obtain that a 3-Lie algebra is a two-dimensional extension of Lie algebras if and only if there is an involutive derivation on = 1 c −1 such that dim 1 =2 or dim −1 = 2, where 1 and −1 are subspaces of with eigenvalues 1 and −1, respectively. We show that there does not exist involutive derivations on nonabelian -Lie algebras with = 2 for ≥1. We also prove that if is a (2 + 2)-dimensional (2 + 1)-Lie algebra with dim 1 =, then there are involutive derivations on if and only if is even, or satisfies 1≤≤+2. We discuss also the existence of involutive derivations on (2 + 3)-dimensional (2 + 1)-Lie algebras. 1. Introduction Derivation is an important tool in studying the structure of n-Lie algebras [1]. e derivation algebra () of an -Lie algebra over the field of real numbers is the Lie algebra of the automorphism group (), which is a Lie group if dim <∞ [2]. Any -Lie algebra-module (, ) is a module of the inner derivation algebra (), which is a linear Lie algebra [3]. Also, derivations have close relationship with extensions of -Lie algebras. e concept of 3-Lie classical Yang-Baxter equations is introduced in [4]. It is known that if there is an involutive derivation on , then (, {, , } ) is a 3-pre-Lie algebra, where {, , } = ((, )()), ∀, , ∈ , and the 3- Lie algebra is a subadjacent 3-Lie algebra of (, {, , } ), and =∑ ∗ ⊗ ( ) − ( )⊗ ∗ is a skew-symmetric solution of the 3-Lie classical Yang-Baxter equation in the 3-Lie algebra ⋉ ∗ ∗ , where { 1 ,⋅⋅⋅ , } is a basis of and { ∗ 1 ,⋅⋅⋅ , ∗ } is the dual basis of ∗ . Due to this importance of involutive derivations on 3- Lie algebras, we investigate in this paper the existence of involutive derivations on finite dimensional n-Lie algebras. More specifically, in Section 2, we discuss the properties of involutive derivations on n-Lie algebras. In Section 3, we study the existence of involutive derivations on (2 + 2)- dimensional (2 + 1)-Lie algebras. In Section 4, we consider the existence of involutive derivations on (2+3)-dimensional (2 + 1)-Lie algebras. In Section 5, we investigate a class of 3-Lie algebras with involutive derivations which are two- dimensional extension of Lie algebras. In the following, we assume that all algebras are over an algebraically closed field F with characteristic zero, is the identity mapping, and Z is the set of integers. For ∈ F and an F -linear mapping on a vector space , denotes the subspace { ∈ | () = }. 2. -Lie Algebras with Involutive Derivations An -Lie algebra [1] is a vector space over a field F equipped with a linear multiplication [,⋅⋅⋅ ,] : ∧ → satisfying, for all 1 ,⋅⋅⋅, , 2 ,⋅⋅⋅ , ∈, [[ 1 ,⋅⋅⋅, ], 2 ,⋅⋅⋅, ] = ∑ =1 [ 1 ,⋅⋅⋅,[ , 2 ,⋅⋅⋅ , ],⋅⋅⋅ , ]. (1) Equation (1) is usually called the generalized Jacobi identity, or Filippov identity. Hindawi International Journal of Mathematics and Mathematical Sciences Volume 2018, Article ID 7202141, 9 pages https://doi.org/10.1155/2018/7202141
Transcript of Structure of -Lie Algebras with Involutive...
Research ArticleStructure of 119899-Lie Algebras with Involutive Derivations
Ruipu Bai 1 Shuai Hou2 and Yansha Gao2
1College of Mathematics and Information Science Hebei University Key Laboratory of Machine Learning andComputational Intelligence of Hebei Province Baoding 071002 China2College of Mathematics and Information Science Hebei University Baoding 071002 China
Correspondence should be addressed to Ruipu Bai bairuipuhbueducn
Received 30 January 2018 Revised 28 June 2018 Accepted 11 July 2018 Published 2 September 2018
Academic Editor Kaiming Zhao
Copyright copy 2018 Ruipu Bai et alThis is an open access article distributed under theCreative CommonsAttribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
We study the structure of 119899-Lie algebras with involutive derivations for 119899 ge 2 We obtain that a 3-Lie algebra119860 is a two-dimensionalextension of Lie algebras if and only if there is an involutive derivation 119863 on 119860 = 1198601 c 119860minus1 such that dim1198601 = 2 or dim119860minus1 =2 where 1198601 and 119860minus1 are subspaces of 119860 with eigenvalues 1 and minus1 respectively We show that there does not exist involutivederivations on nonabelian 119899-Lie algebras with 119899 = 2119904 for 119904 ge 1 We also prove that if 119860 is a (2119904 + 2)-dimensional (2119904 + 1)-Lie algebrawith dim1198601 = 119903 then there are involutive derivations on 119860 if and only if 119903 is even or 119903 satisfies 1 le 119903 le 119904 + 2 We discuss also theexistence of involutive derivations on (2119904 + 3)-dimensional (2119904 + 1)-Lie algebras
1 Introduction
Derivation is an important tool in studying the structure ofn-Lie algebras [1] The derivation algebra 119863119890119903(119860) of an 119899-Liealgebra 119860 over the field of real numbers is the Lie algebraof the automorphism group 119860119906119905(119860) which is a Lie group ifdim119860 lt infin [2] Any 119899-Lie algebra-module (119881 120588) is a moduleof the inner derivation algebra 119886119889(119860) which is a linear Liealgebra [3] Also derivations have close relationship withextensions of 119899-Lie algebras
The concept of 3-Lie classical Yang-Baxter equations isintroduced in [4] It is known that if there is an involutivederivation 119863 on 119860 then (119860 119863) is a 3-pre-Lie algebrawhere 119909 119910 119911119863 = 119863(119886119889(119909 119910)119863(119911)) forall119909 119910 119911 isin 119860 and the 3-Lie algebra 119860 is a subadjacent 3-Lie algebra of (119860 119863) and119903 = sum119894 119890lowast119894 otimes119863(119890119894) minus119863(119890119894) otimes 119890lowast119894 is a skew-symmetric solution ofthe 3-Lie classical Yang-Baxter equation in the 3-Lie algebra119860⋉119886119889lowast119860lowast where 1198901 sdot sdot sdot 119890119898 is a basis of 119860 and 119890lowast1 sdot sdot sdot 119890lowast119898is the dual basis of 119860lowast
Due to this importance of involutive derivations on 3-Lie algebras we investigate in this paper the existence ofinvolutive derivations on finite dimensional n-Lie algebrasMore specifically in Section 2 we discuss the properties ofinvolutive derivations on n-Lie algebras In Section 3 we
study the existence of involutive derivations on (2119904 + 2)-dimensional (2119904 + 1)-Lie algebras In Section 4 we considerthe existence of involutive derivations on (2119904+3)-dimensional(2119904 + 1)-Lie algebras In Section 5 we investigate a classof 3-Lie algebras with involutive derivations which are two-dimensional extension of Lie algebras
In the following we assume that all algebras are over analgebraically closed field F with characteristic zero 119868119889 is theidentity mapping and Z is the set of integers For 120582 isin F andan F-linear mapping 119863 on a vector space 119860 119860120582 denotes thesubspace 119909 isin 119860 | 119863(119909) = 1205821199092 119899-Lie Algebras with Involutive Derivations
An 119899-Lie algebra [1] is a vector space119860 over a field F equippedwith a linear multiplication [ sdot sdot sdot ] and119899119860 997888rarr 119860 satisfyingfor all 1199091 sdot sdot sdot 119909119899 1199102 sdot sdot sdot 119910119899 isin 119860[[1199091 sdot sdot sdot 119909119899] 1199102 sdot sdot sdot 119910119899]
= 119899sum119894=1
[1199091 sdot sdot sdot [119909119894 1199102 sdot sdot sdot 119910119899] sdot sdot sdot 119909119899] (1)
Equation (1) is usually called the generalized Jacobi identityor Filippov identity
HindawiInternational Journal of Mathematics and Mathematical SciencesVolume 2018 Article ID 7202141 9 pageshttpsdoiorg10115520187202141
2 International Journal of Mathematics and Mathematical Sciences
The derived algebra of an 119899-Lie algebra 119860 is a subalgebraof 119860 generated by [1199091 sdot sdot sdot 119909119899] for all 1199091 sdot sdot sdot 119909119899 isin 119860 and isdenoted by 1198601 We use 119885(119860) to denote the center of 119860 thatis 119885(119860) = 119909 | 119909 isin 119860 [119909 119860 sdot sdot sdot 119860] = 0
A derivation of 119860 is an endomorphism of 119860 satisfying
119863([1199091 sdot sdot sdot 119909119899]) = 119899sum119894=1
[1199091 sdot sdot sdot 119863 (119909119894) sdot sdot sdot 119909119899] forall1199091 sdot sdot sdot 119909119899 isin 119860 (2)
If a derivation 119863 satisfies that 1198632 = 119868119889 then 119863 is calledan involutive derivation on 119860 Der(119860) denotes the derivationalgebra of 119860
For 1199091 sdot sdot sdot 119909119899minus1 isin 119860 map ad(1199091 sdot sdot sdot 119909119899minus1) 119860 997888rarr 119860ad (1199091 sdot sdot sdot 119909119899minus1) (119909) = [1199091 sdot sdot sdot 119909119899minus1 119909] forall119909 isin 119860 (3)
is called a leftmultiplication defined by elements 1199091 sdot sdot sdot 119909119899minus1From (1) left multiplications are derivations
The following lemma can be easily verified
Lemma 1 Let 119881 be a finite dimensional vector space over Fand 119863 be an endomorphism of 119881 with 1198632 = 119868119889 Then 119881 canbe decomposed into the direct sum of subspaces 119881 = 1198811 + 119881minus1where 1198811 = V isin 119881 | 119863V = V and 119881minus1 = V isin 119881 | 119863V = minusV
If 119860 is a finite dimensional 119899-Lie algebra with an involu-tive derivation 119863 then we have119860 = 1198601 + 119860minus1 (4)
Lemma 2 Let 119860 be an 119899-Lie algebra over F If 119863 isin 119863119890119903(119860) isan involutive derivation then for all 1199091 sdot sdot sdot 119909119899 isin 119860[1199091 sdot sdot sdot 119909119899] = minus2119899 minus 1sum
119894lt119895
[1199091 sdot sdot sdot 119909119894minus1 119863 (119909119894) 119909119894+1 sdot sdot sdot 119909119895minus1 119863 (119909119895) 119909119895+1 sdot sdot sdot 119909119899] (5)
[119863 (1199091) sdot sdot sdot 119863 (119909119899)] = minus2119899 minus 1sum119894lt119895
[1198631199091 sdot sdot sdot 119863 (119909119894minus1) 119909119894119863 (119909119894+1) sdot sdot sdot 119863 (119909119895minus1) 119909119895 119863 (119909119895+1) sdot sdot sdot 119863 (119909119899)] (6)
Proof If 119863 is an involutive derivation on 119860 then for all1199091 sdot sdot sdot 119909119899 isin 119860[1199091 sdot sdot sdot 119909119899] = 1198632 ([1199091 sdot sdot sdot 119909119899])= 119863( 119899sum
119894=1
[1199091 sdot sdot sdot 119863 (119909119894) sdot sdot sdot 119909119899])= 119899 [1199091 sdot sdot sdot 119909119899]+ 2 sum
1le119894lt119895le119899
[1199091 sdot sdot sdot 119863 (119909119894) sdot sdot sdot 119863 (119909119895) sdot sdot sdot 119909119899] (7)
Equation (5) follows Equation (6) follows from (4) and1198632 =119868119889
Theorem 3 Let 119860 be a finite dimensional 119899-Lie algebra with119899 = 2119904 119904 ge 1 Then there is an involutive derivation 119863 on 119860 ifand only if 119860 is abelian
Proof If 119860 is abelian then the result is trivialConversely let 119863 be an involutive derivation on 119860 By
Lemma 1 119860 = 1198601 + 119860minus1 Then for any 119894 isin Z 1 le 119894 le 1198991199091 sdot sdot sdot 119909119899 isin 1198601 and 1199101 sdot sdot sdot 119910119899 isin 119860minus1119863 ([1199091 sdot sdot sdot 119909119894 1199101 sdot sdot sdot 119910119899minus119894])= 119894 [1199091 sdot sdot sdot 119909119894 1199101 sdot sdot sdot 119910119899minus119894]minus (119899 minus 119894) [1199091 sdot sdot sdot 119909119894 1199101 sdot sdot sdot 119910119899minus119894]= (2119894 minus 2119904) [1199091 sdot sdot sdot 119909119894 1199101 sdot sdot sdot 119910119899minus119894] isin 1198602119894minus2119904119863 ([1199091 sdot sdot sdot 119909119899]) = 2119904 [1199091 sdot sdot sdot 119909119899] 119863 ([1199101 sdot sdot sdot 119910119899]) = minus2119904 [1199101 sdot sdot sdot 119910119899] (8)
Thanks to plusmn2119904 = plusmn1 and 2119894 minus 2119904 = plusmn1 1198602119894minus119899 = 119860plusmn2119904 = 0Therefore 119860 is abelian
Theorem 4 Let 119860 be a finite dimensional 119899-Lie algebra with119899 = 2119904+1 119904 ge 1 and119863 be an involutive derivation on119860 Then1198601 and 119860minus1 are abelian subalgebras and[1198601 sdot sdot sdot 1198601⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
119894
119860minus1 sdot sdot sdot 119860minus1⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟2119904+1minus119894
] = 0forall1 le 119894 le 2119904 119894 = 119904 119904 + 1 (9)
[1198601 sdot sdot sdot 1198601⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟119904
119860minus1 sdot sdot sdot 119860minus1⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟119904+1
] sube 119860minus1 (10)
[1198601 sdot sdot sdot 1198601⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟119904+1
119860minus1 sdot sdot sdot 119860minus1⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟119904
] sube 1198601 (11)
Proof Since 119863 isin 119863119890119903119860 [1198601 sdot sdot sdot 1198601⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟119894
119860minus1 sdot sdot sdot 119860minus1⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟2119904+1minus119894
] sube1198602119894minus2119904minus1 0 le 119894 le 2119904 + 1 If [1198601 sdot sdot sdot 1198601⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟119894
119860minus1 sdot sdot sdot 119860minus1⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟2119904+1minus119894
] = 0then 2119894 minus 2119904 minus 1 = plusmn1 that is 119894 = 119904 + 1 or 119894 = 119904 Therefore[1198601 sdot sdot sdot 1198601] = [119860minus1 sdot sdot sdot 119860minus1] = 0 The result follows
Theorem5 Let119860 be an119898-dimensional 119899-Lie algebra with 119899 =2119904 + 1 119904 ge 1 Then there is an involutive derivation on119860 if andonly if 119860 has the decomposition 119860 = 119861 + 119862 (as direct sum ofsubspaces) and
[119861 sdot sdot sdot 119861⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟119894
119862 sdot sdot sdot 119862⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟2119904+1minus119894
] = 00 le 119894 le 2119904 + 1 119894 = 119904 119904 + 1 (12)
[119861 sdot sdot sdot 119861⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟119904
119862 sdot sdot sdot 119862⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟119904+1
] sube 119862[119861 sdot sdot sdot 119861⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟119904+1
119862 sdot sdot sdot 119862⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟119904
] sube 119861 (13)
International Journal of Mathematics and Mathematical Sciences 3
Proof If there is an involutive derivation 119863 on 119860 then byTheorem 4 119861 = 1198601 and 119862 = 119860minus1 satisfy (12) and (13)
Conversely define an endomorphism 119863 of 119860 by 119863(119909) =119909119863(119910) = minus119910 forall119909 isin 119861 119910 isin 119862 Then 1198632 = 119868119889 119861 = 1198601 and119862 = 119860minus1 By (12) and (13) 119863 is a derivation
Corollary 6 Let 119860 be a (2119904 + 1)-dimensional (2119904 + 1)-Liealgebra with the multiplication [1198901 sdot sdot sdot 1198902119904+1] = 1198901 where1198901 sdot sdot sdot 1198902119904+1 is a basis of 119860 Then the linear mapping 119863 119860 997888rarr 119860 defined by 119863(119890119894) = 119890119894 1 le 119894 le 119904 + 1 119863(119890119895) =minus119890119895 119904 + 2 le 119895 le 2119904 + 1 is an involutive derivation on119860Proof The result follows from a direct computation
3 Involutive Derivations on(119899 + 1)-Dimensional 119899-Lie Algebraswith 119899 = 2119904 + 1
In this section we study involutive derivations on (119899 + 1)-dimensional 119899-Lie algebras over F FromTheorem 3 we onlyneed to discuss the case of 119899 = 2119904 + 1 119904 ge 1Lemma 7 (see [5]) Let 119860 be an (119899 + 1)-dimensional non-abelian 119899-Lie algebra over F 119899 ge 3 Then up to isomorphisms119860 is one and only one of the following possibilities
(1198871) [1198902 1198903 sdot sdot sdot 119890119899+1] = 1198901(1198872) [1198901 1198902 sdot sdot sdot 119890119899] = 1198901(1198881) [1198902 sdot sdot sdot 119890119899+1] = 1198901[1198901 1198903 sdot sdot sdot 119890119899+1] = 1198902
(1198882) [1198902 sdot sdot sdot 119890119899+1] = 1205721198901 + 1198902[1198901 1198903 sdot sdot sdot 119890119899+1] = 1198902 120572 isin F 120572 = 0(1198883) [1198901 1198903 sdot sdot sdot 119890119899+1] = 1198901[1198902 sdot sdot sdot 119890119899+1] = 1198902(119889119903) [1198901 sdot sdot sdot 119890119894 sdot sdot sdot 119890119899+1] = 119890119894 1 le 119894 le 119903
(14)
where 1198901 sdot sdot sdot 119890119899+1 is a basis of119860 3 le 119903 le 119899+1 and 119890119894 meansthat 119890119894 is omitted
Theorem 8 Let 119860 be a (2119904 + 2)-dimensional (2119904 + 1)-Liealgebra over F and dim1198601 = 119903 Then there exists an involutivederivation 119863 on 119860 if and only if 119903 is even or 0 le 119903 le 119904+2Proof If dim1198601 = 119903 le 119904 + 2 then by Lemma 7 and a directcomputation the linear mapping 119863 119860 997888rarr 119860 defined by119863(119890119894) = 119890119894 119863(119890119895) = minus119890119895 1 le 119894 le 119904 + 2 119904 + 3 le 119895 le 2119904 + 2 is aninvolutive derivation on 119860
Now we discuss the case dim1198601 = 119903 ge 119904 + 3 Let1198901 sdot sdot sdot 1198902119904+2 be a basis of 119860 and the multiplication in thebasis be as follows
119890119894 = (minus1)2119904+2+119894 [1198901 sdot sdot sdot 119890119894 sdot sdot sdot 1198902119904+2] = 2119904+2sum119897=1
120573119894119897119890119897120573119894119897 isin 119865 1 le 119894 le 2119904 + 2 (15)
where 120573119894119897 isin F 1 le 119894 119897 le 2119904 + 2 Thanks toTheorem 3 in [1] 119860is a 3-Lie algebra if and only if the (2119904 + 2) times (2119904 + 2)-matrix119861 = (120573119894119897) is symmetric
If 119903 = 2119905 gt 3 2 le 119905 le 119904 + 1 then define the multiplicationon 119860 by
[1198901 sdot sdot sdot 119890119894 sdot sdot sdot 1198902119904+2] = (minus1)119894 1198902119904+3minus1198941 le 119894 le 119905 3 minus 119905 lt 119894 minus 2119904 le 2[1198901 sdot sdot sdot 119890119895 sdot sdot sdot 1198902119904+2] = 0 119905 lt 119895 le 2119904 + 3 minus 119905
(16)
that is 1205731198942119904+3minus119894 = 1205732119904+3minus119894119894 = 1 for 1 le 119894 le 119905 or 2119904 + 3 minus 119905 lt119894 le 2119904 + 2 and others are zero Then 119861 = (120573119894119897) is symmetricTherefore 119860 is a (2119904 + 1)-Lie algebra with the multiplication(16)
Define an endomorphism119863 of119860 by119863119890119894 = 119890119894 1 le 119894 le 119904+1and 119863119890119895 = minus119890119895 for 119904 + 2 le 119895 le 2119904 + 2Then 119863 is an involutivederivation on 119860
For the case dim1198601 = 119903 = 2119905 + 1 ge 119904 + 3 Suppose 119897 =dim1198601 1198971015840 = dim119860minus1
If there is an involutive derivation 119863 on 119860 then byTheorem 4 119897 + 1198971015840 = 2119904 + 2 119904 le 119897 le 119904 + 2 and 119904 le 1198971015840 le 119904 + 2Since dim1198601 = 119903 = 2119905 + 1 ge 119904 + 3 1198601 cap 1198601 = 0 and1198601 cap119860minus1 = 0Therefore dim1198601 = dim119860minus1 = 119904+ 1 Withoutloss of generality we can suppose 1198901 sdot sdot sdot 119890119904+1 sube 1198601 and119890119904+2 sdot sdot sdot 1199092119904+2 sube 119860minus1 By (10) and (11) the (2119904+2)times(2119904+2)-matrix 119861 = (120573119894119897) defined by (15) is nonsymmetric which is acontradiction Therefore if dim1198601 = 119903 = 2119905 + 1 ge 119904 + 3 thenthere do not exist involutive derivations on 119860
By Theorem 8 if 119860 is a 10-dimensional 9-Lie algebrawith dim1198601 = 7 or 9 then there does not exist involutivederivation on 119860 If 1 le dim1198601 = 119903 le 10 and 119903 = 7 9 thenthere are involutive derivations on 1198604 Involutive Derivations on(119899 + 2)-Dimensional 119899-Lie Algebraswith 119899 = 2119904 + 1
By Theorem 3 we only need to discuss the case where 119899 isodd So we suppose that 119860 is a (2119904 + 3)-dimensional (2119904 + 1)-Lie algebra over F 119904 ge 1 and that 119864119905 = Diag(1 sdot sdot sdot 1) is the(119905 times 119905)-unit matrix
Lemma9 (see [6]) Let119860 be a (2119904+3)-dimensional (2119904+1)-Liealgebra over F with a basis 1198901 sdot sdot sdot 1198902119904+3Then119860 is isomorphicto one and only one of the following possibilities
4 International Journal of Mathematics and Mathematical Sciences
(119886) 119860 119894119904 119886119899 119886119887119890119897119894119886119899(119887) dim1198601 = 1
(1198871) [1198902 sdot sdot sdot 1198902119904+2] = 1198901(1198872) [1198901 sdot sdot sdot 1198902119904+1] = 1198901(119888) dim1198601 = 2(1198881) [1198902 sdot sdot sdot 1198902119904+2] = 1198901[1198903 sdot sdot sdot 1198902119904+3] = 1198902
(1198882) [1198902 sdot sdot sdot 1198902119904+2] = 1198901[1198902 1198904 sdot sdot sdot 1198902119904+3] = 1198902[1198901 1198904 sdot sdot sdot 1198902119904+3] = 1198901
(1198883)
[1198902 sdot sdot sdot 1198902119904+2] = 1205721198901 + 1198902[1198901 1198903 sdot sdot sdot 1198902119904+2] = 1198902[1198902 1198904 sdot sdot sdot 1198902119904+3] = 1198902[1198901 1198904 sdot sdot sdot 1198902119904+3] = 1198901(1198884)
[1198902 sdot sdot sdot 1198902119904+2] = 1198901[1198902 1198903 sdot sdot sdot 1198902119904+2] = 1198902[1198902 1198904 sdot sdot sdot 1198902119904+3] = 1198902[1198901 1198904 sdot sdot sdot 1198902119904+3] = 1198901(1198885) [1198902 sdot sdot sdot 1198902119904+2] = 1198901[1198901 1198903 sdot sdot sdot 1198902119904+2] = 1198902
(1198886) [1198902 sdot sdot sdot 1198902119904+2] = 1205721198901 + 1198902[1198901 1198903 sdot sdot sdot 1198902119904+2] = 1198902 120572 isin F 120572 = 0(1198887)[1198901 1198903 sdot sdot sdot 1198902119904+2] = 1198901[1198902 1198903 sdot sdot sdot 1198902119904+2] = 1198902(119889) dim1198601 = 3(1198891)
[1198902 sdot sdot sdot 1198902119904+2] = 1198901[1198902 1198904 sdot sdot sdot 1198902119904+3] = 1198902[1198903 sdot sdot sdot 1198902119904+3] = 1198903(1198892)
[1198902 sdot sdot sdot 1198902119904+2] = 1198901[1198903 sdot sdot sdot 1198902119904+3] = 1198903 + 1205721198902[1198902 1198904 sdot sdot sdot 1198902119904+3] = 1198903[1198901 1198904 sdot sdot sdot 1198902119904+3] = 1198901
(1198893)
[1198902 sdot sdot sdot 1198902119904+2] = 1198901[1198903 1198904 sdot sdot sdot 1198902119904+3] = 1198903[1198902 1198904 sdot sdot sdot 1198902119904+3] = 1198902[1198901 1198904 sdot sdot sdot 1198902119904+3] = 21198901(1198894)
[1198902 sdot sdot sdot 1198902119904+2] = 1198901[1198901 1198903 sdot sdot sdot 1198902119904+2] = 1198902[1198901 1198902 1198904 sdot sdot sdot 1198902119904+2] = 1198903(1198895)
[1198901 1198904 sdot sdot sdot 1198902119904+3] = 1198901[1198902 1198904 sdot sdot sdot 1198902119904+3] = 1198903[1198903 1198904 sdot sdot sdot 1198902119904+3] = 1205731198902 + (1 + 120573) 1198903(1198896)
[1198901 1198904 sdot sdot sdot 1198902119904+3] = 1198901[1198902 1198904 sdot sdot sdot 1198902119904+3] = 1198902[1198903 1198904 sdot sdot sdot 1198902119904+3] = 1198903(1198897)
[1198901 1198904 sdot sdot sdot 1198902119904+3] = 1198902[1198902 1198904 sdot sdot sdot 1198902119904+3] = 1198903[1198903 1198904 sdot sdot sdot 1198902119904+3] = 1199041198901 + 1199051198902 + 1199061198903120573 119904 119905 119906 isin F 120573 = 0 1 119904 = 0(17)
And n-Lie algebras corresponding to the case (1198897) with coef-ficients 119904 119905 119906 and 1199041015840 1199051015840 1199061015840 are isomorphic if and only if thereexists a nonzero element 120582 isin F such that 119904 = 12058231199041015840 119905 = 12058221199051015840119906 = 1205821199061015840 119904 1199041015840 119905 1199051015840 119906 1199061015840 isin F (119903) dim1198601 = 1199034 le 119903 lt 2119904 + 3 119891119900119903 2 le 119895 le 119903 1 le 119894 le 119903
(1199031) [1198902 sdot sdot sdot 1198902119904+2] = 1198901[1198902 sdot sdot sdot 119890119895 sdot sdot sdot 1198902119904+3] = 119890119895
(1199032) [1198901 sdot sdot sdot 119890119894 sdot sdot sdot 1198902119904+2] = 119890119894(18)
Theorem 10 If119860 is a (2119904+3)-dimensional (2119904+1)-Lie algebraover F with dim1198601 = 119903 lt 119904 + 3 then there are involutivederivations on 119860Proof Define linear mappings 119863119895 119860 997888rarr 119860 1 le 119895 le 6 by
1198631 (119890119894) = 119890119894 1 le 119894 le 119904 + 2 or 119894 = 2119904 + 3minus119890119894 otherwise1198632 (119890119894) = 119890119894 1 le 119894 le 119904 + 2minus119890119894 otherwise
International Journal of Mathematics and Mathematical Sciences 5
1198633 (119890119894) = minus119890119894 119904 + 2 le 119894 le 2119904 + 1119890119894 otherwise
1198634 (119890119894) = 119890119894 1 le 119894 le 119904 + 1 or 119894 = 2119904 + 2minus119890119894 otherwise
1198635 (119890119894) = 119890119894 1 le 119894 le 119904 + 1 or 119894 = 2119904 + 3minus119890119894 otherwise
1198636 (119890119894) = 119890119894 1 le 119894 le 119904 + 3minus119890119894 otherwise
(19)
Since dim1198601 = 119903 le 119904 + 2 it is easy to verify that 1198631 isan involutive derivation on the 3-Lie algebras of the cases of(1198871) (119888119894) (119889119895) and (119903119896) where 1 le 119894 le 7 1 le 119895 le 4 and1 le 119896 le 2 1198632 is an involutive derivation on the 3-Lie algebrasof the cases of (1198871) (1198894) and (119888119894) where 5 le 119894 le 7 1198633 1198634and 1198635 are involutive derivations on the 3-Lie algebras of thecase of (1198872) And 1198636 is an involutive derivation on the 3-Lie algebras of the cases of (1198895) (1198896) and (1198897) Also 119863119894 areinvolutive derivations on abelian algebras for 1 le 119894 le 6
Next we discuss the case of dim1198601 = 119903 ge 119904 + 3 Let 119863 bean endomorphism of 119860
119863119890119894 = 2119904+3sum119895=1
119887119894119895119890119895 119887119894119895 isin F 1 le 119894 le 2119904 + 3 (20)
and 119861 = (119887119894119895) be the (2119904 + 3) times (2119904 + 3)-matrix Then
119863(1198901 sdot sdot sdot 1198902119904+3)119879 = 119861 (1198901 sdot sdot sdot 1198902119904+3)119879= (1198611 11986101198612 1198613) (1198901 sdot sdot sdot 1198902119904+3)119879 (21)
where ( 1198611 11986101198612 1198613 ) is the block matrix of 119861 First we discuss(2119904 + 3)-dimensional (2119904 + 1)-Lie algebras of the case (1199031) inLemma 9
Lemma 11 If 119860 is a (2119904 + 3)-dimensional (2119904 + 1)-Lie algebraof the case (1199031) with dim1198601 = 119903 ge 119904 + 3 119904 ge 1Then the linearmapping 119863 is an involutive derivation on 119860 if and only if theblock matrix 119861 = ( 1198611 11986101198612 1198613 ) satisfies that 1198610 = 119874119903times(2119904+3minus119903) (whichis the zero (119903 times (2119904 + 3 minus 119903))-matrix) and11986121 = 11986411990311986123 = 1198642119904+3minus11990311986121198611 + 11986131198612 = 0
(22)
2119904+2sum119895=2
119887119895119895 = 119887112119904+3sum119895=2119895 =119894
119887119895119895 = 119887119894119894 2 le 119894 le 1199031198872119904+3119894 = (minus1)119894+1 1198871198941 2 le 119894 le 119903119887119895119894 = (minus1)119895minus119894minus1 119887119894119895 2 le 119894 119895 le 119903 119894 = 119895
(23)
Proof By (2) and a direct computation 119863 is a derivation of119860 if and only if matrix 119861 has the property
2119904+2sum119895=2
119887119895119895 = 119887112119904+3sum119895=2119895 =119894
119887119895119895 = 119887119894119894 2 le 119894 le 1199031198871119897 = 0 2 le 119897 le 2119904 + 31198872119904+3119894 = (minus1)119894+1 1198871198941 119887119894119897 = 0 2 le 119894 le 119903 119897 ge 119903 + 1119887119895119894 = (minus1)119895minus119894minus1 119887119894119895 2 le 119894 119895 le 119903 119894 = 119895
(24)
Therefore matrix 119861 satisfies (23) and 1198610 = 119874119903times(2119904+3minus119903) And1198632 = 119868119889 if and only if
1198612 = ( 11986121 11986111198610 + 1198610119861311986121198611 + 11986131198612 11986121198610 + 11986123 )= (119864119903 119874119874 1198642119904+3minus119903) (25)
Thanks to 1198610 = 119874119903times(2119904+3minus119903) (22) holdsTheorem 12 Let 119860 be a (2119904 + 3)-dimensional (2119904 + 1)-Liealgebra of the case (1199031) with dim1198601 = 119903 ge 119904 + 3 119904 ge 1 If 119903is odd then there are involutive derivations on 119860Proof Let 119903 = 2119905 + 1 ge 119904 + 3 Then 119905 ge 2 and 119903 ge 5 Suppose119863 is an endomorphism of119860 and the matrix of119863with respectto the basis 1198901 sdot sdot sdot 1198902119904+3 is 119861 = (119887119894119895) = ( 1198611 11986101198612 1198613 ) which satisfies(22) and (23) and 1198610 = 119874119903times(2119904+3minus119903) Then
1198611
= (((((((
11988711 0 0 sdot sdot sdot 0 011988721 11988722 11988723 sdot sdot sdot 1198872119903minus1 119887211990311988731 11988723 11988733 sdot sdot sdot 1198873119903minus1 1198873119903 d119887119903minus11 (minus1)119903 1198872119903minus1 (minus1)119903minus1 1198873119903minus1 sdot sdot sdot 119887119903minus1119903minus1 119887119903minus11199031198871199031 (minus1)119903+1 1198872119903 (minus1)119903 1198873119903 sdot sdot sdot 119887119903minus1119903 119887119903119903
)))))))
6 International Journal of Mathematics and Mathematical Sciences
1198612 = ((
119887119903+11 119887119903+12 119887119903+13 sdot sdot sdot 119887119903+1119903 d1198872119904+21 1198872119904+22 1198872119904+23 sdot sdot sdot 1198872119904+21199031198872119904+31 minus11988721 11988731 sdot sdot sdot (minus1)119903+1 1198871199031
))
(26)
Since sum2119904+2119895=2 119887119895119895 = 11988711 sum2119904+3119895=2119895 =119894 119887119895119895 = 119887119894119894 2 le 119894 le 119903 we haveminus11988711 + 211988722 minus 1198872119904+32119904+3 = 0 (119903 minus 3)11988722 + sum119903+12119904+3 119887119894119894 = 0 11988722 =119887119894119894 3 le 119894 le 119903 Therefore
11988711 = minus1119903 minus 3 ((119903 minus 1) 1198961 + 22119904+3minus119903sum119895=2
119896119895) 119887119894119894 = minus1119903 minus 3 2119904+3minus119903sum
119895=1
119896119895 2 le 119894 le 119903119887119895119895 = 1198962119904+3minus119895+1 119903 + 1 le 119895 le 2119904 + 3 1198962119904+3minus119895+1 isin F
(27)
Suppose
11986121 =((((((((((((
11988811 11988812 sdot sdot sdot 1198881119894 sdot sdot sdot 1198881119895 sdot sdot sdot 1198881119903 d d
d1198881198941 1198881198942 sdot sdot sdot 119888119894119894 sdot sdot sdot 119888119894119895 sdot sdot sdot 119888119894119903 d
d d
1198881198951 1198881198952 sdot sdot sdot 119888119895119894 sdot sdot sdot 119888119895119895 sdot sdot sdot 119888119895119903 d d
d1198881199031 1198881199032 sdot sdot sdot 119888119903119894 sdot sdot sdot 119888119903119895 sdot sdot sdot 119888119903119903
))))))))))))
(28)
By (23)
11988811 = 1198872111198881119897 = 0 2 le 119897 le 119903119888119894119894 = 119894sum119897=1
(minus1)119894+119897minus1 1198872119897119894 + 119903sum119897=119894+1
(minus1)119894+119897minus1 1198872119894119897 2 le 119894 le 119903119888119894119895 = 119894sum119897=1
(minus1)119897+119894+1 119887119897119894119887119897119895 + 119895sum119897=119894+1
119887119894119897119887119897119895+ 119903sum119897=119895+1
(minus1)119897+119895+1 119887119894119897119887119895119897 2 le 119894 lt 119895 le 119903119888119894119895 = 119895sum119897=1
(minus1)119897+119894+1 119887119897119895119887119897119894 + 119894sum119897=119895+1
(minus1)119894+119895 119887119895119897119887119897119894+ 119903sum119897=119894+1
(minus1)119897+119895+1 119887119895119897119887119894119897 1 le 119895 lt 119894 le 119903
(29)
Therefore the endomorphisms 119863 of 119860 which are defined by1198631198901 = 1198901or 1198631198901 = minus1198901
1198631198902 = 119903sum119896=3
119890119896119863119890119894 = (minus1)119894minus1 1198902 + 119894minus1sum
119896=3
(minus1)119894 119890119896 + (minus1)119894minus1 119903sum119896=119894+1
1198901198963 le 119894 le 119903119863119890119895 = (minus1)119895 119890119895 119903 + 1 le 119895 le 2119904 + 31198631198901 = 1198901
or 1198631198901 = minus11989011198631198902 = 119903sum
119896=3
119890119896119863119890119894 = (minus1)119894minus1 1198902 + 119894minus1sum
119896=3
(minus1)119894 119890119896 + (minus1)119894minus1 119903sum119896=119894+1
1198901198963 le 119894 le 119903119863119890119895 = (minus1)119895minus1 119890119895 119903 + 1 le 119895 le 2119904 + 3
(30)
are involutive derivations on 119860Theorem 13 Let 119860 be a (2119904 + 3)-dimensional (2119904 + 1)-Liealgebra of the case (1199031) with dim1198601 = 119903 = 2119904 + 2 (119904 ge 1)then there does not exist an involutive derivation on 119860Proof If 119863 is an involutive derivation on 119860 then byLemma 11 and (23)
11988711 = minus (2119904 + 1) 11989612119904 minus 1 1198872119904+32119904+3 = 1198961119887119894119894 = minus11989612119904 minus 1
2 le 119894 le 119903 1198961 isin F (31)
Thanks to (22) 11988722119904+32119904+3 = 119887211 = 11989621 = 1 Therefore(minus(2119904 + 1)1198961(2119904 minus 1))2 = ((2119904 + 1)(2119904 minus 1))2 = 1 whichis a contradiction
Now we discuss case (1199032)Theorem 14 Let 119860 be a (2119904 + 3)-dimensional (2119904 + 1)-Liealgebra of the case (1199032) with dim1198601 = 119903 ge 119904 + 3 Then thereexist involutive derivations on 119860 if and only if 119903 is evenProof By Lemma 7 119860 = 1198601 + F1198902119904+3 where 1198902s+3 isin 119885(119860)and 1198601 is a (2119904 + 2)-dimensional (2119904 + 1)-Lie subalgebra
International Journal of Mathematics and Mathematical Sciences 7
of 119860 with dim1198601 = dim11986011 = 119903 Then there existinvolutive derivations on119860 if and only if there exist involutivederivations on 1198601
By Theorem 3 in [1] there is a basis 1198901 sdot sdot sdot 1198902119904+2 of 1198601such that[1198901 sdot sdot sdot 119890119895 sdot sdot sdot 1198902119904+2] = 0 119905 lt 119895 le 2119904 + 3 minus 119905[1198901 sdot sdot sdot 119890119894 sdot sdot sdot 1198902119904+2] = (minus1)119894 1198902119904+3minus1198941 le 119894 le 119905 or 2119904 + 3 minus 119905 lt 119894 le 2119904 + 2 (32)
If 119903 is even then 119903 = 2119905 ge 4 By Theorem 8 and (32)endomorphism 1198631 of 1198601 defined by
1198631 (119890119894) = 119890119894 119894 = 1 sdot sdot sdot 119904 + 1minus119890119894 119894 = 119904 + 2 sdot sdot sdot 2119904 + 2 (33)
is an involutive derivation on 1198601 Therefore the endomor-phism 119863 of 119860 defined by119863 (119890119894) = 119890119894 1 le 119894 le 119904 + 1119863 (119890119895) = minus119890119895 119904 + 2 le 119895 le 2119904 + 2119863 (1198902119904+3) = plusmn1198902119904+3
(34)
is involutive derivation on 119860If dim1198601 = 119903 is odd and endomorphism 119863 of 119860 is an
involutive derivation on 119860 then 119903 = 2119905 + 1 gt 4 Suppose119863(119890119894) = sum2119904+3119895=1 119886119894119895119890119895 1 le 119894 le 2119904 + 3 Then[119863 (1198902119904+3) 1198901198941 sdot sdot sdot 1198901198942119904]= 119863 [1198902119904+3 1198901198941 sdot sdot sdot 1198901198942119904]minus 2119904sum119895=1
[1198902119904+3 sdot sdot sdot 119863119890119894119895 sdot sdot sdot 1198901198942119904] = 0 (35)
We get 119863(1198902119904+3) isin 119885(119860) = F1198902119904+3 Since 1198632 = 119868119889 119863(1198902119904+3) =plusmn1198902119904+3 By (32) 1198601 = F1198901 + sdot sdot sdot + F119890119905 + F1198902119904+1 + sdot sdot sdot + F1198902119904+3minus119905and (minus1)119894119863 (1198902119904+3minus119894)
= 2119904+2sum119896=1119896=119894
[1198901 sdot sdot sdot 119890119894 sdot sdot sdot 119863 (119890119896) sdot sdot sdot 1198902119904+2] (36)
where 1 le 119894 le 119905 and 2119904 + 3 minus 119905 lt 119894 le 2119904 + 2 Then 1198861198942119904+3 = 0for 1 le 119894 le 119905 or 2119904 + 3 minus 119905 lt 119894 le 2119904 + 2 and 1198631198601 sube 1198601 Thenthe endomorphism 1198632 of 1198601 defined by1198632 (119890119894) = 119863 (119890119894) 1 le 119894 le 119905 or 2119904 + 3 minus 119905 lt 119894 le 2119904 + 2
1198632 (119890119895) = 119863(119890119895) minus 1198861198952119904+31198902119904+3 = 2119904+2sum119895=1
119886119894119895119890119895119905 lt 119895 le 2119904 + 3 minus 119905
(37)
is an involutive derivation on the (2119904+2)-dimensional (2119904+1)-Lie algebra 1198601 contradiction (Theorem 8) Therefore theredoes not exist involutive derivation on 119860
5 Structure of 3-Lie Algebras withInvolutive Derivations
Let (119871 [ ]) be a Lie algebra over F and 119901 be an element whichis not contained in 119871 Then 119860 = 119871 + F119901 is a 3-Lie algebra inthe multiplication[119909 119910 119911] = 0[119901 119909 119910] = [119909 119910]
for all 119909 119910 119911 isin 119871 (38)
And the 3-Lie algebra (119860 [ ]) is called one-dimensionalextension of 119871Theorem 15 Let 119860 be a 3-Lie algebra then 119860 is one-dimensional extension of a Lie algebra if and only if there existsan involutive derivation 119863 on 119860 such that dim1198601 = 1 ordim119860minus1 = 1Proof If 119860 is an one-dimensional extension of a Lie algebra119871 then 119860 = 119871 + F119901 Define the endomorphism 119863 of 119860 by119863(119901) = minus119901 (or 119901) and 119863(119909) = 119909 (or minus119909) forall119909 isin 119871 Thanks to(38)1198632 = 119868119889 and 119863([119909 119910 119911]) = 0 = [119863119909 119910 119911] + [119909119863119910 119911] +[119909 119910119863119911]119863([119901 119909 119910]) = [119901 119909 119910] = [119863119901 119909 119910] + [119901119863119909 119910] +[119901 119909 119863119910] for all 119909 119910 119911 isin 119871 Therefore 119863 is an involutivederivation on 119860 and dim119860minus1 = 1 (or dim1198601 = 1)
Conversely let 119863 be an involutive derivation on a 3-Liealgebra 119860 and dim119860minus1 = 1 (or dim1198601 = 1) Let 119860minus1 = F119901and 1198601 = 119871 (or 119860minus1 = 1198711198601 = F119901) where 119901 isin 119860minus119871 Thanksto Theorem 3 119871 is a Lie algebra with the multiplication[119909 119910] = [119901 119909 119910] for all 119909 119910 isin 119871 and 119860 is one-dimensionalextension of 119871
Let (119871 [ ]1) and (119871 [ ]2) be Lie algebras and 1199091 sdot sdot sdot 119909119898be a basis of 119871 For convenience denote Lie algebras (119871 [ ]119896)by 119871119896 119896 = 1 2 respectively Suppose 1199011 and 1199012 are twodistinct elements which are not contained in 119871 and 3-Lie algebras (119861 [ ]1) and (119862 [ ]2) are one-dimensionalextensions of Lie algebras 1198711 and 1198712 respectively where119861 = 119871 + F1199011 119862 = 119871 + F1199012 Then 119863119890119903(1198711) and 119863119890119903(1198712) aresubalgebras of 119892119897(119871)Definition 16 Let 1198711 = (119871 [ ]1) and 1198712 = (119871 [ ]2) be twoLie algebras and 1199011 1199012 be two distinct elements which arenot contained in 119871 and 119860 = 119871 + F1199011 + F1199012 Then 3-algebra(119860 [ ]) is called a two-dimensional extension of Lie algebras119871119896 119896 = 1 2 where [ ] 119860 and 119860 and 119860 997888rarr 119860 defined by[119909 119910 1199011] = [119909 119910]1 [119909 119910 1199012] = [119909 119910]2 [119909 119910 119911] = 0[1199011 1199012 119909] = 1205821199091199011 + 1205831199091199012 forall119909 119910 119911 isin 119871 120582119909 120583119909 isin F
(39)
If 119860 is a 3-Lie algebra then 119860 is called a two-dimensionalextension 3-Lie algebra of Lie algebras 119871119896 119896 = 1 2
8 International Journal of Mathematics and Mathematical Sciences
Let 119860 = 119871 + 119882 be a two-dimensional extension of Liealgebras 119871119896 119896 = 1 2 where 119882 = F1199011 + F1199012 Define linearmappings 1198631 1198632 119871 997888rarr 119864119899119889(119871) and 119863 119871 997888rarr 119882 by1198631 (119909) = 119886119889 (1199011 119909) 1198632 (119909) = 119886119889 (1199012 119909) 119863 (119909) = 119886119889 (1199011 1199012) (119909) forall119909 isin 119871
(40)
that is for all 119910 isin 119871 1198631(119909)(119910) = [1199011 119909 119910] = [119909 119910]11198632(119909)(119910) = [1199012 119909 119910] = [119909 119910]2 119863(119909) = [1199011 1199012 119909] We havethe following result
Theorem 17 Let 3-algebra 119860 be a two-dimensional extensionof Lie algebras 1198711 and 1198712 Then 119860 is a 3-Lie algebra if and onlyif linear mappings 1198631 1198632 and 119863 satisfy that 1198631 1198711 997888rarr119863119890119903(1198711) 1198632 1198712 997888rarr 119863119890119903(1198712) are Lie homomorphisms and1198631 (1199093) ([1199091 1199092]2) = [1198631 (1199093) (1199091) 1199092]2+ [1199091 1198631 (1199093) (1199092)]2minus 1205821199093 [1199091 1199092]1 minus 1205831199093 [1199091 1199092]2
(41)
1198632 (1199093) ([1199091 1199092]1) = [1198632 (1199093) (1199091) 1199092]1+ [1199091 1198632 (1199093) (1199092)]1+ 1205821199093 [1199091 1199092]1 + 1205831199093 [1199091 1199092]2 (42)
119863 ([1199091 1199092]1) = (12058311990911205821199092 minus 12058211990911205831199092) 1199011119863 ([1199091 1199092]2) = (12058311990911205821199092 minus 12058211990911205831199092) 1199012 (43)
120582[11990911199092]1 = 120583[11990911199092]2 = 12058311990911205821199092 minus 12058211990911205831199092 120583[11990911199092]1 = 120582[11990911199092]2 = 0 (44)
119863119896 (1199091) (1199092) = minus119863119896 (1199092) (1199091) 119891119900119903 119886119897119897 1199091 1199092 isin 119871 119896 = 1 2 (45)
where 1199091 1199092 1199093 isin 119871 119863(119909119894) = 1205821199091198941199011 + 1205831199091198941199012 119894 = 1 2 3Proof If 119860 is a two-dimensional extension 3-Lie algebrathen by Definition 16 linear mappings 119863119896 satisfy that119863119896(119871119896) sube 119863119890119903(119871119896) and119863119896 are Lie homomorphisms 119896 = 1 2Thanks to (39)1198631 (1199093) ([1199091 1199092]2) = [1199012 [1199011 1199093 1199091] 1199092]+ [1199012 1199091 [1199011 1199093 1199092]]+ [[1199011 1199093 1199012] 1199091 1199092]= [1198631 (1199093) (1199091) 1199092]2+ [1199091 1198631 (1199093) (1199092)]2minus 1205821199093 [1199091 1199092]1 minus 1205831199093 [1199091 1199092]2
(46)
for all 1199091 1199092 1199093 isin 119871 (41) holds Similarly we have (42)Thanks to (39) and (40)119863([1199091 1199092]1) = 119886119889 (1199011 1199012) [1199091 1199092]1= 120582[11990911199092]11199011 + 120583[11990911199092]11199012= (12058311990911205821199092 minus 12058211990911205831199092) 1199011119863 ([1199091 1199092]2) = 119886119889 (1199011 1199012) [1199091 1199092]2= 120582[11990911199092]21199011 + 120583[11990911199092]21199012= (12058311990911205821199092 minus 12058211990911205831199092) 1199012
(47)
Equations (43) and (44) hold Equation (45) follows from (39)and (40) directly
Conversely by (39) forall1199091 1199092 1199093 119909 isin 119871[1199091 1199092 1199093] = 0[1199011 1199091 1199092] = 1198631 (1199091) (1199092) = [1199091 1199092]1 [1199012 1199091 1199092] = 1198632 (1199091) (1199092) = [1199091 1199092]2 [1199011 1199012 119909] = 119863 (119909) = 120582 (119909) 1199011 + 1205831199091199012(48)
Since119863119896(119871) sube 119863119890119903(119871119896) and119863119896 are Lie homomorphisms 119861 =119871 + F1199011 and 119862 = 119871 + F1199012 are 3-Lie algebras which are one-dimensional extension 3-Lie algebras of Lie algebras 119871119896 119896 =1 2 respectively
Next we only need to prove that the multiplication on119860 defined by (39) satisfies (1) For all 119909119894 isin 119871 1 le 119894 le5 that products [[1199091 1199092 1199093] 1199094 1199095] [[119901119895 1199092 1199093] 1199094 1199095][[1199091 1199092 1199093] 1199094 119901119895] and [[1199091 1199092 119901119895] 1199094 119901119895] satisfy (1) 119895 =1 2 follow from that 119861 and 119862 are one-dimensional extension3-Lie algebras of 119871119896 and (39) directlyFrom (41) and (42) it follows that products[[119901119894 1199091 1199092] 119901119895 1199093] 1 le 119894 = 119895 le 2 satisfy (1) It
follows from (43)ndash(45) that products [1199011 1199012 [119901119894 1199091 1199092]][1199091 1199092 [119901119894 1199012 1199093]] and [119901119894 1199091 [1199011 1199012 1199092] 119894 = 1 2 satisfy(1) We omit the computation process
Theorem 18 Let (119860 [ ]) be a 3-Lie algebra Then 119860 is a two-dimensional extension 3-Lie algebra of Lie algebras if and onlyif there is an involutive derivation119879 on119860 such that dim1198601 = 2or dim119860minus1 = 2Proof If 119860 is a two-dimensional extension 3-Lie algebra ofLie algebras Then byTheorem 15 there are Lie algebras 1198711 =(119871 [ ]1) and 1198712 = (119871 [ ]2) such that 119860 = 119871 + 119882 and themultiplication of 119860 is defined by (39) where119882 = F1199011 + F1199012
Define the endomorphism 119879 of 119860 by 119879(119909) = 119909 119879(1199011) =minus1199011 119879(1199012) = minus1199012 or 119879(119909) = minus119909 119879(1199011) = 1199011 119879(1199012) =1199012 forall119909 isin 119871Then1198792 = 119868119889 and1198601 = 119871119860minus1 = 119882 or119860minus1 = 1198711198601 = 119882Thanks to (38) and (41)-(45) 119879 is a derivation of 119860Conversely if there is an involutive derivation 119879 on the3-Lie algebra 119860 such that dim119860minus1 = 2 (or dim1198601 = 2)
By Theorem 4 [1198601 1198601 1198601] = 0 [1198601 1198601 119860minus1] sube 1198601[1198601 119860minus1 119860minus1] sube 119860minus1 Let 119871 = 1198601 and 119860minus1 = F1199011 + F1199012Then [119871 119871 1199011] sube 119871 [119871 119871 1199012] sube 119871 and (119871 [ ]1) and
International Journal of Mathematics and Mathematical Sciences 9
(119871 [ ]2) are Lie algebras where [119909 119910]1 = [119909 119910 1199011] [119909 119910]2 =[119909 119910 1199012] forall119909 119910 isin 119871Thanks toTheorem 17 the 3-Lie algebra119860 is a two-dimensional extension 3-Lie algebra of Lie algebras1198711 and 1198712Data Availability
No data were used to support this study
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
The first named author was supported in part by the NaturalScience Foundation (11371245) and theNatural Science Foun-dation of Hebei Province (A2018201126)
References
[1] V Filippov ldquon-Lie algebrasrdquo SiberianMathematical Journal vol26 no 6 pp 126ndash140 1985
[2] R P Bai and P P Jia ldquoThe real compact n-Lie algebras andinvariant bilinear formsrdquo Acta Mathematica Scientia Series AShuxueWuli Xuebao Chinese Edition vol 27A no 6 pp 1074ndash1081 2007
[3] S Kasymov ldquoOn a theory of n-Lie algebrasrdquo Algebra and Logicvol 26 no 3 pp 277ndash297 1987
[4] C Bai L Guo and Y Sheng ldquoBialgebras the classical Yang-Baxter equation andmain triples for 3-Lie algebrasrdquoMathemat-ical Physics 2016
[5] R Bai and G Song ldquoThe classification of six-dimensional 4-Liealgebrasrdquo Journal of Physics A Mathematical and General vol42 no 3 035207 17 pages 2009
[6] R Bai G Song and Y Zhang ldquoOn classification of n-Liealgebrasrdquo Frontiers of Mathematics in China vol 6 no 4 pp581ndash606 2011
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2 International Journal of Mathematics and Mathematical Sciences
The derived algebra of an 119899-Lie algebra 119860 is a subalgebraof 119860 generated by [1199091 sdot sdot sdot 119909119899] for all 1199091 sdot sdot sdot 119909119899 isin 119860 and isdenoted by 1198601 We use 119885(119860) to denote the center of 119860 thatis 119885(119860) = 119909 | 119909 isin 119860 [119909 119860 sdot sdot sdot 119860] = 0
A derivation of 119860 is an endomorphism of 119860 satisfying
119863([1199091 sdot sdot sdot 119909119899]) = 119899sum119894=1
[1199091 sdot sdot sdot 119863 (119909119894) sdot sdot sdot 119909119899] forall1199091 sdot sdot sdot 119909119899 isin 119860 (2)
If a derivation 119863 satisfies that 1198632 = 119868119889 then 119863 is calledan involutive derivation on 119860 Der(119860) denotes the derivationalgebra of 119860
For 1199091 sdot sdot sdot 119909119899minus1 isin 119860 map ad(1199091 sdot sdot sdot 119909119899minus1) 119860 997888rarr 119860ad (1199091 sdot sdot sdot 119909119899minus1) (119909) = [1199091 sdot sdot sdot 119909119899minus1 119909] forall119909 isin 119860 (3)
is called a leftmultiplication defined by elements 1199091 sdot sdot sdot 119909119899minus1From (1) left multiplications are derivations
The following lemma can be easily verified
Lemma 1 Let 119881 be a finite dimensional vector space over Fand 119863 be an endomorphism of 119881 with 1198632 = 119868119889 Then 119881 canbe decomposed into the direct sum of subspaces 119881 = 1198811 + 119881minus1where 1198811 = V isin 119881 | 119863V = V and 119881minus1 = V isin 119881 | 119863V = minusV
If 119860 is a finite dimensional 119899-Lie algebra with an involu-tive derivation 119863 then we have119860 = 1198601 + 119860minus1 (4)
Lemma 2 Let 119860 be an 119899-Lie algebra over F If 119863 isin 119863119890119903(119860) isan involutive derivation then for all 1199091 sdot sdot sdot 119909119899 isin 119860[1199091 sdot sdot sdot 119909119899] = minus2119899 minus 1sum
119894lt119895
[1199091 sdot sdot sdot 119909119894minus1 119863 (119909119894) 119909119894+1 sdot sdot sdot 119909119895minus1 119863 (119909119895) 119909119895+1 sdot sdot sdot 119909119899] (5)
[119863 (1199091) sdot sdot sdot 119863 (119909119899)] = minus2119899 minus 1sum119894lt119895
[1198631199091 sdot sdot sdot 119863 (119909119894minus1) 119909119894119863 (119909119894+1) sdot sdot sdot 119863 (119909119895minus1) 119909119895 119863 (119909119895+1) sdot sdot sdot 119863 (119909119899)] (6)
Proof If 119863 is an involutive derivation on 119860 then for all1199091 sdot sdot sdot 119909119899 isin 119860[1199091 sdot sdot sdot 119909119899] = 1198632 ([1199091 sdot sdot sdot 119909119899])= 119863( 119899sum
119894=1
[1199091 sdot sdot sdot 119863 (119909119894) sdot sdot sdot 119909119899])= 119899 [1199091 sdot sdot sdot 119909119899]+ 2 sum
1le119894lt119895le119899
[1199091 sdot sdot sdot 119863 (119909119894) sdot sdot sdot 119863 (119909119895) sdot sdot sdot 119909119899] (7)
Equation (5) follows Equation (6) follows from (4) and1198632 =119868119889
Theorem 3 Let 119860 be a finite dimensional 119899-Lie algebra with119899 = 2119904 119904 ge 1 Then there is an involutive derivation 119863 on 119860 ifand only if 119860 is abelian
Proof If 119860 is abelian then the result is trivialConversely let 119863 be an involutive derivation on 119860 By
Lemma 1 119860 = 1198601 + 119860minus1 Then for any 119894 isin Z 1 le 119894 le 1198991199091 sdot sdot sdot 119909119899 isin 1198601 and 1199101 sdot sdot sdot 119910119899 isin 119860minus1119863 ([1199091 sdot sdot sdot 119909119894 1199101 sdot sdot sdot 119910119899minus119894])= 119894 [1199091 sdot sdot sdot 119909119894 1199101 sdot sdot sdot 119910119899minus119894]minus (119899 minus 119894) [1199091 sdot sdot sdot 119909119894 1199101 sdot sdot sdot 119910119899minus119894]= (2119894 minus 2119904) [1199091 sdot sdot sdot 119909119894 1199101 sdot sdot sdot 119910119899minus119894] isin 1198602119894minus2119904119863 ([1199091 sdot sdot sdot 119909119899]) = 2119904 [1199091 sdot sdot sdot 119909119899] 119863 ([1199101 sdot sdot sdot 119910119899]) = minus2119904 [1199101 sdot sdot sdot 119910119899] (8)
Thanks to plusmn2119904 = plusmn1 and 2119894 minus 2119904 = plusmn1 1198602119894minus119899 = 119860plusmn2119904 = 0Therefore 119860 is abelian
Theorem 4 Let 119860 be a finite dimensional 119899-Lie algebra with119899 = 2119904+1 119904 ge 1 and119863 be an involutive derivation on119860 Then1198601 and 119860minus1 are abelian subalgebras and[1198601 sdot sdot sdot 1198601⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
119894
119860minus1 sdot sdot sdot 119860minus1⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟2119904+1minus119894
] = 0forall1 le 119894 le 2119904 119894 = 119904 119904 + 1 (9)
[1198601 sdot sdot sdot 1198601⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟119904
119860minus1 sdot sdot sdot 119860minus1⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟119904+1
] sube 119860minus1 (10)
[1198601 sdot sdot sdot 1198601⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟119904+1
119860minus1 sdot sdot sdot 119860minus1⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟119904
] sube 1198601 (11)
Proof Since 119863 isin 119863119890119903119860 [1198601 sdot sdot sdot 1198601⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟119894
119860minus1 sdot sdot sdot 119860minus1⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟2119904+1minus119894
] sube1198602119894minus2119904minus1 0 le 119894 le 2119904 + 1 If [1198601 sdot sdot sdot 1198601⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟119894
119860minus1 sdot sdot sdot 119860minus1⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟2119904+1minus119894
] = 0then 2119894 minus 2119904 minus 1 = plusmn1 that is 119894 = 119904 + 1 or 119894 = 119904 Therefore[1198601 sdot sdot sdot 1198601] = [119860minus1 sdot sdot sdot 119860minus1] = 0 The result follows
Theorem5 Let119860 be an119898-dimensional 119899-Lie algebra with 119899 =2119904 + 1 119904 ge 1 Then there is an involutive derivation on119860 if andonly if 119860 has the decomposition 119860 = 119861 + 119862 (as direct sum ofsubspaces) and
[119861 sdot sdot sdot 119861⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟119894
119862 sdot sdot sdot 119862⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟2119904+1minus119894
] = 00 le 119894 le 2119904 + 1 119894 = 119904 119904 + 1 (12)
[119861 sdot sdot sdot 119861⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟119904
119862 sdot sdot sdot 119862⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟119904+1
] sube 119862[119861 sdot sdot sdot 119861⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟119904+1
119862 sdot sdot sdot 119862⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟119904
] sube 119861 (13)
International Journal of Mathematics and Mathematical Sciences 3
Proof If there is an involutive derivation 119863 on 119860 then byTheorem 4 119861 = 1198601 and 119862 = 119860minus1 satisfy (12) and (13)
Conversely define an endomorphism 119863 of 119860 by 119863(119909) =119909119863(119910) = minus119910 forall119909 isin 119861 119910 isin 119862 Then 1198632 = 119868119889 119861 = 1198601 and119862 = 119860minus1 By (12) and (13) 119863 is a derivation
Corollary 6 Let 119860 be a (2119904 + 1)-dimensional (2119904 + 1)-Liealgebra with the multiplication [1198901 sdot sdot sdot 1198902119904+1] = 1198901 where1198901 sdot sdot sdot 1198902119904+1 is a basis of 119860 Then the linear mapping 119863 119860 997888rarr 119860 defined by 119863(119890119894) = 119890119894 1 le 119894 le 119904 + 1 119863(119890119895) =minus119890119895 119904 + 2 le 119895 le 2119904 + 1 is an involutive derivation on119860Proof The result follows from a direct computation
3 Involutive Derivations on(119899 + 1)-Dimensional 119899-Lie Algebraswith 119899 = 2119904 + 1
In this section we study involutive derivations on (119899 + 1)-dimensional 119899-Lie algebras over F FromTheorem 3 we onlyneed to discuss the case of 119899 = 2119904 + 1 119904 ge 1Lemma 7 (see [5]) Let 119860 be an (119899 + 1)-dimensional non-abelian 119899-Lie algebra over F 119899 ge 3 Then up to isomorphisms119860 is one and only one of the following possibilities
(1198871) [1198902 1198903 sdot sdot sdot 119890119899+1] = 1198901(1198872) [1198901 1198902 sdot sdot sdot 119890119899] = 1198901(1198881) [1198902 sdot sdot sdot 119890119899+1] = 1198901[1198901 1198903 sdot sdot sdot 119890119899+1] = 1198902
(1198882) [1198902 sdot sdot sdot 119890119899+1] = 1205721198901 + 1198902[1198901 1198903 sdot sdot sdot 119890119899+1] = 1198902 120572 isin F 120572 = 0(1198883) [1198901 1198903 sdot sdot sdot 119890119899+1] = 1198901[1198902 sdot sdot sdot 119890119899+1] = 1198902(119889119903) [1198901 sdot sdot sdot 119890119894 sdot sdot sdot 119890119899+1] = 119890119894 1 le 119894 le 119903
(14)
where 1198901 sdot sdot sdot 119890119899+1 is a basis of119860 3 le 119903 le 119899+1 and 119890119894 meansthat 119890119894 is omitted
Theorem 8 Let 119860 be a (2119904 + 2)-dimensional (2119904 + 1)-Liealgebra over F and dim1198601 = 119903 Then there exists an involutivederivation 119863 on 119860 if and only if 119903 is even or 0 le 119903 le 119904+2Proof If dim1198601 = 119903 le 119904 + 2 then by Lemma 7 and a directcomputation the linear mapping 119863 119860 997888rarr 119860 defined by119863(119890119894) = 119890119894 119863(119890119895) = minus119890119895 1 le 119894 le 119904 + 2 119904 + 3 le 119895 le 2119904 + 2 is aninvolutive derivation on 119860
Now we discuss the case dim1198601 = 119903 ge 119904 + 3 Let1198901 sdot sdot sdot 1198902119904+2 be a basis of 119860 and the multiplication in thebasis be as follows
119890119894 = (minus1)2119904+2+119894 [1198901 sdot sdot sdot 119890119894 sdot sdot sdot 1198902119904+2] = 2119904+2sum119897=1
120573119894119897119890119897120573119894119897 isin 119865 1 le 119894 le 2119904 + 2 (15)
where 120573119894119897 isin F 1 le 119894 119897 le 2119904 + 2 Thanks toTheorem 3 in [1] 119860is a 3-Lie algebra if and only if the (2119904 + 2) times (2119904 + 2)-matrix119861 = (120573119894119897) is symmetric
If 119903 = 2119905 gt 3 2 le 119905 le 119904 + 1 then define the multiplicationon 119860 by
[1198901 sdot sdot sdot 119890119894 sdot sdot sdot 1198902119904+2] = (minus1)119894 1198902119904+3minus1198941 le 119894 le 119905 3 minus 119905 lt 119894 minus 2119904 le 2[1198901 sdot sdot sdot 119890119895 sdot sdot sdot 1198902119904+2] = 0 119905 lt 119895 le 2119904 + 3 minus 119905
(16)
that is 1205731198942119904+3minus119894 = 1205732119904+3minus119894119894 = 1 for 1 le 119894 le 119905 or 2119904 + 3 minus 119905 lt119894 le 2119904 + 2 and others are zero Then 119861 = (120573119894119897) is symmetricTherefore 119860 is a (2119904 + 1)-Lie algebra with the multiplication(16)
Define an endomorphism119863 of119860 by119863119890119894 = 119890119894 1 le 119894 le 119904+1and 119863119890119895 = minus119890119895 for 119904 + 2 le 119895 le 2119904 + 2Then 119863 is an involutivederivation on 119860
For the case dim1198601 = 119903 = 2119905 + 1 ge 119904 + 3 Suppose 119897 =dim1198601 1198971015840 = dim119860minus1
If there is an involutive derivation 119863 on 119860 then byTheorem 4 119897 + 1198971015840 = 2119904 + 2 119904 le 119897 le 119904 + 2 and 119904 le 1198971015840 le 119904 + 2Since dim1198601 = 119903 = 2119905 + 1 ge 119904 + 3 1198601 cap 1198601 = 0 and1198601 cap119860minus1 = 0Therefore dim1198601 = dim119860minus1 = 119904+ 1 Withoutloss of generality we can suppose 1198901 sdot sdot sdot 119890119904+1 sube 1198601 and119890119904+2 sdot sdot sdot 1199092119904+2 sube 119860minus1 By (10) and (11) the (2119904+2)times(2119904+2)-matrix 119861 = (120573119894119897) defined by (15) is nonsymmetric which is acontradiction Therefore if dim1198601 = 119903 = 2119905 + 1 ge 119904 + 3 thenthere do not exist involutive derivations on 119860
By Theorem 8 if 119860 is a 10-dimensional 9-Lie algebrawith dim1198601 = 7 or 9 then there does not exist involutivederivation on 119860 If 1 le dim1198601 = 119903 le 10 and 119903 = 7 9 thenthere are involutive derivations on 1198604 Involutive Derivations on(119899 + 2)-Dimensional 119899-Lie Algebraswith 119899 = 2119904 + 1
By Theorem 3 we only need to discuss the case where 119899 isodd So we suppose that 119860 is a (2119904 + 3)-dimensional (2119904 + 1)-Lie algebra over F 119904 ge 1 and that 119864119905 = Diag(1 sdot sdot sdot 1) is the(119905 times 119905)-unit matrix
Lemma9 (see [6]) Let119860 be a (2119904+3)-dimensional (2119904+1)-Liealgebra over F with a basis 1198901 sdot sdot sdot 1198902119904+3Then119860 is isomorphicto one and only one of the following possibilities
4 International Journal of Mathematics and Mathematical Sciences
(119886) 119860 119894119904 119886119899 119886119887119890119897119894119886119899(119887) dim1198601 = 1
(1198871) [1198902 sdot sdot sdot 1198902119904+2] = 1198901(1198872) [1198901 sdot sdot sdot 1198902119904+1] = 1198901(119888) dim1198601 = 2(1198881) [1198902 sdot sdot sdot 1198902119904+2] = 1198901[1198903 sdot sdot sdot 1198902119904+3] = 1198902
(1198882) [1198902 sdot sdot sdot 1198902119904+2] = 1198901[1198902 1198904 sdot sdot sdot 1198902119904+3] = 1198902[1198901 1198904 sdot sdot sdot 1198902119904+3] = 1198901
(1198883)
[1198902 sdot sdot sdot 1198902119904+2] = 1205721198901 + 1198902[1198901 1198903 sdot sdot sdot 1198902119904+2] = 1198902[1198902 1198904 sdot sdot sdot 1198902119904+3] = 1198902[1198901 1198904 sdot sdot sdot 1198902119904+3] = 1198901(1198884)
[1198902 sdot sdot sdot 1198902119904+2] = 1198901[1198902 1198903 sdot sdot sdot 1198902119904+2] = 1198902[1198902 1198904 sdot sdot sdot 1198902119904+3] = 1198902[1198901 1198904 sdot sdot sdot 1198902119904+3] = 1198901(1198885) [1198902 sdot sdot sdot 1198902119904+2] = 1198901[1198901 1198903 sdot sdot sdot 1198902119904+2] = 1198902
(1198886) [1198902 sdot sdot sdot 1198902119904+2] = 1205721198901 + 1198902[1198901 1198903 sdot sdot sdot 1198902119904+2] = 1198902 120572 isin F 120572 = 0(1198887)[1198901 1198903 sdot sdot sdot 1198902119904+2] = 1198901[1198902 1198903 sdot sdot sdot 1198902119904+2] = 1198902(119889) dim1198601 = 3(1198891)
[1198902 sdot sdot sdot 1198902119904+2] = 1198901[1198902 1198904 sdot sdot sdot 1198902119904+3] = 1198902[1198903 sdot sdot sdot 1198902119904+3] = 1198903(1198892)
[1198902 sdot sdot sdot 1198902119904+2] = 1198901[1198903 sdot sdot sdot 1198902119904+3] = 1198903 + 1205721198902[1198902 1198904 sdot sdot sdot 1198902119904+3] = 1198903[1198901 1198904 sdot sdot sdot 1198902119904+3] = 1198901
(1198893)
[1198902 sdot sdot sdot 1198902119904+2] = 1198901[1198903 1198904 sdot sdot sdot 1198902119904+3] = 1198903[1198902 1198904 sdot sdot sdot 1198902119904+3] = 1198902[1198901 1198904 sdot sdot sdot 1198902119904+3] = 21198901(1198894)
[1198902 sdot sdot sdot 1198902119904+2] = 1198901[1198901 1198903 sdot sdot sdot 1198902119904+2] = 1198902[1198901 1198902 1198904 sdot sdot sdot 1198902119904+2] = 1198903(1198895)
[1198901 1198904 sdot sdot sdot 1198902119904+3] = 1198901[1198902 1198904 sdot sdot sdot 1198902119904+3] = 1198903[1198903 1198904 sdot sdot sdot 1198902119904+3] = 1205731198902 + (1 + 120573) 1198903(1198896)
[1198901 1198904 sdot sdot sdot 1198902119904+3] = 1198901[1198902 1198904 sdot sdot sdot 1198902119904+3] = 1198902[1198903 1198904 sdot sdot sdot 1198902119904+3] = 1198903(1198897)
[1198901 1198904 sdot sdot sdot 1198902119904+3] = 1198902[1198902 1198904 sdot sdot sdot 1198902119904+3] = 1198903[1198903 1198904 sdot sdot sdot 1198902119904+3] = 1199041198901 + 1199051198902 + 1199061198903120573 119904 119905 119906 isin F 120573 = 0 1 119904 = 0(17)
And n-Lie algebras corresponding to the case (1198897) with coef-ficients 119904 119905 119906 and 1199041015840 1199051015840 1199061015840 are isomorphic if and only if thereexists a nonzero element 120582 isin F such that 119904 = 12058231199041015840 119905 = 12058221199051015840119906 = 1205821199061015840 119904 1199041015840 119905 1199051015840 119906 1199061015840 isin F (119903) dim1198601 = 1199034 le 119903 lt 2119904 + 3 119891119900119903 2 le 119895 le 119903 1 le 119894 le 119903
(1199031) [1198902 sdot sdot sdot 1198902119904+2] = 1198901[1198902 sdot sdot sdot 119890119895 sdot sdot sdot 1198902119904+3] = 119890119895
(1199032) [1198901 sdot sdot sdot 119890119894 sdot sdot sdot 1198902119904+2] = 119890119894(18)
Theorem 10 If119860 is a (2119904+3)-dimensional (2119904+1)-Lie algebraover F with dim1198601 = 119903 lt 119904 + 3 then there are involutivederivations on 119860Proof Define linear mappings 119863119895 119860 997888rarr 119860 1 le 119895 le 6 by
1198631 (119890119894) = 119890119894 1 le 119894 le 119904 + 2 or 119894 = 2119904 + 3minus119890119894 otherwise1198632 (119890119894) = 119890119894 1 le 119894 le 119904 + 2minus119890119894 otherwise
International Journal of Mathematics and Mathematical Sciences 5
1198633 (119890119894) = minus119890119894 119904 + 2 le 119894 le 2119904 + 1119890119894 otherwise
1198634 (119890119894) = 119890119894 1 le 119894 le 119904 + 1 or 119894 = 2119904 + 2minus119890119894 otherwise
1198635 (119890119894) = 119890119894 1 le 119894 le 119904 + 1 or 119894 = 2119904 + 3minus119890119894 otherwise
1198636 (119890119894) = 119890119894 1 le 119894 le 119904 + 3minus119890119894 otherwise
(19)
Since dim1198601 = 119903 le 119904 + 2 it is easy to verify that 1198631 isan involutive derivation on the 3-Lie algebras of the cases of(1198871) (119888119894) (119889119895) and (119903119896) where 1 le 119894 le 7 1 le 119895 le 4 and1 le 119896 le 2 1198632 is an involutive derivation on the 3-Lie algebrasof the cases of (1198871) (1198894) and (119888119894) where 5 le 119894 le 7 1198633 1198634and 1198635 are involutive derivations on the 3-Lie algebras of thecase of (1198872) And 1198636 is an involutive derivation on the 3-Lie algebras of the cases of (1198895) (1198896) and (1198897) Also 119863119894 areinvolutive derivations on abelian algebras for 1 le 119894 le 6
Next we discuss the case of dim1198601 = 119903 ge 119904 + 3 Let 119863 bean endomorphism of 119860
119863119890119894 = 2119904+3sum119895=1
119887119894119895119890119895 119887119894119895 isin F 1 le 119894 le 2119904 + 3 (20)
and 119861 = (119887119894119895) be the (2119904 + 3) times (2119904 + 3)-matrix Then
119863(1198901 sdot sdot sdot 1198902119904+3)119879 = 119861 (1198901 sdot sdot sdot 1198902119904+3)119879= (1198611 11986101198612 1198613) (1198901 sdot sdot sdot 1198902119904+3)119879 (21)
where ( 1198611 11986101198612 1198613 ) is the block matrix of 119861 First we discuss(2119904 + 3)-dimensional (2119904 + 1)-Lie algebras of the case (1199031) inLemma 9
Lemma 11 If 119860 is a (2119904 + 3)-dimensional (2119904 + 1)-Lie algebraof the case (1199031) with dim1198601 = 119903 ge 119904 + 3 119904 ge 1Then the linearmapping 119863 is an involutive derivation on 119860 if and only if theblock matrix 119861 = ( 1198611 11986101198612 1198613 ) satisfies that 1198610 = 119874119903times(2119904+3minus119903) (whichis the zero (119903 times (2119904 + 3 minus 119903))-matrix) and11986121 = 11986411990311986123 = 1198642119904+3minus11990311986121198611 + 11986131198612 = 0
(22)
2119904+2sum119895=2
119887119895119895 = 119887112119904+3sum119895=2119895 =119894
119887119895119895 = 119887119894119894 2 le 119894 le 1199031198872119904+3119894 = (minus1)119894+1 1198871198941 2 le 119894 le 119903119887119895119894 = (minus1)119895minus119894minus1 119887119894119895 2 le 119894 119895 le 119903 119894 = 119895
(23)
Proof By (2) and a direct computation 119863 is a derivation of119860 if and only if matrix 119861 has the property
2119904+2sum119895=2
119887119895119895 = 119887112119904+3sum119895=2119895 =119894
119887119895119895 = 119887119894119894 2 le 119894 le 1199031198871119897 = 0 2 le 119897 le 2119904 + 31198872119904+3119894 = (minus1)119894+1 1198871198941 119887119894119897 = 0 2 le 119894 le 119903 119897 ge 119903 + 1119887119895119894 = (minus1)119895minus119894minus1 119887119894119895 2 le 119894 119895 le 119903 119894 = 119895
(24)
Therefore matrix 119861 satisfies (23) and 1198610 = 119874119903times(2119904+3minus119903) And1198632 = 119868119889 if and only if
1198612 = ( 11986121 11986111198610 + 1198610119861311986121198611 + 11986131198612 11986121198610 + 11986123 )= (119864119903 119874119874 1198642119904+3minus119903) (25)
Thanks to 1198610 = 119874119903times(2119904+3minus119903) (22) holdsTheorem 12 Let 119860 be a (2119904 + 3)-dimensional (2119904 + 1)-Liealgebra of the case (1199031) with dim1198601 = 119903 ge 119904 + 3 119904 ge 1 If 119903is odd then there are involutive derivations on 119860Proof Let 119903 = 2119905 + 1 ge 119904 + 3 Then 119905 ge 2 and 119903 ge 5 Suppose119863 is an endomorphism of119860 and the matrix of119863with respectto the basis 1198901 sdot sdot sdot 1198902119904+3 is 119861 = (119887119894119895) = ( 1198611 11986101198612 1198613 ) which satisfies(22) and (23) and 1198610 = 119874119903times(2119904+3minus119903) Then
1198611
= (((((((
11988711 0 0 sdot sdot sdot 0 011988721 11988722 11988723 sdot sdot sdot 1198872119903minus1 119887211990311988731 11988723 11988733 sdot sdot sdot 1198873119903minus1 1198873119903 d119887119903minus11 (minus1)119903 1198872119903minus1 (minus1)119903minus1 1198873119903minus1 sdot sdot sdot 119887119903minus1119903minus1 119887119903minus11199031198871199031 (minus1)119903+1 1198872119903 (minus1)119903 1198873119903 sdot sdot sdot 119887119903minus1119903 119887119903119903
)))))))
6 International Journal of Mathematics and Mathematical Sciences
1198612 = ((
119887119903+11 119887119903+12 119887119903+13 sdot sdot sdot 119887119903+1119903 d1198872119904+21 1198872119904+22 1198872119904+23 sdot sdot sdot 1198872119904+21199031198872119904+31 minus11988721 11988731 sdot sdot sdot (minus1)119903+1 1198871199031
))
(26)
Since sum2119904+2119895=2 119887119895119895 = 11988711 sum2119904+3119895=2119895 =119894 119887119895119895 = 119887119894119894 2 le 119894 le 119903 we haveminus11988711 + 211988722 minus 1198872119904+32119904+3 = 0 (119903 minus 3)11988722 + sum119903+12119904+3 119887119894119894 = 0 11988722 =119887119894119894 3 le 119894 le 119903 Therefore
11988711 = minus1119903 minus 3 ((119903 minus 1) 1198961 + 22119904+3minus119903sum119895=2
119896119895) 119887119894119894 = minus1119903 minus 3 2119904+3minus119903sum
119895=1
119896119895 2 le 119894 le 119903119887119895119895 = 1198962119904+3minus119895+1 119903 + 1 le 119895 le 2119904 + 3 1198962119904+3minus119895+1 isin F
(27)
Suppose
11986121 =((((((((((((
11988811 11988812 sdot sdot sdot 1198881119894 sdot sdot sdot 1198881119895 sdot sdot sdot 1198881119903 d d
d1198881198941 1198881198942 sdot sdot sdot 119888119894119894 sdot sdot sdot 119888119894119895 sdot sdot sdot 119888119894119903 d
d d
1198881198951 1198881198952 sdot sdot sdot 119888119895119894 sdot sdot sdot 119888119895119895 sdot sdot sdot 119888119895119903 d d
d1198881199031 1198881199032 sdot sdot sdot 119888119903119894 sdot sdot sdot 119888119903119895 sdot sdot sdot 119888119903119903
))))))))))))
(28)
By (23)
11988811 = 1198872111198881119897 = 0 2 le 119897 le 119903119888119894119894 = 119894sum119897=1
(minus1)119894+119897minus1 1198872119897119894 + 119903sum119897=119894+1
(minus1)119894+119897minus1 1198872119894119897 2 le 119894 le 119903119888119894119895 = 119894sum119897=1
(minus1)119897+119894+1 119887119897119894119887119897119895 + 119895sum119897=119894+1
119887119894119897119887119897119895+ 119903sum119897=119895+1
(minus1)119897+119895+1 119887119894119897119887119895119897 2 le 119894 lt 119895 le 119903119888119894119895 = 119895sum119897=1
(minus1)119897+119894+1 119887119897119895119887119897119894 + 119894sum119897=119895+1
(minus1)119894+119895 119887119895119897119887119897119894+ 119903sum119897=119894+1
(minus1)119897+119895+1 119887119895119897119887119894119897 1 le 119895 lt 119894 le 119903
(29)
Therefore the endomorphisms 119863 of 119860 which are defined by1198631198901 = 1198901or 1198631198901 = minus1198901
1198631198902 = 119903sum119896=3
119890119896119863119890119894 = (minus1)119894minus1 1198902 + 119894minus1sum
119896=3
(minus1)119894 119890119896 + (minus1)119894minus1 119903sum119896=119894+1
1198901198963 le 119894 le 119903119863119890119895 = (minus1)119895 119890119895 119903 + 1 le 119895 le 2119904 + 31198631198901 = 1198901
or 1198631198901 = minus11989011198631198902 = 119903sum
119896=3
119890119896119863119890119894 = (minus1)119894minus1 1198902 + 119894minus1sum
119896=3
(minus1)119894 119890119896 + (minus1)119894minus1 119903sum119896=119894+1
1198901198963 le 119894 le 119903119863119890119895 = (minus1)119895minus1 119890119895 119903 + 1 le 119895 le 2119904 + 3
(30)
are involutive derivations on 119860Theorem 13 Let 119860 be a (2119904 + 3)-dimensional (2119904 + 1)-Liealgebra of the case (1199031) with dim1198601 = 119903 = 2119904 + 2 (119904 ge 1)then there does not exist an involutive derivation on 119860Proof If 119863 is an involutive derivation on 119860 then byLemma 11 and (23)
11988711 = minus (2119904 + 1) 11989612119904 minus 1 1198872119904+32119904+3 = 1198961119887119894119894 = minus11989612119904 minus 1
2 le 119894 le 119903 1198961 isin F (31)
Thanks to (22) 11988722119904+32119904+3 = 119887211 = 11989621 = 1 Therefore(minus(2119904 + 1)1198961(2119904 minus 1))2 = ((2119904 + 1)(2119904 minus 1))2 = 1 whichis a contradiction
Now we discuss case (1199032)Theorem 14 Let 119860 be a (2119904 + 3)-dimensional (2119904 + 1)-Liealgebra of the case (1199032) with dim1198601 = 119903 ge 119904 + 3 Then thereexist involutive derivations on 119860 if and only if 119903 is evenProof By Lemma 7 119860 = 1198601 + F1198902119904+3 where 1198902s+3 isin 119885(119860)and 1198601 is a (2119904 + 2)-dimensional (2119904 + 1)-Lie subalgebra
International Journal of Mathematics and Mathematical Sciences 7
of 119860 with dim1198601 = dim11986011 = 119903 Then there existinvolutive derivations on119860 if and only if there exist involutivederivations on 1198601
By Theorem 3 in [1] there is a basis 1198901 sdot sdot sdot 1198902119904+2 of 1198601such that[1198901 sdot sdot sdot 119890119895 sdot sdot sdot 1198902119904+2] = 0 119905 lt 119895 le 2119904 + 3 minus 119905[1198901 sdot sdot sdot 119890119894 sdot sdot sdot 1198902119904+2] = (minus1)119894 1198902119904+3minus1198941 le 119894 le 119905 or 2119904 + 3 minus 119905 lt 119894 le 2119904 + 2 (32)
If 119903 is even then 119903 = 2119905 ge 4 By Theorem 8 and (32)endomorphism 1198631 of 1198601 defined by
1198631 (119890119894) = 119890119894 119894 = 1 sdot sdot sdot 119904 + 1minus119890119894 119894 = 119904 + 2 sdot sdot sdot 2119904 + 2 (33)
is an involutive derivation on 1198601 Therefore the endomor-phism 119863 of 119860 defined by119863 (119890119894) = 119890119894 1 le 119894 le 119904 + 1119863 (119890119895) = minus119890119895 119904 + 2 le 119895 le 2119904 + 2119863 (1198902119904+3) = plusmn1198902119904+3
(34)
is involutive derivation on 119860If dim1198601 = 119903 is odd and endomorphism 119863 of 119860 is an
involutive derivation on 119860 then 119903 = 2119905 + 1 gt 4 Suppose119863(119890119894) = sum2119904+3119895=1 119886119894119895119890119895 1 le 119894 le 2119904 + 3 Then[119863 (1198902119904+3) 1198901198941 sdot sdot sdot 1198901198942119904]= 119863 [1198902119904+3 1198901198941 sdot sdot sdot 1198901198942119904]minus 2119904sum119895=1
[1198902119904+3 sdot sdot sdot 119863119890119894119895 sdot sdot sdot 1198901198942119904] = 0 (35)
We get 119863(1198902119904+3) isin 119885(119860) = F1198902119904+3 Since 1198632 = 119868119889 119863(1198902119904+3) =plusmn1198902119904+3 By (32) 1198601 = F1198901 + sdot sdot sdot + F119890119905 + F1198902119904+1 + sdot sdot sdot + F1198902119904+3minus119905and (minus1)119894119863 (1198902119904+3minus119894)
= 2119904+2sum119896=1119896=119894
[1198901 sdot sdot sdot 119890119894 sdot sdot sdot 119863 (119890119896) sdot sdot sdot 1198902119904+2] (36)
where 1 le 119894 le 119905 and 2119904 + 3 minus 119905 lt 119894 le 2119904 + 2 Then 1198861198942119904+3 = 0for 1 le 119894 le 119905 or 2119904 + 3 minus 119905 lt 119894 le 2119904 + 2 and 1198631198601 sube 1198601 Thenthe endomorphism 1198632 of 1198601 defined by1198632 (119890119894) = 119863 (119890119894) 1 le 119894 le 119905 or 2119904 + 3 minus 119905 lt 119894 le 2119904 + 2
1198632 (119890119895) = 119863(119890119895) minus 1198861198952119904+31198902119904+3 = 2119904+2sum119895=1
119886119894119895119890119895119905 lt 119895 le 2119904 + 3 minus 119905
(37)
is an involutive derivation on the (2119904+2)-dimensional (2119904+1)-Lie algebra 1198601 contradiction (Theorem 8) Therefore theredoes not exist involutive derivation on 119860
5 Structure of 3-Lie Algebras withInvolutive Derivations
Let (119871 [ ]) be a Lie algebra over F and 119901 be an element whichis not contained in 119871 Then 119860 = 119871 + F119901 is a 3-Lie algebra inthe multiplication[119909 119910 119911] = 0[119901 119909 119910] = [119909 119910]
for all 119909 119910 119911 isin 119871 (38)
And the 3-Lie algebra (119860 [ ]) is called one-dimensionalextension of 119871Theorem 15 Let 119860 be a 3-Lie algebra then 119860 is one-dimensional extension of a Lie algebra if and only if there existsan involutive derivation 119863 on 119860 such that dim1198601 = 1 ordim119860minus1 = 1Proof If 119860 is an one-dimensional extension of a Lie algebra119871 then 119860 = 119871 + F119901 Define the endomorphism 119863 of 119860 by119863(119901) = minus119901 (or 119901) and 119863(119909) = 119909 (or minus119909) forall119909 isin 119871 Thanks to(38)1198632 = 119868119889 and 119863([119909 119910 119911]) = 0 = [119863119909 119910 119911] + [119909119863119910 119911] +[119909 119910119863119911]119863([119901 119909 119910]) = [119901 119909 119910] = [119863119901 119909 119910] + [119901119863119909 119910] +[119901 119909 119863119910] for all 119909 119910 119911 isin 119871 Therefore 119863 is an involutivederivation on 119860 and dim119860minus1 = 1 (or dim1198601 = 1)
Conversely let 119863 be an involutive derivation on a 3-Liealgebra 119860 and dim119860minus1 = 1 (or dim1198601 = 1) Let 119860minus1 = F119901and 1198601 = 119871 (or 119860minus1 = 1198711198601 = F119901) where 119901 isin 119860minus119871 Thanksto Theorem 3 119871 is a Lie algebra with the multiplication[119909 119910] = [119901 119909 119910] for all 119909 119910 isin 119871 and 119860 is one-dimensionalextension of 119871
Let (119871 [ ]1) and (119871 [ ]2) be Lie algebras and 1199091 sdot sdot sdot 119909119898be a basis of 119871 For convenience denote Lie algebras (119871 [ ]119896)by 119871119896 119896 = 1 2 respectively Suppose 1199011 and 1199012 are twodistinct elements which are not contained in 119871 and 3-Lie algebras (119861 [ ]1) and (119862 [ ]2) are one-dimensionalextensions of Lie algebras 1198711 and 1198712 respectively where119861 = 119871 + F1199011 119862 = 119871 + F1199012 Then 119863119890119903(1198711) and 119863119890119903(1198712) aresubalgebras of 119892119897(119871)Definition 16 Let 1198711 = (119871 [ ]1) and 1198712 = (119871 [ ]2) be twoLie algebras and 1199011 1199012 be two distinct elements which arenot contained in 119871 and 119860 = 119871 + F1199011 + F1199012 Then 3-algebra(119860 [ ]) is called a two-dimensional extension of Lie algebras119871119896 119896 = 1 2 where [ ] 119860 and 119860 and 119860 997888rarr 119860 defined by[119909 119910 1199011] = [119909 119910]1 [119909 119910 1199012] = [119909 119910]2 [119909 119910 119911] = 0[1199011 1199012 119909] = 1205821199091199011 + 1205831199091199012 forall119909 119910 119911 isin 119871 120582119909 120583119909 isin F
(39)
If 119860 is a 3-Lie algebra then 119860 is called a two-dimensionalextension 3-Lie algebra of Lie algebras 119871119896 119896 = 1 2
8 International Journal of Mathematics and Mathematical Sciences
Let 119860 = 119871 + 119882 be a two-dimensional extension of Liealgebras 119871119896 119896 = 1 2 where 119882 = F1199011 + F1199012 Define linearmappings 1198631 1198632 119871 997888rarr 119864119899119889(119871) and 119863 119871 997888rarr 119882 by1198631 (119909) = 119886119889 (1199011 119909) 1198632 (119909) = 119886119889 (1199012 119909) 119863 (119909) = 119886119889 (1199011 1199012) (119909) forall119909 isin 119871
(40)
that is for all 119910 isin 119871 1198631(119909)(119910) = [1199011 119909 119910] = [119909 119910]11198632(119909)(119910) = [1199012 119909 119910] = [119909 119910]2 119863(119909) = [1199011 1199012 119909] We havethe following result
Theorem 17 Let 3-algebra 119860 be a two-dimensional extensionof Lie algebras 1198711 and 1198712 Then 119860 is a 3-Lie algebra if and onlyif linear mappings 1198631 1198632 and 119863 satisfy that 1198631 1198711 997888rarr119863119890119903(1198711) 1198632 1198712 997888rarr 119863119890119903(1198712) are Lie homomorphisms and1198631 (1199093) ([1199091 1199092]2) = [1198631 (1199093) (1199091) 1199092]2+ [1199091 1198631 (1199093) (1199092)]2minus 1205821199093 [1199091 1199092]1 minus 1205831199093 [1199091 1199092]2
(41)
1198632 (1199093) ([1199091 1199092]1) = [1198632 (1199093) (1199091) 1199092]1+ [1199091 1198632 (1199093) (1199092)]1+ 1205821199093 [1199091 1199092]1 + 1205831199093 [1199091 1199092]2 (42)
119863 ([1199091 1199092]1) = (12058311990911205821199092 minus 12058211990911205831199092) 1199011119863 ([1199091 1199092]2) = (12058311990911205821199092 minus 12058211990911205831199092) 1199012 (43)
120582[11990911199092]1 = 120583[11990911199092]2 = 12058311990911205821199092 minus 12058211990911205831199092 120583[11990911199092]1 = 120582[11990911199092]2 = 0 (44)
119863119896 (1199091) (1199092) = minus119863119896 (1199092) (1199091) 119891119900119903 119886119897119897 1199091 1199092 isin 119871 119896 = 1 2 (45)
where 1199091 1199092 1199093 isin 119871 119863(119909119894) = 1205821199091198941199011 + 1205831199091198941199012 119894 = 1 2 3Proof If 119860 is a two-dimensional extension 3-Lie algebrathen by Definition 16 linear mappings 119863119896 satisfy that119863119896(119871119896) sube 119863119890119903(119871119896) and119863119896 are Lie homomorphisms 119896 = 1 2Thanks to (39)1198631 (1199093) ([1199091 1199092]2) = [1199012 [1199011 1199093 1199091] 1199092]+ [1199012 1199091 [1199011 1199093 1199092]]+ [[1199011 1199093 1199012] 1199091 1199092]= [1198631 (1199093) (1199091) 1199092]2+ [1199091 1198631 (1199093) (1199092)]2minus 1205821199093 [1199091 1199092]1 minus 1205831199093 [1199091 1199092]2
(46)
for all 1199091 1199092 1199093 isin 119871 (41) holds Similarly we have (42)Thanks to (39) and (40)119863([1199091 1199092]1) = 119886119889 (1199011 1199012) [1199091 1199092]1= 120582[11990911199092]11199011 + 120583[11990911199092]11199012= (12058311990911205821199092 minus 12058211990911205831199092) 1199011119863 ([1199091 1199092]2) = 119886119889 (1199011 1199012) [1199091 1199092]2= 120582[11990911199092]21199011 + 120583[11990911199092]21199012= (12058311990911205821199092 minus 12058211990911205831199092) 1199012
(47)
Equations (43) and (44) hold Equation (45) follows from (39)and (40) directly
Conversely by (39) forall1199091 1199092 1199093 119909 isin 119871[1199091 1199092 1199093] = 0[1199011 1199091 1199092] = 1198631 (1199091) (1199092) = [1199091 1199092]1 [1199012 1199091 1199092] = 1198632 (1199091) (1199092) = [1199091 1199092]2 [1199011 1199012 119909] = 119863 (119909) = 120582 (119909) 1199011 + 1205831199091199012(48)
Since119863119896(119871) sube 119863119890119903(119871119896) and119863119896 are Lie homomorphisms 119861 =119871 + F1199011 and 119862 = 119871 + F1199012 are 3-Lie algebras which are one-dimensional extension 3-Lie algebras of Lie algebras 119871119896 119896 =1 2 respectively
Next we only need to prove that the multiplication on119860 defined by (39) satisfies (1) For all 119909119894 isin 119871 1 le 119894 le5 that products [[1199091 1199092 1199093] 1199094 1199095] [[119901119895 1199092 1199093] 1199094 1199095][[1199091 1199092 1199093] 1199094 119901119895] and [[1199091 1199092 119901119895] 1199094 119901119895] satisfy (1) 119895 =1 2 follow from that 119861 and 119862 are one-dimensional extension3-Lie algebras of 119871119896 and (39) directlyFrom (41) and (42) it follows that products[[119901119894 1199091 1199092] 119901119895 1199093] 1 le 119894 = 119895 le 2 satisfy (1) It
follows from (43)ndash(45) that products [1199011 1199012 [119901119894 1199091 1199092]][1199091 1199092 [119901119894 1199012 1199093]] and [119901119894 1199091 [1199011 1199012 1199092] 119894 = 1 2 satisfy(1) We omit the computation process
Theorem 18 Let (119860 [ ]) be a 3-Lie algebra Then 119860 is a two-dimensional extension 3-Lie algebra of Lie algebras if and onlyif there is an involutive derivation119879 on119860 such that dim1198601 = 2or dim119860minus1 = 2Proof If 119860 is a two-dimensional extension 3-Lie algebra ofLie algebras Then byTheorem 15 there are Lie algebras 1198711 =(119871 [ ]1) and 1198712 = (119871 [ ]2) such that 119860 = 119871 + 119882 and themultiplication of 119860 is defined by (39) where119882 = F1199011 + F1199012
Define the endomorphism 119879 of 119860 by 119879(119909) = 119909 119879(1199011) =minus1199011 119879(1199012) = minus1199012 or 119879(119909) = minus119909 119879(1199011) = 1199011 119879(1199012) =1199012 forall119909 isin 119871Then1198792 = 119868119889 and1198601 = 119871119860minus1 = 119882 or119860minus1 = 1198711198601 = 119882Thanks to (38) and (41)-(45) 119879 is a derivation of 119860Conversely if there is an involutive derivation 119879 on the3-Lie algebra 119860 such that dim119860minus1 = 2 (or dim1198601 = 2)
By Theorem 4 [1198601 1198601 1198601] = 0 [1198601 1198601 119860minus1] sube 1198601[1198601 119860minus1 119860minus1] sube 119860minus1 Let 119871 = 1198601 and 119860minus1 = F1199011 + F1199012Then [119871 119871 1199011] sube 119871 [119871 119871 1199012] sube 119871 and (119871 [ ]1) and
International Journal of Mathematics and Mathematical Sciences 9
(119871 [ ]2) are Lie algebras where [119909 119910]1 = [119909 119910 1199011] [119909 119910]2 =[119909 119910 1199012] forall119909 119910 isin 119871Thanks toTheorem 17 the 3-Lie algebra119860 is a two-dimensional extension 3-Lie algebra of Lie algebras1198711 and 1198712Data Availability
No data were used to support this study
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
The first named author was supported in part by the NaturalScience Foundation (11371245) and theNatural Science Foun-dation of Hebei Province (A2018201126)
References
[1] V Filippov ldquon-Lie algebrasrdquo SiberianMathematical Journal vol26 no 6 pp 126ndash140 1985
[2] R P Bai and P P Jia ldquoThe real compact n-Lie algebras andinvariant bilinear formsrdquo Acta Mathematica Scientia Series AShuxueWuli Xuebao Chinese Edition vol 27A no 6 pp 1074ndash1081 2007
[3] S Kasymov ldquoOn a theory of n-Lie algebrasrdquo Algebra and Logicvol 26 no 3 pp 277ndash297 1987
[4] C Bai L Guo and Y Sheng ldquoBialgebras the classical Yang-Baxter equation andmain triples for 3-Lie algebrasrdquoMathemat-ical Physics 2016
[5] R Bai and G Song ldquoThe classification of six-dimensional 4-Liealgebrasrdquo Journal of Physics A Mathematical and General vol42 no 3 035207 17 pages 2009
[6] R Bai G Song and Y Zhang ldquoOn classification of n-Liealgebrasrdquo Frontiers of Mathematics in China vol 6 no 4 pp581ndash606 2011
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International Journal of Mathematics and Mathematical Sciences 3
Proof If there is an involutive derivation 119863 on 119860 then byTheorem 4 119861 = 1198601 and 119862 = 119860minus1 satisfy (12) and (13)
Conversely define an endomorphism 119863 of 119860 by 119863(119909) =119909119863(119910) = minus119910 forall119909 isin 119861 119910 isin 119862 Then 1198632 = 119868119889 119861 = 1198601 and119862 = 119860minus1 By (12) and (13) 119863 is a derivation
Corollary 6 Let 119860 be a (2119904 + 1)-dimensional (2119904 + 1)-Liealgebra with the multiplication [1198901 sdot sdot sdot 1198902119904+1] = 1198901 where1198901 sdot sdot sdot 1198902119904+1 is a basis of 119860 Then the linear mapping 119863 119860 997888rarr 119860 defined by 119863(119890119894) = 119890119894 1 le 119894 le 119904 + 1 119863(119890119895) =minus119890119895 119904 + 2 le 119895 le 2119904 + 1 is an involutive derivation on119860Proof The result follows from a direct computation
3 Involutive Derivations on(119899 + 1)-Dimensional 119899-Lie Algebraswith 119899 = 2119904 + 1
In this section we study involutive derivations on (119899 + 1)-dimensional 119899-Lie algebras over F FromTheorem 3 we onlyneed to discuss the case of 119899 = 2119904 + 1 119904 ge 1Lemma 7 (see [5]) Let 119860 be an (119899 + 1)-dimensional non-abelian 119899-Lie algebra over F 119899 ge 3 Then up to isomorphisms119860 is one and only one of the following possibilities
(1198871) [1198902 1198903 sdot sdot sdot 119890119899+1] = 1198901(1198872) [1198901 1198902 sdot sdot sdot 119890119899] = 1198901(1198881) [1198902 sdot sdot sdot 119890119899+1] = 1198901[1198901 1198903 sdot sdot sdot 119890119899+1] = 1198902
(1198882) [1198902 sdot sdot sdot 119890119899+1] = 1205721198901 + 1198902[1198901 1198903 sdot sdot sdot 119890119899+1] = 1198902 120572 isin F 120572 = 0(1198883) [1198901 1198903 sdot sdot sdot 119890119899+1] = 1198901[1198902 sdot sdot sdot 119890119899+1] = 1198902(119889119903) [1198901 sdot sdot sdot 119890119894 sdot sdot sdot 119890119899+1] = 119890119894 1 le 119894 le 119903
(14)
where 1198901 sdot sdot sdot 119890119899+1 is a basis of119860 3 le 119903 le 119899+1 and 119890119894 meansthat 119890119894 is omitted
Theorem 8 Let 119860 be a (2119904 + 2)-dimensional (2119904 + 1)-Liealgebra over F and dim1198601 = 119903 Then there exists an involutivederivation 119863 on 119860 if and only if 119903 is even or 0 le 119903 le 119904+2Proof If dim1198601 = 119903 le 119904 + 2 then by Lemma 7 and a directcomputation the linear mapping 119863 119860 997888rarr 119860 defined by119863(119890119894) = 119890119894 119863(119890119895) = minus119890119895 1 le 119894 le 119904 + 2 119904 + 3 le 119895 le 2119904 + 2 is aninvolutive derivation on 119860
Now we discuss the case dim1198601 = 119903 ge 119904 + 3 Let1198901 sdot sdot sdot 1198902119904+2 be a basis of 119860 and the multiplication in thebasis be as follows
119890119894 = (minus1)2119904+2+119894 [1198901 sdot sdot sdot 119890119894 sdot sdot sdot 1198902119904+2] = 2119904+2sum119897=1
120573119894119897119890119897120573119894119897 isin 119865 1 le 119894 le 2119904 + 2 (15)
where 120573119894119897 isin F 1 le 119894 119897 le 2119904 + 2 Thanks toTheorem 3 in [1] 119860is a 3-Lie algebra if and only if the (2119904 + 2) times (2119904 + 2)-matrix119861 = (120573119894119897) is symmetric
If 119903 = 2119905 gt 3 2 le 119905 le 119904 + 1 then define the multiplicationon 119860 by
[1198901 sdot sdot sdot 119890119894 sdot sdot sdot 1198902119904+2] = (minus1)119894 1198902119904+3minus1198941 le 119894 le 119905 3 minus 119905 lt 119894 minus 2119904 le 2[1198901 sdot sdot sdot 119890119895 sdot sdot sdot 1198902119904+2] = 0 119905 lt 119895 le 2119904 + 3 minus 119905
(16)
that is 1205731198942119904+3minus119894 = 1205732119904+3minus119894119894 = 1 for 1 le 119894 le 119905 or 2119904 + 3 minus 119905 lt119894 le 2119904 + 2 and others are zero Then 119861 = (120573119894119897) is symmetricTherefore 119860 is a (2119904 + 1)-Lie algebra with the multiplication(16)
Define an endomorphism119863 of119860 by119863119890119894 = 119890119894 1 le 119894 le 119904+1and 119863119890119895 = minus119890119895 for 119904 + 2 le 119895 le 2119904 + 2Then 119863 is an involutivederivation on 119860
For the case dim1198601 = 119903 = 2119905 + 1 ge 119904 + 3 Suppose 119897 =dim1198601 1198971015840 = dim119860minus1
If there is an involutive derivation 119863 on 119860 then byTheorem 4 119897 + 1198971015840 = 2119904 + 2 119904 le 119897 le 119904 + 2 and 119904 le 1198971015840 le 119904 + 2Since dim1198601 = 119903 = 2119905 + 1 ge 119904 + 3 1198601 cap 1198601 = 0 and1198601 cap119860minus1 = 0Therefore dim1198601 = dim119860minus1 = 119904+ 1 Withoutloss of generality we can suppose 1198901 sdot sdot sdot 119890119904+1 sube 1198601 and119890119904+2 sdot sdot sdot 1199092119904+2 sube 119860minus1 By (10) and (11) the (2119904+2)times(2119904+2)-matrix 119861 = (120573119894119897) defined by (15) is nonsymmetric which is acontradiction Therefore if dim1198601 = 119903 = 2119905 + 1 ge 119904 + 3 thenthere do not exist involutive derivations on 119860
By Theorem 8 if 119860 is a 10-dimensional 9-Lie algebrawith dim1198601 = 7 or 9 then there does not exist involutivederivation on 119860 If 1 le dim1198601 = 119903 le 10 and 119903 = 7 9 thenthere are involutive derivations on 1198604 Involutive Derivations on(119899 + 2)-Dimensional 119899-Lie Algebraswith 119899 = 2119904 + 1
By Theorem 3 we only need to discuss the case where 119899 isodd So we suppose that 119860 is a (2119904 + 3)-dimensional (2119904 + 1)-Lie algebra over F 119904 ge 1 and that 119864119905 = Diag(1 sdot sdot sdot 1) is the(119905 times 119905)-unit matrix
Lemma9 (see [6]) Let119860 be a (2119904+3)-dimensional (2119904+1)-Liealgebra over F with a basis 1198901 sdot sdot sdot 1198902119904+3Then119860 is isomorphicto one and only one of the following possibilities
4 International Journal of Mathematics and Mathematical Sciences
(119886) 119860 119894119904 119886119899 119886119887119890119897119894119886119899(119887) dim1198601 = 1
(1198871) [1198902 sdot sdot sdot 1198902119904+2] = 1198901(1198872) [1198901 sdot sdot sdot 1198902119904+1] = 1198901(119888) dim1198601 = 2(1198881) [1198902 sdot sdot sdot 1198902119904+2] = 1198901[1198903 sdot sdot sdot 1198902119904+3] = 1198902
(1198882) [1198902 sdot sdot sdot 1198902119904+2] = 1198901[1198902 1198904 sdot sdot sdot 1198902119904+3] = 1198902[1198901 1198904 sdot sdot sdot 1198902119904+3] = 1198901
(1198883)
[1198902 sdot sdot sdot 1198902119904+2] = 1205721198901 + 1198902[1198901 1198903 sdot sdot sdot 1198902119904+2] = 1198902[1198902 1198904 sdot sdot sdot 1198902119904+3] = 1198902[1198901 1198904 sdot sdot sdot 1198902119904+3] = 1198901(1198884)
[1198902 sdot sdot sdot 1198902119904+2] = 1198901[1198902 1198903 sdot sdot sdot 1198902119904+2] = 1198902[1198902 1198904 sdot sdot sdot 1198902119904+3] = 1198902[1198901 1198904 sdot sdot sdot 1198902119904+3] = 1198901(1198885) [1198902 sdot sdot sdot 1198902119904+2] = 1198901[1198901 1198903 sdot sdot sdot 1198902119904+2] = 1198902
(1198886) [1198902 sdot sdot sdot 1198902119904+2] = 1205721198901 + 1198902[1198901 1198903 sdot sdot sdot 1198902119904+2] = 1198902 120572 isin F 120572 = 0(1198887)[1198901 1198903 sdot sdot sdot 1198902119904+2] = 1198901[1198902 1198903 sdot sdot sdot 1198902119904+2] = 1198902(119889) dim1198601 = 3(1198891)
[1198902 sdot sdot sdot 1198902119904+2] = 1198901[1198902 1198904 sdot sdot sdot 1198902119904+3] = 1198902[1198903 sdot sdot sdot 1198902119904+3] = 1198903(1198892)
[1198902 sdot sdot sdot 1198902119904+2] = 1198901[1198903 sdot sdot sdot 1198902119904+3] = 1198903 + 1205721198902[1198902 1198904 sdot sdot sdot 1198902119904+3] = 1198903[1198901 1198904 sdot sdot sdot 1198902119904+3] = 1198901
(1198893)
[1198902 sdot sdot sdot 1198902119904+2] = 1198901[1198903 1198904 sdot sdot sdot 1198902119904+3] = 1198903[1198902 1198904 sdot sdot sdot 1198902119904+3] = 1198902[1198901 1198904 sdot sdot sdot 1198902119904+3] = 21198901(1198894)
[1198902 sdot sdot sdot 1198902119904+2] = 1198901[1198901 1198903 sdot sdot sdot 1198902119904+2] = 1198902[1198901 1198902 1198904 sdot sdot sdot 1198902119904+2] = 1198903(1198895)
[1198901 1198904 sdot sdot sdot 1198902119904+3] = 1198901[1198902 1198904 sdot sdot sdot 1198902119904+3] = 1198903[1198903 1198904 sdot sdot sdot 1198902119904+3] = 1205731198902 + (1 + 120573) 1198903(1198896)
[1198901 1198904 sdot sdot sdot 1198902119904+3] = 1198901[1198902 1198904 sdot sdot sdot 1198902119904+3] = 1198902[1198903 1198904 sdot sdot sdot 1198902119904+3] = 1198903(1198897)
[1198901 1198904 sdot sdot sdot 1198902119904+3] = 1198902[1198902 1198904 sdot sdot sdot 1198902119904+3] = 1198903[1198903 1198904 sdot sdot sdot 1198902119904+3] = 1199041198901 + 1199051198902 + 1199061198903120573 119904 119905 119906 isin F 120573 = 0 1 119904 = 0(17)
And n-Lie algebras corresponding to the case (1198897) with coef-ficients 119904 119905 119906 and 1199041015840 1199051015840 1199061015840 are isomorphic if and only if thereexists a nonzero element 120582 isin F such that 119904 = 12058231199041015840 119905 = 12058221199051015840119906 = 1205821199061015840 119904 1199041015840 119905 1199051015840 119906 1199061015840 isin F (119903) dim1198601 = 1199034 le 119903 lt 2119904 + 3 119891119900119903 2 le 119895 le 119903 1 le 119894 le 119903
(1199031) [1198902 sdot sdot sdot 1198902119904+2] = 1198901[1198902 sdot sdot sdot 119890119895 sdot sdot sdot 1198902119904+3] = 119890119895
(1199032) [1198901 sdot sdot sdot 119890119894 sdot sdot sdot 1198902119904+2] = 119890119894(18)
Theorem 10 If119860 is a (2119904+3)-dimensional (2119904+1)-Lie algebraover F with dim1198601 = 119903 lt 119904 + 3 then there are involutivederivations on 119860Proof Define linear mappings 119863119895 119860 997888rarr 119860 1 le 119895 le 6 by
1198631 (119890119894) = 119890119894 1 le 119894 le 119904 + 2 or 119894 = 2119904 + 3minus119890119894 otherwise1198632 (119890119894) = 119890119894 1 le 119894 le 119904 + 2minus119890119894 otherwise
International Journal of Mathematics and Mathematical Sciences 5
1198633 (119890119894) = minus119890119894 119904 + 2 le 119894 le 2119904 + 1119890119894 otherwise
1198634 (119890119894) = 119890119894 1 le 119894 le 119904 + 1 or 119894 = 2119904 + 2minus119890119894 otherwise
1198635 (119890119894) = 119890119894 1 le 119894 le 119904 + 1 or 119894 = 2119904 + 3minus119890119894 otherwise
1198636 (119890119894) = 119890119894 1 le 119894 le 119904 + 3minus119890119894 otherwise
(19)
Since dim1198601 = 119903 le 119904 + 2 it is easy to verify that 1198631 isan involutive derivation on the 3-Lie algebras of the cases of(1198871) (119888119894) (119889119895) and (119903119896) where 1 le 119894 le 7 1 le 119895 le 4 and1 le 119896 le 2 1198632 is an involutive derivation on the 3-Lie algebrasof the cases of (1198871) (1198894) and (119888119894) where 5 le 119894 le 7 1198633 1198634and 1198635 are involutive derivations on the 3-Lie algebras of thecase of (1198872) And 1198636 is an involutive derivation on the 3-Lie algebras of the cases of (1198895) (1198896) and (1198897) Also 119863119894 areinvolutive derivations on abelian algebras for 1 le 119894 le 6
Next we discuss the case of dim1198601 = 119903 ge 119904 + 3 Let 119863 bean endomorphism of 119860
119863119890119894 = 2119904+3sum119895=1
119887119894119895119890119895 119887119894119895 isin F 1 le 119894 le 2119904 + 3 (20)
and 119861 = (119887119894119895) be the (2119904 + 3) times (2119904 + 3)-matrix Then
119863(1198901 sdot sdot sdot 1198902119904+3)119879 = 119861 (1198901 sdot sdot sdot 1198902119904+3)119879= (1198611 11986101198612 1198613) (1198901 sdot sdot sdot 1198902119904+3)119879 (21)
where ( 1198611 11986101198612 1198613 ) is the block matrix of 119861 First we discuss(2119904 + 3)-dimensional (2119904 + 1)-Lie algebras of the case (1199031) inLemma 9
Lemma 11 If 119860 is a (2119904 + 3)-dimensional (2119904 + 1)-Lie algebraof the case (1199031) with dim1198601 = 119903 ge 119904 + 3 119904 ge 1Then the linearmapping 119863 is an involutive derivation on 119860 if and only if theblock matrix 119861 = ( 1198611 11986101198612 1198613 ) satisfies that 1198610 = 119874119903times(2119904+3minus119903) (whichis the zero (119903 times (2119904 + 3 minus 119903))-matrix) and11986121 = 11986411990311986123 = 1198642119904+3minus11990311986121198611 + 11986131198612 = 0
(22)
2119904+2sum119895=2
119887119895119895 = 119887112119904+3sum119895=2119895 =119894
119887119895119895 = 119887119894119894 2 le 119894 le 1199031198872119904+3119894 = (minus1)119894+1 1198871198941 2 le 119894 le 119903119887119895119894 = (minus1)119895minus119894minus1 119887119894119895 2 le 119894 119895 le 119903 119894 = 119895
(23)
Proof By (2) and a direct computation 119863 is a derivation of119860 if and only if matrix 119861 has the property
2119904+2sum119895=2
119887119895119895 = 119887112119904+3sum119895=2119895 =119894
119887119895119895 = 119887119894119894 2 le 119894 le 1199031198871119897 = 0 2 le 119897 le 2119904 + 31198872119904+3119894 = (minus1)119894+1 1198871198941 119887119894119897 = 0 2 le 119894 le 119903 119897 ge 119903 + 1119887119895119894 = (minus1)119895minus119894minus1 119887119894119895 2 le 119894 119895 le 119903 119894 = 119895
(24)
Therefore matrix 119861 satisfies (23) and 1198610 = 119874119903times(2119904+3minus119903) And1198632 = 119868119889 if and only if
1198612 = ( 11986121 11986111198610 + 1198610119861311986121198611 + 11986131198612 11986121198610 + 11986123 )= (119864119903 119874119874 1198642119904+3minus119903) (25)
Thanks to 1198610 = 119874119903times(2119904+3minus119903) (22) holdsTheorem 12 Let 119860 be a (2119904 + 3)-dimensional (2119904 + 1)-Liealgebra of the case (1199031) with dim1198601 = 119903 ge 119904 + 3 119904 ge 1 If 119903is odd then there are involutive derivations on 119860Proof Let 119903 = 2119905 + 1 ge 119904 + 3 Then 119905 ge 2 and 119903 ge 5 Suppose119863 is an endomorphism of119860 and the matrix of119863with respectto the basis 1198901 sdot sdot sdot 1198902119904+3 is 119861 = (119887119894119895) = ( 1198611 11986101198612 1198613 ) which satisfies(22) and (23) and 1198610 = 119874119903times(2119904+3minus119903) Then
1198611
= (((((((
11988711 0 0 sdot sdot sdot 0 011988721 11988722 11988723 sdot sdot sdot 1198872119903minus1 119887211990311988731 11988723 11988733 sdot sdot sdot 1198873119903minus1 1198873119903 d119887119903minus11 (minus1)119903 1198872119903minus1 (minus1)119903minus1 1198873119903minus1 sdot sdot sdot 119887119903minus1119903minus1 119887119903minus11199031198871199031 (minus1)119903+1 1198872119903 (minus1)119903 1198873119903 sdot sdot sdot 119887119903minus1119903 119887119903119903
)))))))
6 International Journal of Mathematics and Mathematical Sciences
1198612 = ((
119887119903+11 119887119903+12 119887119903+13 sdot sdot sdot 119887119903+1119903 d1198872119904+21 1198872119904+22 1198872119904+23 sdot sdot sdot 1198872119904+21199031198872119904+31 minus11988721 11988731 sdot sdot sdot (minus1)119903+1 1198871199031
))
(26)
Since sum2119904+2119895=2 119887119895119895 = 11988711 sum2119904+3119895=2119895 =119894 119887119895119895 = 119887119894119894 2 le 119894 le 119903 we haveminus11988711 + 211988722 minus 1198872119904+32119904+3 = 0 (119903 minus 3)11988722 + sum119903+12119904+3 119887119894119894 = 0 11988722 =119887119894119894 3 le 119894 le 119903 Therefore
11988711 = minus1119903 minus 3 ((119903 minus 1) 1198961 + 22119904+3minus119903sum119895=2
119896119895) 119887119894119894 = minus1119903 minus 3 2119904+3minus119903sum
119895=1
119896119895 2 le 119894 le 119903119887119895119895 = 1198962119904+3minus119895+1 119903 + 1 le 119895 le 2119904 + 3 1198962119904+3minus119895+1 isin F
(27)
Suppose
11986121 =((((((((((((
11988811 11988812 sdot sdot sdot 1198881119894 sdot sdot sdot 1198881119895 sdot sdot sdot 1198881119903 d d
d1198881198941 1198881198942 sdot sdot sdot 119888119894119894 sdot sdot sdot 119888119894119895 sdot sdot sdot 119888119894119903 d
d d
1198881198951 1198881198952 sdot sdot sdot 119888119895119894 sdot sdot sdot 119888119895119895 sdot sdot sdot 119888119895119903 d d
d1198881199031 1198881199032 sdot sdot sdot 119888119903119894 sdot sdot sdot 119888119903119895 sdot sdot sdot 119888119903119903
))))))))))))
(28)
By (23)
11988811 = 1198872111198881119897 = 0 2 le 119897 le 119903119888119894119894 = 119894sum119897=1
(minus1)119894+119897minus1 1198872119897119894 + 119903sum119897=119894+1
(minus1)119894+119897minus1 1198872119894119897 2 le 119894 le 119903119888119894119895 = 119894sum119897=1
(minus1)119897+119894+1 119887119897119894119887119897119895 + 119895sum119897=119894+1
119887119894119897119887119897119895+ 119903sum119897=119895+1
(minus1)119897+119895+1 119887119894119897119887119895119897 2 le 119894 lt 119895 le 119903119888119894119895 = 119895sum119897=1
(minus1)119897+119894+1 119887119897119895119887119897119894 + 119894sum119897=119895+1
(minus1)119894+119895 119887119895119897119887119897119894+ 119903sum119897=119894+1
(minus1)119897+119895+1 119887119895119897119887119894119897 1 le 119895 lt 119894 le 119903
(29)
Therefore the endomorphisms 119863 of 119860 which are defined by1198631198901 = 1198901or 1198631198901 = minus1198901
1198631198902 = 119903sum119896=3
119890119896119863119890119894 = (minus1)119894minus1 1198902 + 119894minus1sum
119896=3
(minus1)119894 119890119896 + (minus1)119894minus1 119903sum119896=119894+1
1198901198963 le 119894 le 119903119863119890119895 = (minus1)119895 119890119895 119903 + 1 le 119895 le 2119904 + 31198631198901 = 1198901
or 1198631198901 = minus11989011198631198902 = 119903sum
119896=3
119890119896119863119890119894 = (minus1)119894minus1 1198902 + 119894minus1sum
119896=3
(minus1)119894 119890119896 + (minus1)119894minus1 119903sum119896=119894+1
1198901198963 le 119894 le 119903119863119890119895 = (minus1)119895minus1 119890119895 119903 + 1 le 119895 le 2119904 + 3
(30)
are involutive derivations on 119860Theorem 13 Let 119860 be a (2119904 + 3)-dimensional (2119904 + 1)-Liealgebra of the case (1199031) with dim1198601 = 119903 = 2119904 + 2 (119904 ge 1)then there does not exist an involutive derivation on 119860Proof If 119863 is an involutive derivation on 119860 then byLemma 11 and (23)
11988711 = minus (2119904 + 1) 11989612119904 minus 1 1198872119904+32119904+3 = 1198961119887119894119894 = minus11989612119904 minus 1
2 le 119894 le 119903 1198961 isin F (31)
Thanks to (22) 11988722119904+32119904+3 = 119887211 = 11989621 = 1 Therefore(minus(2119904 + 1)1198961(2119904 minus 1))2 = ((2119904 + 1)(2119904 minus 1))2 = 1 whichis a contradiction
Now we discuss case (1199032)Theorem 14 Let 119860 be a (2119904 + 3)-dimensional (2119904 + 1)-Liealgebra of the case (1199032) with dim1198601 = 119903 ge 119904 + 3 Then thereexist involutive derivations on 119860 if and only if 119903 is evenProof By Lemma 7 119860 = 1198601 + F1198902119904+3 where 1198902s+3 isin 119885(119860)and 1198601 is a (2119904 + 2)-dimensional (2119904 + 1)-Lie subalgebra
International Journal of Mathematics and Mathematical Sciences 7
of 119860 with dim1198601 = dim11986011 = 119903 Then there existinvolutive derivations on119860 if and only if there exist involutivederivations on 1198601
By Theorem 3 in [1] there is a basis 1198901 sdot sdot sdot 1198902119904+2 of 1198601such that[1198901 sdot sdot sdot 119890119895 sdot sdot sdot 1198902119904+2] = 0 119905 lt 119895 le 2119904 + 3 minus 119905[1198901 sdot sdot sdot 119890119894 sdot sdot sdot 1198902119904+2] = (minus1)119894 1198902119904+3minus1198941 le 119894 le 119905 or 2119904 + 3 minus 119905 lt 119894 le 2119904 + 2 (32)
If 119903 is even then 119903 = 2119905 ge 4 By Theorem 8 and (32)endomorphism 1198631 of 1198601 defined by
1198631 (119890119894) = 119890119894 119894 = 1 sdot sdot sdot 119904 + 1minus119890119894 119894 = 119904 + 2 sdot sdot sdot 2119904 + 2 (33)
is an involutive derivation on 1198601 Therefore the endomor-phism 119863 of 119860 defined by119863 (119890119894) = 119890119894 1 le 119894 le 119904 + 1119863 (119890119895) = minus119890119895 119904 + 2 le 119895 le 2119904 + 2119863 (1198902119904+3) = plusmn1198902119904+3
(34)
is involutive derivation on 119860If dim1198601 = 119903 is odd and endomorphism 119863 of 119860 is an
involutive derivation on 119860 then 119903 = 2119905 + 1 gt 4 Suppose119863(119890119894) = sum2119904+3119895=1 119886119894119895119890119895 1 le 119894 le 2119904 + 3 Then[119863 (1198902119904+3) 1198901198941 sdot sdot sdot 1198901198942119904]= 119863 [1198902119904+3 1198901198941 sdot sdot sdot 1198901198942119904]minus 2119904sum119895=1
[1198902119904+3 sdot sdot sdot 119863119890119894119895 sdot sdot sdot 1198901198942119904] = 0 (35)
We get 119863(1198902119904+3) isin 119885(119860) = F1198902119904+3 Since 1198632 = 119868119889 119863(1198902119904+3) =plusmn1198902119904+3 By (32) 1198601 = F1198901 + sdot sdot sdot + F119890119905 + F1198902119904+1 + sdot sdot sdot + F1198902119904+3minus119905and (minus1)119894119863 (1198902119904+3minus119894)
= 2119904+2sum119896=1119896=119894
[1198901 sdot sdot sdot 119890119894 sdot sdot sdot 119863 (119890119896) sdot sdot sdot 1198902119904+2] (36)
where 1 le 119894 le 119905 and 2119904 + 3 minus 119905 lt 119894 le 2119904 + 2 Then 1198861198942119904+3 = 0for 1 le 119894 le 119905 or 2119904 + 3 minus 119905 lt 119894 le 2119904 + 2 and 1198631198601 sube 1198601 Thenthe endomorphism 1198632 of 1198601 defined by1198632 (119890119894) = 119863 (119890119894) 1 le 119894 le 119905 or 2119904 + 3 minus 119905 lt 119894 le 2119904 + 2
1198632 (119890119895) = 119863(119890119895) minus 1198861198952119904+31198902119904+3 = 2119904+2sum119895=1
119886119894119895119890119895119905 lt 119895 le 2119904 + 3 minus 119905
(37)
is an involutive derivation on the (2119904+2)-dimensional (2119904+1)-Lie algebra 1198601 contradiction (Theorem 8) Therefore theredoes not exist involutive derivation on 119860
5 Structure of 3-Lie Algebras withInvolutive Derivations
Let (119871 [ ]) be a Lie algebra over F and 119901 be an element whichis not contained in 119871 Then 119860 = 119871 + F119901 is a 3-Lie algebra inthe multiplication[119909 119910 119911] = 0[119901 119909 119910] = [119909 119910]
for all 119909 119910 119911 isin 119871 (38)
And the 3-Lie algebra (119860 [ ]) is called one-dimensionalextension of 119871Theorem 15 Let 119860 be a 3-Lie algebra then 119860 is one-dimensional extension of a Lie algebra if and only if there existsan involutive derivation 119863 on 119860 such that dim1198601 = 1 ordim119860minus1 = 1Proof If 119860 is an one-dimensional extension of a Lie algebra119871 then 119860 = 119871 + F119901 Define the endomorphism 119863 of 119860 by119863(119901) = minus119901 (or 119901) and 119863(119909) = 119909 (or minus119909) forall119909 isin 119871 Thanks to(38)1198632 = 119868119889 and 119863([119909 119910 119911]) = 0 = [119863119909 119910 119911] + [119909119863119910 119911] +[119909 119910119863119911]119863([119901 119909 119910]) = [119901 119909 119910] = [119863119901 119909 119910] + [119901119863119909 119910] +[119901 119909 119863119910] for all 119909 119910 119911 isin 119871 Therefore 119863 is an involutivederivation on 119860 and dim119860minus1 = 1 (or dim1198601 = 1)
Conversely let 119863 be an involutive derivation on a 3-Liealgebra 119860 and dim119860minus1 = 1 (or dim1198601 = 1) Let 119860minus1 = F119901and 1198601 = 119871 (or 119860minus1 = 1198711198601 = F119901) where 119901 isin 119860minus119871 Thanksto Theorem 3 119871 is a Lie algebra with the multiplication[119909 119910] = [119901 119909 119910] for all 119909 119910 isin 119871 and 119860 is one-dimensionalextension of 119871
Let (119871 [ ]1) and (119871 [ ]2) be Lie algebras and 1199091 sdot sdot sdot 119909119898be a basis of 119871 For convenience denote Lie algebras (119871 [ ]119896)by 119871119896 119896 = 1 2 respectively Suppose 1199011 and 1199012 are twodistinct elements which are not contained in 119871 and 3-Lie algebras (119861 [ ]1) and (119862 [ ]2) are one-dimensionalextensions of Lie algebras 1198711 and 1198712 respectively where119861 = 119871 + F1199011 119862 = 119871 + F1199012 Then 119863119890119903(1198711) and 119863119890119903(1198712) aresubalgebras of 119892119897(119871)Definition 16 Let 1198711 = (119871 [ ]1) and 1198712 = (119871 [ ]2) be twoLie algebras and 1199011 1199012 be two distinct elements which arenot contained in 119871 and 119860 = 119871 + F1199011 + F1199012 Then 3-algebra(119860 [ ]) is called a two-dimensional extension of Lie algebras119871119896 119896 = 1 2 where [ ] 119860 and 119860 and 119860 997888rarr 119860 defined by[119909 119910 1199011] = [119909 119910]1 [119909 119910 1199012] = [119909 119910]2 [119909 119910 119911] = 0[1199011 1199012 119909] = 1205821199091199011 + 1205831199091199012 forall119909 119910 119911 isin 119871 120582119909 120583119909 isin F
(39)
If 119860 is a 3-Lie algebra then 119860 is called a two-dimensionalextension 3-Lie algebra of Lie algebras 119871119896 119896 = 1 2
8 International Journal of Mathematics and Mathematical Sciences
Let 119860 = 119871 + 119882 be a two-dimensional extension of Liealgebras 119871119896 119896 = 1 2 where 119882 = F1199011 + F1199012 Define linearmappings 1198631 1198632 119871 997888rarr 119864119899119889(119871) and 119863 119871 997888rarr 119882 by1198631 (119909) = 119886119889 (1199011 119909) 1198632 (119909) = 119886119889 (1199012 119909) 119863 (119909) = 119886119889 (1199011 1199012) (119909) forall119909 isin 119871
(40)
that is for all 119910 isin 119871 1198631(119909)(119910) = [1199011 119909 119910] = [119909 119910]11198632(119909)(119910) = [1199012 119909 119910] = [119909 119910]2 119863(119909) = [1199011 1199012 119909] We havethe following result
Theorem 17 Let 3-algebra 119860 be a two-dimensional extensionof Lie algebras 1198711 and 1198712 Then 119860 is a 3-Lie algebra if and onlyif linear mappings 1198631 1198632 and 119863 satisfy that 1198631 1198711 997888rarr119863119890119903(1198711) 1198632 1198712 997888rarr 119863119890119903(1198712) are Lie homomorphisms and1198631 (1199093) ([1199091 1199092]2) = [1198631 (1199093) (1199091) 1199092]2+ [1199091 1198631 (1199093) (1199092)]2minus 1205821199093 [1199091 1199092]1 minus 1205831199093 [1199091 1199092]2
(41)
1198632 (1199093) ([1199091 1199092]1) = [1198632 (1199093) (1199091) 1199092]1+ [1199091 1198632 (1199093) (1199092)]1+ 1205821199093 [1199091 1199092]1 + 1205831199093 [1199091 1199092]2 (42)
119863 ([1199091 1199092]1) = (12058311990911205821199092 minus 12058211990911205831199092) 1199011119863 ([1199091 1199092]2) = (12058311990911205821199092 minus 12058211990911205831199092) 1199012 (43)
120582[11990911199092]1 = 120583[11990911199092]2 = 12058311990911205821199092 minus 12058211990911205831199092 120583[11990911199092]1 = 120582[11990911199092]2 = 0 (44)
119863119896 (1199091) (1199092) = minus119863119896 (1199092) (1199091) 119891119900119903 119886119897119897 1199091 1199092 isin 119871 119896 = 1 2 (45)
where 1199091 1199092 1199093 isin 119871 119863(119909119894) = 1205821199091198941199011 + 1205831199091198941199012 119894 = 1 2 3Proof If 119860 is a two-dimensional extension 3-Lie algebrathen by Definition 16 linear mappings 119863119896 satisfy that119863119896(119871119896) sube 119863119890119903(119871119896) and119863119896 are Lie homomorphisms 119896 = 1 2Thanks to (39)1198631 (1199093) ([1199091 1199092]2) = [1199012 [1199011 1199093 1199091] 1199092]+ [1199012 1199091 [1199011 1199093 1199092]]+ [[1199011 1199093 1199012] 1199091 1199092]= [1198631 (1199093) (1199091) 1199092]2+ [1199091 1198631 (1199093) (1199092)]2minus 1205821199093 [1199091 1199092]1 minus 1205831199093 [1199091 1199092]2
(46)
for all 1199091 1199092 1199093 isin 119871 (41) holds Similarly we have (42)Thanks to (39) and (40)119863([1199091 1199092]1) = 119886119889 (1199011 1199012) [1199091 1199092]1= 120582[11990911199092]11199011 + 120583[11990911199092]11199012= (12058311990911205821199092 minus 12058211990911205831199092) 1199011119863 ([1199091 1199092]2) = 119886119889 (1199011 1199012) [1199091 1199092]2= 120582[11990911199092]21199011 + 120583[11990911199092]21199012= (12058311990911205821199092 minus 12058211990911205831199092) 1199012
(47)
Equations (43) and (44) hold Equation (45) follows from (39)and (40) directly
Conversely by (39) forall1199091 1199092 1199093 119909 isin 119871[1199091 1199092 1199093] = 0[1199011 1199091 1199092] = 1198631 (1199091) (1199092) = [1199091 1199092]1 [1199012 1199091 1199092] = 1198632 (1199091) (1199092) = [1199091 1199092]2 [1199011 1199012 119909] = 119863 (119909) = 120582 (119909) 1199011 + 1205831199091199012(48)
Since119863119896(119871) sube 119863119890119903(119871119896) and119863119896 are Lie homomorphisms 119861 =119871 + F1199011 and 119862 = 119871 + F1199012 are 3-Lie algebras which are one-dimensional extension 3-Lie algebras of Lie algebras 119871119896 119896 =1 2 respectively
Next we only need to prove that the multiplication on119860 defined by (39) satisfies (1) For all 119909119894 isin 119871 1 le 119894 le5 that products [[1199091 1199092 1199093] 1199094 1199095] [[119901119895 1199092 1199093] 1199094 1199095][[1199091 1199092 1199093] 1199094 119901119895] and [[1199091 1199092 119901119895] 1199094 119901119895] satisfy (1) 119895 =1 2 follow from that 119861 and 119862 are one-dimensional extension3-Lie algebras of 119871119896 and (39) directlyFrom (41) and (42) it follows that products[[119901119894 1199091 1199092] 119901119895 1199093] 1 le 119894 = 119895 le 2 satisfy (1) It
follows from (43)ndash(45) that products [1199011 1199012 [119901119894 1199091 1199092]][1199091 1199092 [119901119894 1199012 1199093]] and [119901119894 1199091 [1199011 1199012 1199092] 119894 = 1 2 satisfy(1) We omit the computation process
Theorem 18 Let (119860 [ ]) be a 3-Lie algebra Then 119860 is a two-dimensional extension 3-Lie algebra of Lie algebras if and onlyif there is an involutive derivation119879 on119860 such that dim1198601 = 2or dim119860minus1 = 2Proof If 119860 is a two-dimensional extension 3-Lie algebra ofLie algebras Then byTheorem 15 there are Lie algebras 1198711 =(119871 [ ]1) and 1198712 = (119871 [ ]2) such that 119860 = 119871 + 119882 and themultiplication of 119860 is defined by (39) where119882 = F1199011 + F1199012
Define the endomorphism 119879 of 119860 by 119879(119909) = 119909 119879(1199011) =minus1199011 119879(1199012) = minus1199012 or 119879(119909) = minus119909 119879(1199011) = 1199011 119879(1199012) =1199012 forall119909 isin 119871Then1198792 = 119868119889 and1198601 = 119871119860minus1 = 119882 or119860minus1 = 1198711198601 = 119882Thanks to (38) and (41)-(45) 119879 is a derivation of 119860Conversely if there is an involutive derivation 119879 on the3-Lie algebra 119860 such that dim119860minus1 = 2 (or dim1198601 = 2)
By Theorem 4 [1198601 1198601 1198601] = 0 [1198601 1198601 119860minus1] sube 1198601[1198601 119860minus1 119860minus1] sube 119860minus1 Let 119871 = 1198601 and 119860minus1 = F1199011 + F1199012Then [119871 119871 1199011] sube 119871 [119871 119871 1199012] sube 119871 and (119871 [ ]1) and
International Journal of Mathematics and Mathematical Sciences 9
(119871 [ ]2) are Lie algebras where [119909 119910]1 = [119909 119910 1199011] [119909 119910]2 =[119909 119910 1199012] forall119909 119910 isin 119871Thanks toTheorem 17 the 3-Lie algebra119860 is a two-dimensional extension 3-Lie algebra of Lie algebras1198711 and 1198712Data Availability
No data were used to support this study
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
The first named author was supported in part by the NaturalScience Foundation (11371245) and theNatural Science Foun-dation of Hebei Province (A2018201126)
References
[1] V Filippov ldquon-Lie algebrasrdquo SiberianMathematical Journal vol26 no 6 pp 126ndash140 1985
[2] R P Bai and P P Jia ldquoThe real compact n-Lie algebras andinvariant bilinear formsrdquo Acta Mathematica Scientia Series AShuxueWuli Xuebao Chinese Edition vol 27A no 6 pp 1074ndash1081 2007
[3] S Kasymov ldquoOn a theory of n-Lie algebrasrdquo Algebra and Logicvol 26 no 3 pp 277ndash297 1987
[4] C Bai L Guo and Y Sheng ldquoBialgebras the classical Yang-Baxter equation andmain triples for 3-Lie algebrasrdquoMathemat-ical Physics 2016
[5] R Bai and G Song ldquoThe classification of six-dimensional 4-Liealgebrasrdquo Journal of Physics A Mathematical and General vol42 no 3 035207 17 pages 2009
[6] R Bai G Song and Y Zhang ldquoOn classification of n-Liealgebrasrdquo Frontiers of Mathematics in China vol 6 no 4 pp581ndash606 2011
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4 International Journal of Mathematics and Mathematical Sciences
(119886) 119860 119894119904 119886119899 119886119887119890119897119894119886119899(119887) dim1198601 = 1
(1198871) [1198902 sdot sdot sdot 1198902119904+2] = 1198901(1198872) [1198901 sdot sdot sdot 1198902119904+1] = 1198901(119888) dim1198601 = 2(1198881) [1198902 sdot sdot sdot 1198902119904+2] = 1198901[1198903 sdot sdot sdot 1198902119904+3] = 1198902
(1198882) [1198902 sdot sdot sdot 1198902119904+2] = 1198901[1198902 1198904 sdot sdot sdot 1198902119904+3] = 1198902[1198901 1198904 sdot sdot sdot 1198902119904+3] = 1198901
(1198883)
[1198902 sdot sdot sdot 1198902119904+2] = 1205721198901 + 1198902[1198901 1198903 sdot sdot sdot 1198902119904+2] = 1198902[1198902 1198904 sdot sdot sdot 1198902119904+3] = 1198902[1198901 1198904 sdot sdot sdot 1198902119904+3] = 1198901(1198884)
[1198902 sdot sdot sdot 1198902119904+2] = 1198901[1198902 1198903 sdot sdot sdot 1198902119904+2] = 1198902[1198902 1198904 sdot sdot sdot 1198902119904+3] = 1198902[1198901 1198904 sdot sdot sdot 1198902119904+3] = 1198901(1198885) [1198902 sdot sdot sdot 1198902119904+2] = 1198901[1198901 1198903 sdot sdot sdot 1198902119904+2] = 1198902
(1198886) [1198902 sdot sdot sdot 1198902119904+2] = 1205721198901 + 1198902[1198901 1198903 sdot sdot sdot 1198902119904+2] = 1198902 120572 isin F 120572 = 0(1198887)[1198901 1198903 sdot sdot sdot 1198902119904+2] = 1198901[1198902 1198903 sdot sdot sdot 1198902119904+2] = 1198902(119889) dim1198601 = 3(1198891)
[1198902 sdot sdot sdot 1198902119904+2] = 1198901[1198902 1198904 sdot sdot sdot 1198902119904+3] = 1198902[1198903 sdot sdot sdot 1198902119904+3] = 1198903(1198892)
[1198902 sdot sdot sdot 1198902119904+2] = 1198901[1198903 sdot sdot sdot 1198902119904+3] = 1198903 + 1205721198902[1198902 1198904 sdot sdot sdot 1198902119904+3] = 1198903[1198901 1198904 sdot sdot sdot 1198902119904+3] = 1198901
(1198893)
[1198902 sdot sdot sdot 1198902119904+2] = 1198901[1198903 1198904 sdot sdot sdot 1198902119904+3] = 1198903[1198902 1198904 sdot sdot sdot 1198902119904+3] = 1198902[1198901 1198904 sdot sdot sdot 1198902119904+3] = 21198901(1198894)
[1198902 sdot sdot sdot 1198902119904+2] = 1198901[1198901 1198903 sdot sdot sdot 1198902119904+2] = 1198902[1198901 1198902 1198904 sdot sdot sdot 1198902119904+2] = 1198903(1198895)
[1198901 1198904 sdot sdot sdot 1198902119904+3] = 1198901[1198902 1198904 sdot sdot sdot 1198902119904+3] = 1198903[1198903 1198904 sdot sdot sdot 1198902119904+3] = 1205731198902 + (1 + 120573) 1198903(1198896)
[1198901 1198904 sdot sdot sdot 1198902119904+3] = 1198901[1198902 1198904 sdot sdot sdot 1198902119904+3] = 1198902[1198903 1198904 sdot sdot sdot 1198902119904+3] = 1198903(1198897)
[1198901 1198904 sdot sdot sdot 1198902119904+3] = 1198902[1198902 1198904 sdot sdot sdot 1198902119904+3] = 1198903[1198903 1198904 sdot sdot sdot 1198902119904+3] = 1199041198901 + 1199051198902 + 1199061198903120573 119904 119905 119906 isin F 120573 = 0 1 119904 = 0(17)
And n-Lie algebras corresponding to the case (1198897) with coef-ficients 119904 119905 119906 and 1199041015840 1199051015840 1199061015840 are isomorphic if and only if thereexists a nonzero element 120582 isin F such that 119904 = 12058231199041015840 119905 = 12058221199051015840119906 = 1205821199061015840 119904 1199041015840 119905 1199051015840 119906 1199061015840 isin F (119903) dim1198601 = 1199034 le 119903 lt 2119904 + 3 119891119900119903 2 le 119895 le 119903 1 le 119894 le 119903
(1199031) [1198902 sdot sdot sdot 1198902119904+2] = 1198901[1198902 sdot sdot sdot 119890119895 sdot sdot sdot 1198902119904+3] = 119890119895
(1199032) [1198901 sdot sdot sdot 119890119894 sdot sdot sdot 1198902119904+2] = 119890119894(18)
Theorem 10 If119860 is a (2119904+3)-dimensional (2119904+1)-Lie algebraover F with dim1198601 = 119903 lt 119904 + 3 then there are involutivederivations on 119860Proof Define linear mappings 119863119895 119860 997888rarr 119860 1 le 119895 le 6 by
1198631 (119890119894) = 119890119894 1 le 119894 le 119904 + 2 or 119894 = 2119904 + 3minus119890119894 otherwise1198632 (119890119894) = 119890119894 1 le 119894 le 119904 + 2minus119890119894 otherwise
International Journal of Mathematics and Mathematical Sciences 5
1198633 (119890119894) = minus119890119894 119904 + 2 le 119894 le 2119904 + 1119890119894 otherwise
1198634 (119890119894) = 119890119894 1 le 119894 le 119904 + 1 or 119894 = 2119904 + 2minus119890119894 otherwise
1198635 (119890119894) = 119890119894 1 le 119894 le 119904 + 1 or 119894 = 2119904 + 3minus119890119894 otherwise
1198636 (119890119894) = 119890119894 1 le 119894 le 119904 + 3minus119890119894 otherwise
(19)
Since dim1198601 = 119903 le 119904 + 2 it is easy to verify that 1198631 isan involutive derivation on the 3-Lie algebras of the cases of(1198871) (119888119894) (119889119895) and (119903119896) where 1 le 119894 le 7 1 le 119895 le 4 and1 le 119896 le 2 1198632 is an involutive derivation on the 3-Lie algebrasof the cases of (1198871) (1198894) and (119888119894) where 5 le 119894 le 7 1198633 1198634and 1198635 are involutive derivations on the 3-Lie algebras of thecase of (1198872) And 1198636 is an involutive derivation on the 3-Lie algebras of the cases of (1198895) (1198896) and (1198897) Also 119863119894 areinvolutive derivations on abelian algebras for 1 le 119894 le 6
Next we discuss the case of dim1198601 = 119903 ge 119904 + 3 Let 119863 bean endomorphism of 119860
119863119890119894 = 2119904+3sum119895=1
119887119894119895119890119895 119887119894119895 isin F 1 le 119894 le 2119904 + 3 (20)
and 119861 = (119887119894119895) be the (2119904 + 3) times (2119904 + 3)-matrix Then
119863(1198901 sdot sdot sdot 1198902119904+3)119879 = 119861 (1198901 sdot sdot sdot 1198902119904+3)119879= (1198611 11986101198612 1198613) (1198901 sdot sdot sdot 1198902119904+3)119879 (21)
where ( 1198611 11986101198612 1198613 ) is the block matrix of 119861 First we discuss(2119904 + 3)-dimensional (2119904 + 1)-Lie algebras of the case (1199031) inLemma 9
Lemma 11 If 119860 is a (2119904 + 3)-dimensional (2119904 + 1)-Lie algebraof the case (1199031) with dim1198601 = 119903 ge 119904 + 3 119904 ge 1Then the linearmapping 119863 is an involutive derivation on 119860 if and only if theblock matrix 119861 = ( 1198611 11986101198612 1198613 ) satisfies that 1198610 = 119874119903times(2119904+3minus119903) (whichis the zero (119903 times (2119904 + 3 minus 119903))-matrix) and11986121 = 11986411990311986123 = 1198642119904+3minus11990311986121198611 + 11986131198612 = 0
(22)
2119904+2sum119895=2
119887119895119895 = 119887112119904+3sum119895=2119895 =119894
119887119895119895 = 119887119894119894 2 le 119894 le 1199031198872119904+3119894 = (minus1)119894+1 1198871198941 2 le 119894 le 119903119887119895119894 = (minus1)119895minus119894minus1 119887119894119895 2 le 119894 119895 le 119903 119894 = 119895
(23)
Proof By (2) and a direct computation 119863 is a derivation of119860 if and only if matrix 119861 has the property
2119904+2sum119895=2
119887119895119895 = 119887112119904+3sum119895=2119895 =119894
119887119895119895 = 119887119894119894 2 le 119894 le 1199031198871119897 = 0 2 le 119897 le 2119904 + 31198872119904+3119894 = (minus1)119894+1 1198871198941 119887119894119897 = 0 2 le 119894 le 119903 119897 ge 119903 + 1119887119895119894 = (minus1)119895minus119894minus1 119887119894119895 2 le 119894 119895 le 119903 119894 = 119895
(24)
Therefore matrix 119861 satisfies (23) and 1198610 = 119874119903times(2119904+3minus119903) And1198632 = 119868119889 if and only if
1198612 = ( 11986121 11986111198610 + 1198610119861311986121198611 + 11986131198612 11986121198610 + 11986123 )= (119864119903 119874119874 1198642119904+3minus119903) (25)
Thanks to 1198610 = 119874119903times(2119904+3minus119903) (22) holdsTheorem 12 Let 119860 be a (2119904 + 3)-dimensional (2119904 + 1)-Liealgebra of the case (1199031) with dim1198601 = 119903 ge 119904 + 3 119904 ge 1 If 119903is odd then there are involutive derivations on 119860Proof Let 119903 = 2119905 + 1 ge 119904 + 3 Then 119905 ge 2 and 119903 ge 5 Suppose119863 is an endomorphism of119860 and the matrix of119863with respectto the basis 1198901 sdot sdot sdot 1198902119904+3 is 119861 = (119887119894119895) = ( 1198611 11986101198612 1198613 ) which satisfies(22) and (23) and 1198610 = 119874119903times(2119904+3minus119903) Then
1198611
= (((((((
11988711 0 0 sdot sdot sdot 0 011988721 11988722 11988723 sdot sdot sdot 1198872119903minus1 119887211990311988731 11988723 11988733 sdot sdot sdot 1198873119903minus1 1198873119903 d119887119903minus11 (minus1)119903 1198872119903minus1 (minus1)119903minus1 1198873119903minus1 sdot sdot sdot 119887119903minus1119903minus1 119887119903minus11199031198871199031 (minus1)119903+1 1198872119903 (minus1)119903 1198873119903 sdot sdot sdot 119887119903minus1119903 119887119903119903
)))))))
6 International Journal of Mathematics and Mathematical Sciences
1198612 = ((
119887119903+11 119887119903+12 119887119903+13 sdot sdot sdot 119887119903+1119903 d1198872119904+21 1198872119904+22 1198872119904+23 sdot sdot sdot 1198872119904+21199031198872119904+31 minus11988721 11988731 sdot sdot sdot (minus1)119903+1 1198871199031
))
(26)
Since sum2119904+2119895=2 119887119895119895 = 11988711 sum2119904+3119895=2119895 =119894 119887119895119895 = 119887119894119894 2 le 119894 le 119903 we haveminus11988711 + 211988722 minus 1198872119904+32119904+3 = 0 (119903 minus 3)11988722 + sum119903+12119904+3 119887119894119894 = 0 11988722 =119887119894119894 3 le 119894 le 119903 Therefore
11988711 = minus1119903 minus 3 ((119903 minus 1) 1198961 + 22119904+3minus119903sum119895=2
119896119895) 119887119894119894 = minus1119903 minus 3 2119904+3minus119903sum
119895=1
119896119895 2 le 119894 le 119903119887119895119895 = 1198962119904+3minus119895+1 119903 + 1 le 119895 le 2119904 + 3 1198962119904+3minus119895+1 isin F
(27)
Suppose
11986121 =((((((((((((
11988811 11988812 sdot sdot sdot 1198881119894 sdot sdot sdot 1198881119895 sdot sdot sdot 1198881119903 d d
d1198881198941 1198881198942 sdot sdot sdot 119888119894119894 sdot sdot sdot 119888119894119895 sdot sdot sdot 119888119894119903 d
d d
1198881198951 1198881198952 sdot sdot sdot 119888119895119894 sdot sdot sdot 119888119895119895 sdot sdot sdot 119888119895119903 d d
d1198881199031 1198881199032 sdot sdot sdot 119888119903119894 sdot sdot sdot 119888119903119895 sdot sdot sdot 119888119903119903
))))))))))))
(28)
By (23)
11988811 = 1198872111198881119897 = 0 2 le 119897 le 119903119888119894119894 = 119894sum119897=1
(minus1)119894+119897minus1 1198872119897119894 + 119903sum119897=119894+1
(minus1)119894+119897minus1 1198872119894119897 2 le 119894 le 119903119888119894119895 = 119894sum119897=1
(minus1)119897+119894+1 119887119897119894119887119897119895 + 119895sum119897=119894+1
119887119894119897119887119897119895+ 119903sum119897=119895+1
(minus1)119897+119895+1 119887119894119897119887119895119897 2 le 119894 lt 119895 le 119903119888119894119895 = 119895sum119897=1
(minus1)119897+119894+1 119887119897119895119887119897119894 + 119894sum119897=119895+1
(minus1)119894+119895 119887119895119897119887119897119894+ 119903sum119897=119894+1
(minus1)119897+119895+1 119887119895119897119887119894119897 1 le 119895 lt 119894 le 119903
(29)
Therefore the endomorphisms 119863 of 119860 which are defined by1198631198901 = 1198901or 1198631198901 = minus1198901
1198631198902 = 119903sum119896=3
119890119896119863119890119894 = (minus1)119894minus1 1198902 + 119894minus1sum
119896=3
(minus1)119894 119890119896 + (minus1)119894minus1 119903sum119896=119894+1
1198901198963 le 119894 le 119903119863119890119895 = (minus1)119895 119890119895 119903 + 1 le 119895 le 2119904 + 31198631198901 = 1198901
or 1198631198901 = minus11989011198631198902 = 119903sum
119896=3
119890119896119863119890119894 = (minus1)119894minus1 1198902 + 119894minus1sum
119896=3
(minus1)119894 119890119896 + (minus1)119894minus1 119903sum119896=119894+1
1198901198963 le 119894 le 119903119863119890119895 = (minus1)119895minus1 119890119895 119903 + 1 le 119895 le 2119904 + 3
(30)
are involutive derivations on 119860Theorem 13 Let 119860 be a (2119904 + 3)-dimensional (2119904 + 1)-Liealgebra of the case (1199031) with dim1198601 = 119903 = 2119904 + 2 (119904 ge 1)then there does not exist an involutive derivation on 119860Proof If 119863 is an involutive derivation on 119860 then byLemma 11 and (23)
11988711 = minus (2119904 + 1) 11989612119904 minus 1 1198872119904+32119904+3 = 1198961119887119894119894 = minus11989612119904 minus 1
2 le 119894 le 119903 1198961 isin F (31)
Thanks to (22) 11988722119904+32119904+3 = 119887211 = 11989621 = 1 Therefore(minus(2119904 + 1)1198961(2119904 minus 1))2 = ((2119904 + 1)(2119904 minus 1))2 = 1 whichis a contradiction
Now we discuss case (1199032)Theorem 14 Let 119860 be a (2119904 + 3)-dimensional (2119904 + 1)-Liealgebra of the case (1199032) with dim1198601 = 119903 ge 119904 + 3 Then thereexist involutive derivations on 119860 if and only if 119903 is evenProof By Lemma 7 119860 = 1198601 + F1198902119904+3 where 1198902s+3 isin 119885(119860)and 1198601 is a (2119904 + 2)-dimensional (2119904 + 1)-Lie subalgebra
International Journal of Mathematics and Mathematical Sciences 7
of 119860 with dim1198601 = dim11986011 = 119903 Then there existinvolutive derivations on119860 if and only if there exist involutivederivations on 1198601
By Theorem 3 in [1] there is a basis 1198901 sdot sdot sdot 1198902119904+2 of 1198601such that[1198901 sdot sdot sdot 119890119895 sdot sdot sdot 1198902119904+2] = 0 119905 lt 119895 le 2119904 + 3 minus 119905[1198901 sdot sdot sdot 119890119894 sdot sdot sdot 1198902119904+2] = (minus1)119894 1198902119904+3minus1198941 le 119894 le 119905 or 2119904 + 3 minus 119905 lt 119894 le 2119904 + 2 (32)
If 119903 is even then 119903 = 2119905 ge 4 By Theorem 8 and (32)endomorphism 1198631 of 1198601 defined by
1198631 (119890119894) = 119890119894 119894 = 1 sdot sdot sdot 119904 + 1minus119890119894 119894 = 119904 + 2 sdot sdot sdot 2119904 + 2 (33)
is an involutive derivation on 1198601 Therefore the endomor-phism 119863 of 119860 defined by119863 (119890119894) = 119890119894 1 le 119894 le 119904 + 1119863 (119890119895) = minus119890119895 119904 + 2 le 119895 le 2119904 + 2119863 (1198902119904+3) = plusmn1198902119904+3
(34)
is involutive derivation on 119860If dim1198601 = 119903 is odd and endomorphism 119863 of 119860 is an
involutive derivation on 119860 then 119903 = 2119905 + 1 gt 4 Suppose119863(119890119894) = sum2119904+3119895=1 119886119894119895119890119895 1 le 119894 le 2119904 + 3 Then[119863 (1198902119904+3) 1198901198941 sdot sdot sdot 1198901198942119904]= 119863 [1198902119904+3 1198901198941 sdot sdot sdot 1198901198942119904]minus 2119904sum119895=1
[1198902119904+3 sdot sdot sdot 119863119890119894119895 sdot sdot sdot 1198901198942119904] = 0 (35)
We get 119863(1198902119904+3) isin 119885(119860) = F1198902119904+3 Since 1198632 = 119868119889 119863(1198902119904+3) =plusmn1198902119904+3 By (32) 1198601 = F1198901 + sdot sdot sdot + F119890119905 + F1198902119904+1 + sdot sdot sdot + F1198902119904+3minus119905and (minus1)119894119863 (1198902119904+3minus119894)
= 2119904+2sum119896=1119896=119894
[1198901 sdot sdot sdot 119890119894 sdot sdot sdot 119863 (119890119896) sdot sdot sdot 1198902119904+2] (36)
where 1 le 119894 le 119905 and 2119904 + 3 minus 119905 lt 119894 le 2119904 + 2 Then 1198861198942119904+3 = 0for 1 le 119894 le 119905 or 2119904 + 3 minus 119905 lt 119894 le 2119904 + 2 and 1198631198601 sube 1198601 Thenthe endomorphism 1198632 of 1198601 defined by1198632 (119890119894) = 119863 (119890119894) 1 le 119894 le 119905 or 2119904 + 3 minus 119905 lt 119894 le 2119904 + 2
1198632 (119890119895) = 119863(119890119895) minus 1198861198952119904+31198902119904+3 = 2119904+2sum119895=1
119886119894119895119890119895119905 lt 119895 le 2119904 + 3 minus 119905
(37)
is an involutive derivation on the (2119904+2)-dimensional (2119904+1)-Lie algebra 1198601 contradiction (Theorem 8) Therefore theredoes not exist involutive derivation on 119860
5 Structure of 3-Lie Algebras withInvolutive Derivations
Let (119871 [ ]) be a Lie algebra over F and 119901 be an element whichis not contained in 119871 Then 119860 = 119871 + F119901 is a 3-Lie algebra inthe multiplication[119909 119910 119911] = 0[119901 119909 119910] = [119909 119910]
for all 119909 119910 119911 isin 119871 (38)
And the 3-Lie algebra (119860 [ ]) is called one-dimensionalextension of 119871Theorem 15 Let 119860 be a 3-Lie algebra then 119860 is one-dimensional extension of a Lie algebra if and only if there existsan involutive derivation 119863 on 119860 such that dim1198601 = 1 ordim119860minus1 = 1Proof If 119860 is an one-dimensional extension of a Lie algebra119871 then 119860 = 119871 + F119901 Define the endomorphism 119863 of 119860 by119863(119901) = minus119901 (or 119901) and 119863(119909) = 119909 (or minus119909) forall119909 isin 119871 Thanks to(38)1198632 = 119868119889 and 119863([119909 119910 119911]) = 0 = [119863119909 119910 119911] + [119909119863119910 119911] +[119909 119910119863119911]119863([119901 119909 119910]) = [119901 119909 119910] = [119863119901 119909 119910] + [119901119863119909 119910] +[119901 119909 119863119910] for all 119909 119910 119911 isin 119871 Therefore 119863 is an involutivederivation on 119860 and dim119860minus1 = 1 (or dim1198601 = 1)
Conversely let 119863 be an involutive derivation on a 3-Liealgebra 119860 and dim119860minus1 = 1 (or dim1198601 = 1) Let 119860minus1 = F119901and 1198601 = 119871 (or 119860minus1 = 1198711198601 = F119901) where 119901 isin 119860minus119871 Thanksto Theorem 3 119871 is a Lie algebra with the multiplication[119909 119910] = [119901 119909 119910] for all 119909 119910 isin 119871 and 119860 is one-dimensionalextension of 119871
Let (119871 [ ]1) and (119871 [ ]2) be Lie algebras and 1199091 sdot sdot sdot 119909119898be a basis of 119871 For convenience denote Lie algebras (119871 [ ]119896)by 119871119896 119896 = 1 2 respectively Suppose 1199011 and 1199012 are twodistinct elements which are not contained in 119871 and 3-Lie algebras (119861 [ ]1) and (119862 [ ]2) are one-dimensionalextensions of Lie algebras 1198711 and 1198712 respectively where119861 = 119871 + F1199011 119862 = 119871 + F1199012 Then 119863119890119903(1198711) and 119863119890119903(1198712) aresubalgebras of 119892119897(119871)Definition 16 Let 1198711 = (119871 [ ]1) and 1198712 = (119871 [ ]2) be twoLie algebras and 1199011 1199012 be two distinct elements which arenot contained in 119871 and 119860 = 119871 + F1199011 + F1199012 Then 3-algebra(119860 [ ]) is called a two-dimensional extension of Lie algebras119871119896 119896 = 1 2 where [ ] 119860 and 119860 and 119860 997888rarr 119860 defined by[119909 119910 1199011] = [119909 119910]1 [119909 119910 1199012] = [119909 119910]2 [119909 119910 119911] = 0[1199011 1199012 119909] = 1205821199091199011 + 1205831199091199012 forall119909 119910 119911 isin 119871 120582119909 120583119909 isin F
(39)
If 119860 is a 3-Lie algebra then 119860 is called a two-dimensionalextension 3-Lie algebra of Lie algebras 119871119896 119896 = 1 2
8 International Journal of Mathematics and Mathematical Sciences
Let 119860 = 119871 + 119882 be a two-dimensional extension of Liealgebras 119871119896 119896 = 1 2 where 119882 = F1199011 + F1199012 Define linearmappings 1198631 1198632 119871 997888rarr 119864119899119889(119871) and 119863 119871 997888rarr 119882 by1198631 (119909) = 119886119889 (1199011 119909) 1198632 (119909) = 119886119889 (1199012 119909) 119863 (119909) = 119886119889 (1199011 1199012) (119909) forall119909 isin 119871
(40)
that is for all 119910 isin 119871 1198631(119909)(119910) = [1199011 119909 119910] = [119909 119910]11198632(119909)(119910) = [1199012 119909 119910] = [119909 119910]2 119863(119909) = [1199011 1199012 119909] We havethe following result
Theorem 17 Let 3-algebra 119860 be a two-dimensional extensionof Lie algebras 1198711 and 1198712 Then 119860 is a 3-Lie algebra if and onlyif linear mappings 1198631 1198632 and 119863 satisfy that 1198631 1198711 997888rarr119863119890119903(1198711) 1198632 1198712 997888rarr 119863119890119903(1198712) are Lie homomorphisms and1198631 (1199093) ([1199091 1199092]2) = [1198631 (1199093) (1199091) 1199092]2+ [1199091 1198631 (1199093) (1199092)]2minus 1205821199093 [1199091 1199092]1 minus 1205831199093 [1199091 1199092]2
(41)
1198632 (1199093) ([1199091 1199092]1) = [1198632 (1199093) (1199091) 1199092]1+ [1199091 1198632 (1199093) (1199092)]1+ 1205821199093 [1199091 1199092]1 + 1205831199093 [1199091 1199092]2 (42)
119863 ([1199091 1199092]1) = (12058311990911205821199092 minus 12058211990911205831199092) 1199011119863 ([1199091 1199092]2) = (12058311990911205821199092 minus 12058211990911205831199092) 1199012 (43)
120582[11990911199092]1 = 120583[11990911199092]2 = 12058311990911205821199092 minus 12058211990911205831199092 120583[11990911199092]1 = 120582[11990911199092]2 = 0 (44)
119863119896 (1199091) (1199092) = minus119863119896 (1199092) (1199091) 119891119900119903 119886119897119897 1199091 1199092 isin 119871 119896 = 1 2 (45)
where 1199091 1199092 1199093 isin 119871 119863(119909119894) = 1205821199091198941199011 + 1205831199091198941199012 119894 = 1 2 3Proof If 119860 is a two-dimensional extension 3-Lie algebrathen by Definition 16 linear mappings 119863119896 satisfy that119863119896(119871119896) sube 119863119890119903(119871119896) and119863119896 are Lie homomorphisms 119896 = 1 2Thanks to (39)1198631 (1199093) ([1199091 1199092]2) = [1199012 [1199011 1199093 1199091] 1199092]+ [1199012 1199091 [1199011 1199093 1199092]]+ [[1199011 1199093 1199012] 1199091 1199092]= [1198631 (1199093) (1199091) 1199092]2+ [1199091 1198631 (1199093) (1199092)]2minus 1205821199093 [1199091 1199092]1 minus 1205831199093 [1199091 1199092]2
(46)
for all 1199091 1199092 1199093 isin 119871 (41) holds Similarly we have (42)Thanks to (39) and (40)119863([1199091 1199092]1) = 119886119889 (1199011 1199012) [1199091 1199092]1= 120582[11990911199092]11199011 + 120583[11990911199092]11199012= (12058311990911205821199092 minus 12058211990911205831199092) 1199011119863 ([1199091 1199092]2) = 119886119889 (1199011 1199012) [1199091 1199092]2= 120582[11990911199092]21199011 + 120583[11990911199092]21199012= (12058311990911205821199092 minus 12058211990911205831199092) 1199012
(47)
Equations (43) and (44) hold Equation (45) follows from (39)and (40) directly
Conversely by (39) forall1199091 1199092 1199093 119909 isin 119871[1199091 1199092 1199093] = 0[1199011 1199091 1199092] = 1198631 (1199091) (1199092) = [1199091 1199092]1 [1199012 1199091 1199092] = 1198632 (1199091) (1199092) = [1199091 1199092]2 [1199011 1199012 119909] = 119863 (119909) = 120582 (119909) 1199011 + 1205831199091199012(48)
Since119863119896(119871) sube 119863119890119903(119871119896) and119863119896 are Lie homomorphisms 119861 =119871 + F1199011 and 119862 = 119871 + F1199012 are 3-Lie algebras which are one-dimensional extension 3-Lie algebras of Lie algebras 119871119896 119896 =1 2 respectively
Next we only need to prove that the multiplication on119860 defined by (39) satisfies (1) For all 119909119894 isin 119871 1 le 119894 le5 that products [[1199091 1199092 1199093] 1199094 1199095] [[119901119895 1199092 1199093] 1199094 1199095][[1199091 1199092 1199093] 1199094 119901119895] and [[1199091 1199092 119901119895] 1199094 119901119895] satisfy (1) 119895 =1 2 follow from that 119861 and 119862 are one-dimensional extension3-Lie algebras of 119871119896 and (39) directlyFrom (41) and (42) it follows that products[[119901119894 1199091 1199092] 119901119895 1199093] 1 le 119894 = 119895 le 2 satisfy (1) It
follows from (43)ndash(45) that products [1199011 1199012 [119901119894 1199091 1199092]][1199091 1199092 [119901119894 1199012 1199093]] and [119901119894 1199091 [1199011 1199012 1199092] 119894 = 1 2 satisfy(1) We omit the computation process
Theorem 18 Let (119860 [ ]) be a 3-Lie algebra Then 119860 is a two-dimensional extension 3-Lie algebra of Lie algebras if and onlyif there is an involutive derivation119879 on119860 such that dim1198601 = 2or dim119860minus1 = 2Proof If 119860 is a two-dimensional extension 3-Lie algebra ofLie algebras Then byTheorem 15 there are Lie algebras 1198711 =(119871 [ ]1) and 1198712 = (119871 [ ]2) such that 119860 = 119871 + 119882 and themultiplication of 119860 is defined by (39) where119882 = F1199011 + F1199012
Define the endomorphism 119879 of 119860 by 119879(119909) = 119909 119879(1199011) =minus1199011 119879(1199012) = minus1199012 or 119879(119909) = minus119909 119879(1199011) = 1199011 119879(1199012) =1199012 forall119909 isin 119871Then1198792 = 119868119889 and1198601 = 119871119860minus1 = 119882 or119860minus1 = 1198711198601 = 119882Thanks to (38) and (41)-(45) 119879 is a derivation of 119860Conversely if there is an involutive derivation 119879 on the3-Lie algebra 119860 such that dim119860minus1 = 2 (or dim1198601 = 2)
By Theorem 4 [1198601 1198601 1198601] = 0 [1198601 1198601 119860minus1] sube 1198601[1198601 119860minus1 119860minus1] sube 119860minus1 Let 119871 = 1198601 and 119860minus1 = F1199011 + F1199012Then [119871 119871 1199011] sube 119871 [119871 119871 1199012] sube 119871 and (119871 [ ]1) and
International Journal of Mathematics and Mathematical Sciences 9
(119871 [ ]2) are Lie algebras where [119909 119910]1 = [119909 119910 1199011] [119909 119910]2 =[119909 119910 1199012] forall119909 119910 isin 119871Thanks toTheorem 17 the 3-Lie algebra119860 is a two-dimensional extension 3-Lie algebra of Lie algebras1198711 and 1198712Data Availability
No data were used to support this study
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
The first named author was supported in part by the NaturalScience Foundation (11371245) and theNatural Science Foun-dation of Hebei Province (A2018201126)
References
[1] V Filippov ldquon-Lie algebrasrdquo SiberianMathematical Journal vol26 no 6 pp 126ndash140 1985
[2] R P Bai and P P Jia ldquoThe real compact n-Lie algebras andinvariant bilinear formsrdquo Acta Mathematica Scientia Series AShuxueWuli Xuebao Chinese Edition vol 27A no 6 pp 1074ndash1081 2007
[3] S Kasymov ldquoOn a theory of n-Lie algebrasrdquo Algebra and Logicvol 26 no 3 pp 277ndash297 1987
[4] C Bai L Guo and Y Sheng ldquoBialgebras the classical Yang-Baxter equation andmain triples for 3-Lie algebrasrdquoMathemat-ical Physics 2016
[5] R Bai and G Song ldquoThe classification of six-dimensional 4-Liealgebrasrdquo Journal of Physics A Mathematical and General vol42 no 3 035207 17 pages 2009
[6] R Bai G Song and Y Zhang ldquoOn classification of n-Liealgebrasrdquo Frontiers of Mathematics in China vol 6 no 4 pp581ndash606 2011
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International Journal of Mathematics and Mathematical Sciences 5
1198633 (119890119894) = minus119890119894 119904 + 2 le 119894 le 2119904 + 1119890119894 otherwise
1198634 (119890119894) = 119890119894 1 le 119894 le 119904 + 1 or 119894 = 2119904 + 2minus119890119894 otherwise
1198635 (119890119894) = 119890119894 1 le 119894 le 119904 + 1 or 119894 = 2119904 + 3minus119890119894 otherwise
1198636 (119890119894) = 119890119894 1 le 119894 le 119904 + 3minus119890119894 otherwise
(19)
Since dim1198601 = 119903 le 119904 + 2 it is easy to verify that 1198631 isan involutive derivation on the 3-Lie algebras of the cases of(1198871) (119888119894) (119889119895) and (119903119896) where 1 le 119894 le 7 1 le 119895 le 4 and1 le 119896 le 2 1198632 is an involutive derivation on the 3-Lie algebrasof the cases of (1198871) (1198894) and (119888119894) where 5 le 119894 le 7 1198633 1198634and 1198635 are involutive derivations on the 3-Lie algebras of thecase of (1198872) And 1198636 is an involutive derivation on the 3-Lie algebras of the cases of (1198895) (1198896) and (1198897) Also 119863119894 areinvolutive derivations on abelian algebras for 1 le 119894 le 6
Next we discuss the case of dim1198601 = 119903 ge 119904 + 3 Let 119863 bean endomorphism of 119860
119863119890119894 = 2119904+3sum119895=1
119887119894119895119890119895 119887119894119895 isin F 1 le 119894 le 2119904 + 3 (20)
and 119861 = (119887119894119895) be the (2119904 + 3) times (2119904 + 3)-matrix Then
119863(1198901 sdot sdot sdot 1198902119904+3)119879 = 119861 (1198901 sdot sdot sdot 1198902119904+3)119879= (1198611 11986101198612 1198613) (1198901 sdot sdot sdot 1198902119904+3)119879 (21)
where ( 1198611 11986101198612 1198613 ) is the block matrix of 119861 First we discuss(2119904 + 3)-dimensional (2119904 + 1)-Lie algebras of the case (1199031) inLemma 9
Lemma 11 If 119860 is a (2119904 + 3)-dimensional (2119904 + 1)-Lie algebraof the case (1199031) with dim1198601 = 119903 ge 119904 + 3 119904 ge 1Then the linearmapping 119863 is an involutive derivation on 119860 if and only if theblock matrix 119861 = ( 1198611 11986101198612 1198613 ) satisfies that 1198610 = 119874119903times(2119904+3minus119903) (whichis the zero (119903 times (2119904 + 3 minus 119903))-matrix) and11986121 = 11986411990311986123 = 1198642119904+3minus11990311986121198611 + 11986131198612 = 0
(22)
2119904+2sum119895=2
119887119895119895 = 119887112119904+3sum119895=2119895 =119894
119887119895119895 = 119887119894119894 2 le 119894 le 1199031198872119904+3119894 = (minus1)119894+1 1198871198941 2 le 119894 le 119903119887119895119894 = (minus1)119895minus119894minus1 119887119894119895 2 le 119894 119895 le 119903 119894 = 119895
(23)
Proof By (2) and a direct computation 119863 is a derivation of119860 if and only if matrix 119861 has the property
2119904+2sum119895=2
119887119895119895 = 119887112119904+3sum119895=2119895 =119894
119887119895119895 = 119887119894119894 2 le 119894 le 1199031198871119897 = 0 2 le 119897 le 2119904 + 31198872119904+3119894 = (minus1)119894+1 1198871198941 119887119894119897 = 0 2 le 119894 le 119903 119897 ge 119903 + 1119887119895119894 = (minus1)119895minus119894minus1 119887119894119895 2 le 119894 119895 le 119903 119894 = 119895
(24)
Therefore matrix 119861 satisfies (23) and 1198610 = 119874119903times(2119904+3minus119903) And1198632 = 119868119889 if and only if
1198612 = ( 11986121 11986111198610 + 1198610119861311986121198611 + 11986131198612 11986121198610 + 11986123 )= (119864119903 119874119874 1198642119904+3minus119903) (25)
Thanks to 1198610 = 119874119903times(2119904+3minus119903) (22) holdsTheorem 12 Let 119860 be a (2119904 + 3)-dimensional (2119904 + 1)-Liealgebra of the case (1199031) with dim1198601 = 119903 ge 119904 + 3 119904 ge 1 If 119903is odd then there are involutive derivations on 119860Proof Let 119903 = 2119905 + 1 ge 119904 + 3 Then 119905 ge 2 and 119903 ge 5 Suppose119863 is an endomorphism of119860 and the matrix of119863with respectto the basis 1198901 sdot sdot sdot 1198902119904+3 is 119861 = (119887119894119895) = ( 1198611 11986101198612 1198613 ) which satisfies(22) and (23) and 1198610 = 119874119903times(2119904+3minus119903) Then
1198611
= (((((((
11988711 0 0 sdot sdot sdot 0 011988721 11988722 11988723 sdot sdot sdot 1198872119903minus1 119887211990311988731 11988723 11988733 sdot sdot sdot 1198873119903minus1 1198873119903 d119887119903minus11 (minus1)119903 1198872119903minus1 (minus1)119903minus1 1198873119903minus1 sdot sdot sdot 119887119903minus1119903minus1 119887119903minus11199031198871199031 (minus1)119903+1 1198872119903 (minus1)119903 1198873119903 sdot sdot sdot 119887119903minus1119903 119887119903119903
)))))))
6 International Journal of Mathematics and Mathematical Sciences
1198612 = ((
119887119903+11 119887119903+12 119887119903+13 sdot sdot sdot 119887119903+1119903 d1198872119904+21 1198872119904+22 1198872119904+23 sdot sdot sdot 1198872119904+21199031198872119904+31 minus11988721 11988731 sdot sdot sdot (minus1)119903+1 1198871199031
))
(26)
Since sum2119904+2119895=2 119887119895119895 = 11988711 sum2119904+3119895=2119895 =119894 119887119895119895 = 119887119894119894 2 le 119894 le 119903 we haveminus11988711 + 211988722 minus 1198872119904+32119904+3 = 0 (119903 minus 3)11988722 + sum119903+12119904+3 119887119894119894 = 0 11988722 =119887119894119894 3 le 119894 le 119903 Therefore
11988711 = minus1119903 minus 3 ((119903 minus 1) 1198961 + 22119904+3minus119903sum119895=2
119896119895) 119887119894119894 = minus1119903 minus 3 2119904+3minus119903sum
119895=1
119896119895 2 le 119894 le 119903119887119895119895 = 1198962119904+3minus119895+1 119903 + 1 le 119895 le 2119904 + 3 1198962119904+3minus119895+1 isin F
(27)
Suppose
11986121 =((((((((((((
11988811 11988812 sdot sdot sdot 1198881119894 sdot sdot sdot 1198881119895 sdot sdot sdot 1198881119903 d d
d1198881198941 1198881198942 sdot sdot sdot 119888119894119894 sdot sdot sdot 119888119894119895 sdot sdot sdot 119888119894119903 d
d d
1198881198951 1198881198952 sdot sdot sdot 119888119895119894 sdot sdot sdot 119888119895119895 sdot sdot sdot 119888119895119903 d d
d1198881199031 1198881199032 sdot sdot sdot 119888119903119894 sdot sdot sdot 119888119903119895 sdot sdot sdot 119888119903119903
))))))))))))
(28)
By (23)
11988811 = 1198872111198881119897 = 0 2 le 119897 le 119903119888119894119894 = 119894sum119897=1
(minus1)119894+119897minus1 1198872119897119894 + 119903sum119897=119894+1
(minus1)119894+119897minus1 1198872119894119897 2 le 119894 le 119903119888119894119895 = 119894sum119897=1
(minus1)119897+119894+1 119887119897119894119887119897119895 + 119895sum119897=119894+1
119887119894119897119887119897119895+ 119903sum119897=119895+1
(minus1)119897+119895+1 119887119894119897119887119895119897 2 le 119894 lt 119895 le 119903119888119894119895 = 119895sum119897=1
(minus1)119897+119894+1 119887119897119895119887119897119894 + 119894sum119897=119895+1
(minus1)119894+119895 119887119895119897119887119897119894+ 119903sum119897=119894+1
(minus1)119897+119895+1 119887119895119897119887119894119897 1 le 119895 lt 119894 le 119903
(29)
Therefore the endomorphisms 119863 of 119860 which are defined by1198631198901 = 1198901or 1198631198901 = minus1198901
1198631198902 = 119903sum119896=3
119890119896119863119890119894 = (minus1)119894minus1 1198902 + 119894minus1sum
119896=3
(minus1)119894 119890119896 + (minus1)119894minus1 119903sum119896=119894+1
1198901198963 le 119894 le 119903119863119890119895 = (minus1)119895 119890119895 119903 + 1 le 119895 le 2119904 + 31198631198901 = 1198901
or 1198631198901 = minus11989011198631198902 = 119903sum
119896=3
119890119896119863119890119894 = (minus1)119894minus1 1198902 + 119894minus1sum
119896=3
(minus1)119894 119890119896 + (minus1)119894minus1 119903sum119896=119894+1
1198901198963 le 119894 le 119903119863119890119895 = (minus1)119895minus1 119890119895 119903 + 1 le 119895 le 2119904 + 3
(30)
are involutive derivations on 119860Theorem 13 Let 119860 be a (2119904 + 3)-dimensional (2119904 + 1)-Liealgebra of the case (1199031) with dim1198601 = 119903 = 2119904 + 2 (119904 ge 1)then there does not exist an involutive derivation on 119860Proof If 119863 is an involutive derivation on 119860 then byLemma 11 and (23)
11988711 = minus (2119904 + 1) 11989612119904 minus 1 1198872119904+32119904+3 = 1198961119887119894119894 = minus11989612119904 minus 1
2 le 119894 le 119903 1198961 isin F (31)
Thanks to (22) 11988722119904+32119904+3 = 119887211 = 11989621 = 1 Therefore(minus(2119904 + 1)1198961(2119904 minus 1))2 = ((2119904 + 1)(2119904 minus 1))2 = 1 whichis a contradiction
Now we discuss case (1199032)Theorem 14 Let 119860 be a (2119904 + 3)-dimensional (2119904 + 1)-Liealgebra of the case (1199032) with dim1198601 = 119903 ge 119904 + 3 Then thereexist involutive derivations on 119860 if and only if 119903 is evenProof By Lemma 7 119860 = 1198601 + F1198902119904+3 where 1198902s+3 isin 119885(119860)and 1198601 is a (2119904 + 2)-dimensional (2119904 + 1)-Lie subalgebra
International Journal of Mathematics and Mathematical Sciences 7
of 119860 with dim1198601 = dim11986011 = 119903 Then there existinvolutive derivations on119860 if and only if there exist involutivederivations on 1198601
By Theorem 3 in [1] there is a basis 1198901 sdot sdot sdot 1198902119904+2 of 1198601such that[1198901 sdot sdot sdot 119890119895 sdot sdot sdot 1198902119904+2] = 0 119905 lt 119895 le 2119904 + 3 minus 119905[1198901 sdot sdot sdot 119890119894 sdot sdot sdot 1198902119904+2] = (minus1)119894 1198902119904+3minus1198941 le 119894 le 119905 or 2119904 + 3 minus 119905 lt 119894 le 2119904 + 2 (32)
If 119903 is even then 119903 = 2119905 ge 4 By Theorem 8 and (32)endomorphism 1198631 of 1198601 defined by
1198631 (119890119894) = 119890119894 119894 = 1 sdot sdot sdot 119904 + 1minus119890119894 119894 = 119904 + 2 sdot sdot sdot 2119904 + 2 (33)
is an involutive derivation on 1198601 Therefore the endomor-phism 119863 of 119860 defined by119863 (119890119894) = 119890119894 1 le 119894 le 119904 + 1119863 (119890119895) = minus119890119895 119904 + 2 le 119895 le 2119904 + 2119863 (1198902119904+3) = plusmn1198902119904+3
(34)
is involutive derivation on 119860If dim1198601 = 119903 is odd and endomorphism 119863 of 119860 is an
involutive derivation on 119860 then 119903 = 2119905 + 1 gt 4 Suppose119863(119890119894) = sum2119904+3119895=1 119886119894119895119890119895 1 le 119894 le 2119904 + 3 Then[119863 (1198902119904+3) 1198901198941 sdot sdot sdot 1198901198942119904]= 119863 [1198902119904+3 1198901198941 sdot sdot sdot 1198901198942119904]minus 2119904sum119895=1
[1198902119904+3 sdot sdot sdot 119863119890119894119895 sdot sdot sdot 1198901198942119904] = 0 (35)
We get 119863(1198902119904+3) isin 119885(119860) = F1198902119904+3 Since 1198632 = 119868119889 119863(1198902119904+3) =plusmn1198902119904+3 By (32) 1198601 = F1198901 + sdot sdot sdot + F119890119905 + F1198902119904+1 + sdot sdot sdot + F1198902119904+3minus119905and (minus1)119894119863 (1198902119904+3minus119894)
= 2119904+2sum119896=1119896=119894
[1198901 sdot sdot sdot 119890119894 sdot sdot sdot 119863 (119890119896) sdot sdot sdot 1198902119904+2] (36)
where 1 le 119894 le 119905 and 2119904 + 3 minus 119905 lt 119894 le 2119904 + 2 Then 1198861198942119904+3 = 0for 1 le 119894 le 119905 or 2119904 + 3 minus 119905 lt 119894 le 2119904 + 2 and 1198631198601 sube 1198601 Thenthe endomorphism 1198632 of 1198601 defined by1198632 (119890119894) = 119863 (119890119894) 1 le 119894 le 119905 or 2119904 + 3 minus 119905 lt 119894 le 2119904 + 2
1198632 (119890119895) = 119863(119890119895) minus 1198861198952119904+31198902119904+3 = 2119904+2sum119895=1
119886119894119895119890119895119905 lt 119895 le 2119904 + 3 minus 119905
(37)
is an involutive derivation on the (2119904+2)-dimensional (2119904+1)-Lie algebra 1198601 contradiction (Theorem 8) Therefore theredoes not exist involutive derivation on 119860
5 Structure of 3-Lie Algebras withInvolutive Derivations
Let (119871 [ ]) be a Lie algebra over F and 119901 be an element whichis not contained in 119871 Then 119860 = 119871 + F119901 is a 3-Lie algebra inthe multiplication[119909 119910 119911] = 0[119901 119909 119910] = [119909 119910]
for all 119909 119910 119911 isin 119871 (38)
And the 3-Lie algebra (119860 [ ]) is called one-dimensionalextension of 119871Theorem 15 Let 119860 be a 3-Lie algebra then 119860 is one-dimensional extension of a Lie algebra if and only if there existsan involutive derivation 119863 on 119860 such that dim1198601 = 1 ordim119860minus1 = 1Proof If 119860 is an one-dimensional extension of a Lie algebra119871 then 119860 = 119871 + F119901 Define the endomorphism 119863 of 119860 by119863(119901) = minus119901 (or 119901) and 119863(119909) = 119909 (or minus119909) forall119909 isin 119871 Thanks to(38)1198632 = 119868119889 and 119863([119909 119910 119911]) = 0 = [119863119909 119910 119911] + [119909119863119910 119911] +[119909 119910119863119911]119863([119901 119909 119910]) = [119901 119909 119910] = [119863119901 119909 119910] + [119901119863119909 119910] +[119901 119909 119863119910] for all 119909 119910 119911 isin 119871 Therefore 119863 is an involutivederivation on 119860 and dim119860minus1 = 1 (or dim1198601 = 1)
Conversely let 119863 be an involutive derivation on a 3-Liealgebra 119860 and dim119860minus1 = 1 (or dim1198601 = 1) Let 119860minus1 = F119901and 1198601 = 119871 (or 119860minus1 = 1198711198601 = F119901) where 119901 isin 119860minus119871 Thanksto Theorem 3 119871 is a Lie algebra with the multiplication[119909 119910] = [119901 119909 119910] for all 119909 119910 isin 119871 and 119860 is one-dimensionalextension of 119871
Let (119871 [ ]1) and (119871 [ ]2) be Lie algebras and 1199091 sdot sdot sdot 119909119898be a basis of 119871 For convenience denote Lie algebras (119871 [ ]119896)by 119871119896 119896 = 1 2 respectively Suppose 1199011 and 1199012 are twodistinct elements which are not contained in 119871 and 3-Lie algebras (119861 [ ]1) and (119862 [ ]2) are one-dimensionalextensions of Lie algebras 1198711 and 1198712 respectively where119861 = 119871 + F1199011 119862 = 119871 + F1199012 Then 119863119890119903(1198711) and 119863119890119903(1198712) aresubalgebras of 119892119897(119871)Definition 16 Let 1198711 = (119871 [ ]1) and 1198712 = (119871 [ ]2) be twoLie algebras and 1199011 1199012 be two distinct elements which arenot contained in 119871 and 119860 = 119871 + F1199011 + F1199012 Then 3-algebra(119860 [ ]) is called a two-dimensional extension of Lie algebras119871119896 119896 = 1 2 where [ ] 119860 and 119860 and 119860 997888rarr 119860 defined by[119909 119910 1199011] = [119909 119910]1 [119909 119910 1199012] = [119909 119910]2 [119909 119910 119911] = 0[1199011 1199012 119909] = 1205821199091199011 + 1205831199091199012 forall119909 119910 119911 isin 119871 120582119909 120583119909 isin F
(39)
If 119860 is a 3-Lie algebra then 119860 is called a two-dimensionalextension 3-Lie algebra of Lie algebras 119871119896 119896 = 1 2
8 International Journal of Mathematics and Mathematical Sciences
Let 119860 = 119871 + 119882 be a two-dimensional extension of Liealgebras 119871119896 119896 = 1 2 where 119882 = F1199011 + F1199012 Define linearmappings 1198631 1198632 119871 997888rarr 119864119899119889(119871) and 119863 119871 997888rarr 119882 by1198631 (119909) = 119886119889 (1199011 119909) 1198632 (119909) = 119886119889 (1199012 119909) 119863 (119909) = 119886119889 (1199011 1199012) (119909) forall119909 isin 119871
(40)
that is for all 119910 isin 119871 1198631(119909)(119910) = [1199011 119909 119910] = [119909 119910]11198632(119909)(119910) = [1199012 119909 119910] = [119909 119910]2 119863(119909) = [1199011 1199012 119909] We havethe following result
Theorem 17 Let 3-algebra 119860 be a two-dimensional extensionof Lie algebras 1198711 and 1198712 Then 119860 is a 3-Lie algebra if and onlyif linear mappings 1198631 1198632 and 119863 satisfy that 1198631 1198711 997888rarr119863119890119903(1198711) 1198632 1198712 997888rarr 119863119890119903(1198712) are Lie homomorphisms and1198631 (1199093) ([1199091 1199092]2) = [1198631 (1199093) (1199091) 1199092]2+ [1199091 1198631 (1199093) (1199092)]2minus 1205821199093 [1199091 1199092]1 minus 1205831199093 [1199091 1199092]2
(41)
1198632 (1199093) ([1199091 1199092]1) = [1198632 (1199093) (1199091) 1199092]1+ [1199091 1198632 (1199093) (1199092)]1+ 1205821199093 [1199091 1199092]1 + 1205831199093 [1199091 1199092]2 (42)
119863 ([1199091 1199092]1) = (12058311990911205821199092 minus 12058211990911205831199092) 1199011119863 ([1199091 1199092]2) = (12058311990911205821199092 minus 12058211990911205831199092) 1199012 (43)
120582[11990911199092]1 = 120583[11990911199092]2 = 12058311990911205821199092 minus 12058211990911205831199092 120583[11990911199092]1 = 120582[11990911199092]2 = 0 (44)
119863119896 (1199091) (1199092) = minus119863119896 (1199092) (1199091) 119891119900119903 119886119897119897 1199091 1199092 isin 119871 119896 = 1 2 (45)
where 1199091 1199092 1199093 isin 119871 119863(119909119894) = 1205821199091198941199011 + 1205831199091198941199012 119894 = 1 2 3Proof If 119860 is a two-dimensional extension 3-Lie algebrathen by Definition 16 linear mappings 119863119896 satisfy that119863119896(119871119896) sube 119863119890119903(119871119896) and119863119896 are Lie homomorphisms 119896 = 1 2Thanks to (39)1198631 (1199093) ([1199091 1199092]2) = [1199012 [1199011 1199093 1199091] 1199092]+ [1199012 1199091 [1199011 1199093 1199092]]+ [[1199011 1199093 1199012] 1199091 1199092]= [1198631 (1199093) (1199091) 1199092]2+ [1199091 1198631 (1199093) (1199092)]2minus 1205821199093 [1199091 1199092]1 minus 1205831199093 [1199091 1199092]2
(46)
for all 1199091 1199092 1199093 isin 119871 (41) holds Similarly we have (42)Thanks to (39) and (40)119863([1199091 1199092]1) = 119886119889 (1199011 1199012) [1199091 1199092]1= 120582[11990911199092]11199011 + 120583[11990911199092]11199012= (12058311990911205821199092 minus 12058211990911205831199092) 1199011119863 ([1199091 1199092]2) = 119886119889 (1199011 1199012) [1199091 1199092]2= 120582[11990911199092]21199011 + 120583[11990911199092]21199012= (12058311990911205821199092 minus 12058211990911205831199092) 1199012
(47)
Equations (43) and (44) hold Equation (45) follows from (39)and (40) directly
Conversely by (39) forall1199091 1199092 1199093 119909 isin 119871[1199091 1199092 1199093] = 0[1199011 1199091 1199092] = 1198631 (1199091) (1199092) = [1199091 1199092]1 [1199012 1199091 1199092] = 1198632 (1199091) (1199092) = [1199091 1199092]2 [1199011 1199012 119909] = 119863 (119909) = 120582 (119909) 1199011 + 1205831199091199012(48)
Since119863119896(119871) sube 119863119890119903(119871119896) and119863119896 are Lie homomorphisms 119861 =119871 + F1199011 and 119862 = 119871 + F1199012 are 3-Lie algebras which are one-dimensional extension 3-Lie algebras of Lie algebras 119871119896 119896 =1 2 respectively
Next we only need to prove that the multiplication on119860 defined by (39) satisfies (1) For all 119909119894 isin 119871 1 le 119894 le5 that products [[1199091 1199092 1199093] 1199094 1199095] [[119901119895 1199092 1199093] 1199094 1199095][[1199091 1199092 1199093] 1199094 119901119895] and [[1199091 1199092 119901119895] 1199094 119901119895] satisfy (1) 119895 =1 2 follow from that 119861 and 119862 are one-dimensional extension3-Lie algebras of 119871119896 and (39) directlyFrom (41) and (42) it follows that products[[119901119894 1199091 1199092] 119901119895 1199093] 1 le 119894 = 119895 le 2 satisfy (1) It
follows from (43)ndash(45) that products [1199011 1199012 [119901119894 1199091 1199092]][1199091 1199092 [119901119894 1199012 1199093]] and [119901119894 1199091 [1199011 1199012 1199092] 119894 = 1 2 satisfy(1) We omit the computation process
Theorem 18 Let (119860 [ ]) be a 3-Lie algebra Then 119860 is a two-dimensional extension 3-Lie algebra of Lie algebras if and onlyif there is an involutive derivation119879 on119860 such that dim1198601 = 2or dim119860minus1 = 2Proof If 119860 is a two-dimensional extension 3-Lie algebra ofLie algebras Then byTheorem 15 there are Lie algebras 1198711 =(119871 [ ]1) and 1198712 = (119871 [ ]2) such that 119860 = 119871 + 119882 and themultiplication of 119860 is defined by (39) where119882 = F1199011 + F1199012
Define the endomorphism 119879 of 119860 by 119879(119909) = 119909 119879(1199011) =minus1199011 119879(1199012) = minus1199012 or 119879(119909) = minus119909 119879(1199011) = 1199011 119879(1199012) =1199012 forall119909 isin 119871Then1198792 = 119868119889 and1198601 = 119871119860minus1 = 119882 or119860minus1 = 1198711198601 = 119882Thanks to (38) and (41)-(45) 119879 is a derivation of 119860Conversely if there is an involutive derivation 119879 on the3-Lie algebra 119860 such that dim119860minus1 = 2 (or dim1198601 = 2)
By Theorem 4 [1198601 1198601 1198601] = 0 [1198601 1198601 119860minus1] sube 1198601[1198601 119860minus1 119860minus1] sube 119860minus1 Let 119871 = 1198601 and 119860minus1 = F1199011 + F1199012Then [119871 119871 1199011] sube 119871 [119871 119871 1199012] sube 119871 and (119871 [ ]1) and
International Journal of Mathematics and Mathematical Sciences 9
(119871 [ ]2) are Lie algebras where [119909 119910]1 = [119909 119910 1199011] [119909 119910]2 =[119909 119910 1199012] forall119909 119910 isin 119871Thanks toTheorem 17 the 3-Lie algebra119860 is a two-dimensional extension 3-Lie algebra of Lie algebras1198711 and 1198712Data Availability
No data were used to support this study
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
The first named author was supported in part by the NaturalScience Foundation (11371245) and theNatural Science Foun-dation of Hebei Province (A2018201126)
References
[1] V Filippov ldquon-Lie algebrasrdquo SiberianMathematical Journal vol26 no 6 pp 126ndash140 1985
[2] R P Bai and P P Jia ldquoThe real compact n-Lie algebras andinvariant bilinear formsrdquo Acta Mathematica Scientia Series AShuxueWuli Xuebao Chinese Edition vol 27A no 6 pp 1074ndash1081 2007
[3] S Kasymov ldquoOn a theory of n-Lie algebrasrdquo Algebra and Logicvol 26 no 3 pp 277ndash297 1987
[4] C Bai L Guo and Y Sheng ldquoBialgebras the classical Yang-Baxter equation andmain triples for 3-Lie algebrasrdquoMathemat-ical Physics 2016
[5] R Bai and G Song ldquoThe classification of six-dimensional 4-Liealgebrasrdquo Journal of Physics A Mathematical and General vol42 no 3 035207 17 pages 2009
[6] R Bai G Song and Y Zhang ldquoOn classification of n-Liealgebrasrdquo Frontiers of Mathematics in China vol 6 no 4 pp581ndash606 2011
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6 International Journal of Mathematics and Mathematical Sciences
1198612 = ((
119887119903+11 119887119903+12 119887119903+13 sdot sdot sdot 119887119903+1119903 d1198872119904+21 1198872119904+22 1198872119904+23 sdot sdot sdot 1198872119904+21199031198872119904+31 minus11988721 11988731 sdot sdot sdot (minus1)119903+1 1198871199031
))
(26)
Since sum2119904+2119895=2 119887119895119895 = 11988711 sum2119904+3119895=2119895 =119894 119887119895119895 = 119887119894119894 2 le 119894 le 119903 we haveminus11988711 + 211988722 minus 1198872119904+32119904+3 = 0 (119903 minus 3)11988722 + sum119903+12119904+3 119887119894119894 = 0 11988722 =119887119894119894 3 le 119894 le 119903 Therefore
11988711 = minus1119903 minus 3 ((119903 minus 1) 1198961 + 22119904+3minus119903sum119895=2
119896119895) 119887119894119894 = minus1119903 minus 3 2119904+3minus119903sum
119895=1
119896119895 2 le 119894 le 119903119887119895119895 = 1198962119904+3minus119895+1 119903 + 1 le 119895 le 2119904 + 3 1198962119904+3minus119895+1 isin F
(27)
Suppose
11986121 =((((((((((((
11988811 11988812 sdot sdot sdot 1198881119894 sdot sdot sdot 1198881119895 sdot sdot sdot 1198881119903 d d
d1198881198941 1198881198942 sdot sdot sdot 119888119894119894 sdot sdot sdot 119888119894119895 sdot sdot sdot 119888119894119903 d
d d
1198881198951 1198881198952 sdot sdot sdot 119888119895119894 sdot sdot sdot 119888119895119895 sdot sdot sdot 119888119895119903 d d
d1198881199031 1198881199032 sdot sdot sdot 119888119903119894 sdot sdot sdot 119888119903119895 sdot sdot sdot 119888119903119903
))))))))))))
(28)
By (23)
11988811 = 1198872111198881119897 = 0 2 le 119897 le 119903119888119894119894 = 119894sum119897=1
(minus1)119894+119897minus1 1198872119897119894 + 119903sum119897=119894+1
(minus1)119894+119897minus1 1198872119894119897 2 le 119894 le 119903119888119894119895 = 119894sum119897=1
(minus1)119897+119894+1 119887119897119894119887119897119895 + 119895sum119897=119894+1
119887119894119897119887119897119895+ 119903sum119897=119895+1
(minus1)119897+119895+1 119887119894119897119887119895119897 2 le 119894 lt 119895 le 119903119888119894119895 = 119895sum119897=1
(minus1)119897+119894+1 119887119897119895119887119897119894 + 119894sum119897=119895+1
(minus1)119894+119895 119887119895119897119887119897119894+ 119903sum119897=119894+1
(minus1)119897+119895+1 119887119895119897119887119894119897 1 le 119895 lt 119894 le 119903
(29)
Therefore the endomorphisms 119863 of 119860 which are defined by1198631198901 = 1198901or 1198631198901 = minus1198901
1198631198902 = 119903sum119896=3
119890119896119863119890119894 = (minus1)119894minus1 1198902 + 119894minus1sum
119896=3
(minus1)119894 119890119896 + (minus1)119894minus1 119903sum119896=119894+1
1198901198963 le 119894 le 119903119863119890119895 = (minus1)119895 119890119895 119903 + 1 le 119895 le 2119904 + 31198631198901 = 1198901
or 1198631198901 = minus11989011198631198902 = 119903sum
119896=3
119890119896119863119890119894 = (minus1)119894minus1 1198902 + 119894minus1sum
119896=3
(minus1)119894 119890119896 + (minus1)119894minus1 119903sum119896=119894+1
1198901198963 le 119894 le 119903119863119890119895 = (minus1)119895minus1 119890119895 119903 + 1 le 119895 le 2119904 + 3
(30)
are involutive derivations on 119860Theorem 13 Let 119860 be a (2119904 + 3)-dimensional (2119904 + 1)-Liealgebra of the case (1199031) with dim1198601 = 119903 = 2119904 + 2 (119904 ge 1)then there does not exist an involutive derivation on 119860Proof If 119863 is an involutive derivation on 119860 then byLemma 11 and (23)
11988711 = minus (2119904 + 1) 11989612119904 minus 1 1198872119904+32119904+3 = 1198961119887119894119894 = minus11989612119904 minus 1
2 le 119894 le 119903 1198961 isin F (31)
Thanks to (22) 11988722119904+32119904+3 = 119887211 = 11989621 = 1 Therefore(minus(2119904 + 1)1198961(2119904 minus 1))2 = ((2119904 + 1)(2119904 minus 1))2 = 1 whichis a contradiction
Now we discuss case (1199032)Theorem 14 Let 119860 be a (2119904 + 3)-dimensional (2119904 + 1)-Liealgebra of the case (1199032) with dim1198601 = 119903 ge 119904 + 3 Then thereexist involutive derivations on 119860 if and only if 119903 is evenProof By Lemma 7 119860 = 1198601 + F1198902119904+3 where 1198902s+3 isin 119885(119860)and 1198601 is a (2119904 + 2)-dimensional (2119904 + 1)-Lie subalgebra
International Journal of Mathematics and Mathematical Sciences 7
of 119860 with dim1198601 = dim11986011 = 119903 Then there existinvolutive derivations on119860 if and only if there exist involutivederivations on 1198601
By Theorem 3 in [1] there is a basis 1198901 sdot sdot sdot 1198902119904+2 of 1198601such that[1198901 sdot sdot sdot 119890119895 sdot sdot sdot 1198902119904+2] = 0 119905 lt 119895 le 2119904 + 3 minus 119905[1198901 sdot sdot sdot 119890119894 sdot sdot sdot 1198902119904+2] = (minus1)119894 1198902119904+3minus1198941 le 119894 le 119905 or 2119904 + 3 minus 119905 lt 119894 le 2119904 + 2 (32)
If 119903 is even then 119903 = 2119905 ge 4 By Theorem 8 and (32)endomorphism 1198631 of 1198601 defined by
1198631 (119890119894) = 119890119894 119894 = 1 sdot sdot sdot 119904 + 1minus119890119894 119894 = 119904 + 2 sdot sdot sdot 2119904 + 2 (33)
is an involutive derivation on 1198601 Therefore the endomor-phism 119863 of 119860 defined by119863 (119890119894) = 119890119894 1 le 119894 le 119904 + 1119863 (119890119895) = minus119890119895 119904 + 2 le 119895 le 2119904 + 2119863 (1198902119904+3) = plusmn1198902119904+3
(34)
is involutive derivation on 119860If dim1198601 = 119903 is odd and endomorphism 119863 of 119860 is an
involutive derivation on 119860 then 119903 = 2119905 + 1 gt 4 Suppose119863(119890119894) = sum2119904+3119895=1 119886119894119895119890119895 1 le 119894 le 2119904 + 3 Then[119863 (1198902119904+3) 1198901198941 sdot sdot sdot 1198901198942119904]= 119863 [1198902119904+3 1198901198941 sdot sdot sdot 1198901198942119904]minus 2119904sum119895=1
[1198902119904+3 sdot sdot sdot 119863119890119894119895 sdot sdot sdot 1198901198942119904] = 0 (35)
We get 119863(1198902119904+3) isin 119885(119860) = F1198902119904+3 Since 1198632 = 119868119889 119863(1198902119904+3) =plusmn1198902119904+3 By (32) 1198601 = F1198901 + sdot sdot sdot + F119890119905 + F1198902119904+1 + sdot sdot sdot + F1198902119904+3minus119905and (minus1)119894119863 (1198902119904+3minus119894)
= 2119904+2sum119896=1119896=119894
[1198901 sdot sdot sdot 119890119894 sdot sdot sdot 119863 (119890119896) sdot sdot sdot 1198902119904+2] (36)
where 1 le 119894 le 119905 and 2119904 + 3 minus 119905 lt 119894 le 2119904 + 2 Then 1198861198942119904+3 = 0for 1 le 119894 le 119905 or 2119904 + 3 minus 119905 lt 119894 le 2119904 + 2 and 1198631198601 sube 1198601 Thenthe endomorphism 1198632 of 1198601 defined by1198632 (119890119894) = 119863 (119890119894) 1 le 119894 le 119905 or 2119904 + 3 minus 119905 lt 119894 le 2119904 + 2
1198632 (119890119895) = 119863(119890119895) minus 1198861198952119904+31198902119904+3 = 2119904+2sum119895=1
119886119894119895119890119895119905 lt 119895 le 2119904 + 3 minus 119905
(37)
is an involutive derivation on the (2119904+2)-dimensional (2119904+1)-Lie algebra 1198601 contradiction (Theorem 8) Therefore theredoes not exist involutive derivation on 119860
5 Structure of 3-Lie Algebras withInvolutive Derivations
Let (119871 [ ]) be a Lie algebra over F and 119901 be an element whichis not contained in 119871 Then 119860 = 119871 + F119901 is a 3-Lie algebra inthe multiplication[119909 119910 119911] = 0[119901 119909 119910] = [119909 119910]
for all 119909 119910 119911 isin 119871 (38)
And the 3-Lie algebra (119860 [ ]) is called one-dimensionalextension of 119871Theorem 15 Let 119860 be a 3-Lie algebra then 119860 is one-dimensional extension of a Lie algebra if and only if there existsan involutive derivation 119863 on 119860 such that dim1198601 = 1 ordim119860minus1 = 1Proof If 119860 is an one-dimensional extension of a Lie algebra119871 then 119860 = 119871 + F119901 Define the endomorphism 119863 of 119860 by119863(119901) = minus119901 (or 119901) and 119863(119909) = 119909 (or minus119909) forall119909 isin 119871 Thanks to(38)1198632 = 119868119889 and 119863([119909 119910 119911]) = 0 = [119863119909 119910 119911] + [119909119863119910 119911] +[119909 119910119863119911]119863([119901 119909 119910]) = [119901 119909 119910] = [119863119901 119909 119910] + [119901119863119909 119910] +[119901 119909 119863119910] for all 119909 119910 119911 isin 119871 Therefore 119863 is an involutivederivation on 119860 and dim119860minus1 = 1 (or dim1198601 = 1)
Conversely let 119863 be an involutive derivation on a 3-Liealgebra 119860 and dim119860minus1 = 1 (or dim1198601 = 1) Let 119860minus1 = F119901and 1198601 = 119871 (or 119860minus1 = 1198711198601 = F119901) where 119901 isin 119860minus119871 Thanksto Theorem 3 119871 is a Lie algebra with the multiplication[119909 119910] = [119901 119909 119910] for all 119909 119910 isin 119871 and 119860 is one-dimensionalextension of 119871
Let (119871 [ ]1) and (119871 [ ]2) be Lie algebras and 1199091 sdot sdot sdot 119909119898be a basis of 119871 For convenience denote Lie algebras (119871 [ ]119896)by 119871119896 119896 = 1 2 respectively Suppose 1199011 and 1199012 are twodistinct elements which are not contained in 119871 and 3-Lie algebras (119861 [ ]1) and (119862 [ ]2) are one-dimensionalextensions of Lie algebras 1198711 and 1198712 respectively where119861 = 119871 + F1199011 119862 = 119871 + F1199012 Then 119863119890119903(1198711) and 119863119890119903(1198712) aresubalgebras of 119892119897(119871)Definition 16 Let 1198711 = (119871 [ ]1) and 1198712 = (119871 [ ]2) be twoLie algebras and 1199011 1199012 be two distinct elements which arenot contained in 119871 and 119860 = 119871 + F1199011 + F1199012 Then 3-algebra(119860 [ ]) is called a two-dimensional extension of Lie algebras119871119896 119896 = 1 2 where [ ] 119860 and 119860 and 119860 997888rarr 119860 defined by[119909 119910 1199011] = [119909 119910]1 [119909 119910 1199012] = [119909 119910]2 [119909 119910 119911] = 0[1199011 1199012 119909] = 1205821199091199011 + 1205831199091199012 forall119909 119910 119911 isin 119871 120582119909 120583119909 isin F
(39)
If 119860 is a 3-Lie algebra then 119860 is called a two-dimensionalextension 3-Lie algebra of Lie algebras 119871119896 119896 = 1 2
8 International Journal of Mathematics and Mathematical Sciences
Let 119860 = 119871 + 119882 be a two-dimensional extension of Liealgebras 119871119896 119896 = 1 2 where 119882 = F1199011 + F1199012 Define linearmappings 1198631 1198632 119871 997888rarr 119864119899119889(119871) and 119863 119871 997888rarr 119882 by1198631 (119909) = 119886119889 (1199011 119909) 1198632 (119909) = 119886119889 (1199012 119909) 119863 (119909) = 119886119889 (1199011 1199012) (119909) forall119909 isin 119871
(40)
that is for all 119910 isin 119871 1198631(119909)(119910) = [1199011 119909 119910] = [119909 119910]11198632(119909)(119910) = [1199012 119909 119910] = [119909 119910]2 119863(119909) = [1199011 1199012 119909] We havethe following result
Theorem 17 Let 3-algebra 119860 be a two-dimensional extensionof Lie algebras 1198711 and 1198712 Then 119860 is a 3-Lie algebra if and onlyif linear mappings 1198631 1198632 and 119863 satisfy that 1198631 1198711 997888rarr119863119890119903(1198711) 1198632 1198712 997888rarr 119863119890119903(1198712) are Lie homomorphisms and1198631 (1199093) ([1199091 1199092]2) = [1198631 (1199093) (1199091) 1199092]2+ [1199091 1198631 (1199093) (1199092)]2minus 1205821199093 [1199091 1199092]1 minus 1205831199093 [1199091 1199092]2
(41)
1198632 (1199093) ([1199091 1199092]1) = [1198632 (1199093) (1199091) 1199092]1+ [1199091 1198632 (1199093) (1199092)]1+ 1205821199093 [1199091 1199092]1 + 1205831199093 [1199091 1199092]2 (42)
119863 ([1199091 1199092]1) = (12058311990911205821199092 minus 12058211990911205831199092) 1199011119863 ([1199091 1199092]2) = (12058311990911205821199092 minus 12058211990911205831199092) 1199012 (43)
120582[11990911199092]1 = 120583[11990911199092]2 = 12058311990911205821199092 minus 12058211990911205831199092 120583[11990911199092]1 = 120582[11990911199092]2 = 0 (44)
119863119896 (1199091) (1199092) = minus119863119896 (1199092) (1199091) 119891119900119903 119886119897119897 1199091 1199092 isin 119871 119896 = 1 2 (45)
where 1199091 1199092 1199093 isin 119871 119863(119909119894) = 1205821199091198941199011 + 1205831199091198941199012 119894 = 1 2 3Proof If 119860 is a two-dimensional extension 3-Lie algebrathen by Definition 16 linear mappings 119863119896 satisfy that119863119896(119871119896) sube 119863119890119903(119871119896) and119863119896 are Lie homomorphisms 119896 = 1 2Thanks to (39)1198631 (1199093) ([1199091 1199092]2) = [1199012 [1199011 1199093 1199091] 1199092]+ [1199012 1199091 [1199011 1199093 1199092]]+ [[1199011 1199093 1199012] 1199091 1199092]= [1198631 (1199093) (1199091) 1199092]2+ [1199091 1198631 (1199093) (1199092)]2minus 1205821199093 [1199091 1199092]1 minus 1205831199093 [1199091 1199092]2
(46)
for all 1199091 1199092 1199093 isin 119871 (41) holds Similarly we have (42)Thanks to (39) and (40)119863([1199091 1199092]1) = 119886119889 (1199011 1199012) [1199091 1199092]1= 120582[11990911199092]11199011 + 120583[11990911199092]11199012= (12058311990911205821199092 minus 12058211990911205831199092) 1199011119863 ([1199091 1199092]2) = 119886119889 (1199011 1199012) [1199091 1199092]2= 120582[11990911199092]21199011 + 120583[11990911199092]21199012= (12058311990911205821199092 minus 12058211990911205831199092) 1199012
(47)
Equations (43) and (44) hold Equation (45) follows from (39)and (40) directly
Conversely by (39) forall1199091 1199092 1199093 119909 isin 119871[1199091 1199092 1199093] = 0[1199011 1199091 1199092] = 1198631 (1199091) (1199092) = [1199091 1199092]1 [1199012 1199091 1199092] = 1198632 (1199091) (1199092) = [1199091 1199092]2 [1199011 1199012 119909] = 119863 (119909) = 120582 (119909) 1199011 + 1205831199091199012(48)
Since119863119896(119871) sube 119863119890119903(119871119896) and119863119896 are Lie homomorphisms 119861 =119871 + F1199011 and 119862 = 119871 + F1199012 are 3-Lie algebras which are one-dimensional extension 3-Lie algebras of Lie algebras 119871119896 119896 =1 2 respectively
Next we only need to prove that the multiplication on119860 defined by (39) satisfies (1) For all 119909119894 isin 119871 1 le 119894 le5 that products [[1199091 1199092 1199093] 1199094 1199095] [[119901119895 1199092 1199093] 1199094 1199095][[1199091 1199092 1199093] 1199094 119901119895] and [[1199091 1199092 119901119895] 1199094 119901119895] satisfy (1) 119895 =1 2 follow from that 119861 and 119862 are one-dimensional extension3-Lie algebras of 119871119896 and (39) directlyFrom (41) and (42) it follows that products[[119901119894 1199091 1199092] 119901119895 1199093] 1 le 119894 = 119895 le 2 satisfy (1) It
follows from (43)ndash(45) that products [1199011 1199012 [119901119894 1199091 1199092]][1199091 1199092 [119901119894 1199012 1199093]] and [119901119894 1199091 [1199011 1199012 1199092] 119894 = 1 2 satisfy(1) We omit the computation process
Theorem 18 Let (119860 [ ]) be a 3-Lie algebra Then 119860 is a two-dimensional extension 3-Lie algebra of Lie algebras if and onlyif there is an involutive derivation119879 on119860 such that dim1198601 = 2or dim119860minus1 = 2Proof If 119860 is a two-dimensional extension 3-Lie algebra ofLie algebras Then byTheorem 15 there are Lie algebras 1198711 =(119871 [ ]1) and 1198712 = (119871 [ ]2) such that 119860 = 119871 + 119882 and themultiplication of 119860 is defined by (39) where119882 = F1199011 + F1199012
Define the endomorphism 119879 of 119860 by 119879(119909) = 119909 119879(1199011) =minus1199011 119879(1199012) = minus1199012 or 119879(119909) = minus119909 119879(1199011) = 1199011 119879(1199012) =1199012 forall119909 isin 119871Then1198792 = 119868119889 and1198601 = 119871119860minus1 = 119882 or119860minus1 = 1198711198601 = 119882Thanks to (38) and (41)-(45) 119879 is a derivation of 119860Conversely if there is an involutive derivation 119879 on the3-Lie algebra 119860 such that dim119860minus1 = 2 (or dim1198601 = 2)
By Theorem 4 [1198601 1198601 1198601] = 0 [1198601 1198601 119860minus1] sube 1198601[1198601 119860minus1 119860minus1] sube 119860minus1 Let 119871 = 1198601 and 119860minus1 = F1199011 + F1199012Then [119871 119871 1199011] sube 119871 [119871 119871 1199012] sube 119871 and (119871 [ ]1) and
International Journal of Mathematics and Mathematical Sciences 9
(119871 [ ]2) are Lie algebras where [119909 119910]1 = [119909 119910 1199011] [119909 119910]2 =[119909 119910 1199012] forall119909 119910 isin 119871Thanks toTheorem 17 the 3-Lie algebra119860 is a two-dimensional extension 3-Lie algebra of Lie algebras1198711 and 1198712Data Availability
No data were used to support this study
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
The first named author was supported in part by the NaturalScience Foundation (11371245) and theNatural Science Foun-dation of Hebei Province (A2018201126)
References
[1] V Filippov ldquon-Lie algebrasrdquo SiberianMathematical Journal vol26 no 6 pp 126ndash140 1985
[2] R P Bai and P P Jia ldquoThe real compact n-Lie algebras andinvariant bilinear formsrdquo Acta Mathematica Scientia Series AShuxueWuli Xuebao Chinese Edition vol 27A no 6 pp 1074ndash1081 2007
[3] S Kasymov ldquoOn a theory of n-Lie algebrasrdquo Algebra and Logicvol 26 no 3 pp 277ndash297 1987
[4] C Bai L Guo and Y Sheng ldquoBialgebras the classical Yang-Baxter equation andmain triples for 3-Lie algebrasrdquoMathemat-ical Physics 2016
[5] R Bai and G Song ldquoThe classification of six-dimensional 4-Liealgebrasrdquo Journal of Physics A Mathematical and General vol42 no 3 035207 17 pages 2009
[6] R Bai G Song and Y Zhang ldquoOn classification of n-Liealgebrasrdquo Frontiers of Mathematics in China vol 6 no 4 pp581ndash606 2011
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
International Journal of Mathematics and Mathematical Sciences 7
of 119860 with dim1198601 = dim11986011 = 119903 Then there existinvolutive derivations on119860 if and only if there exist involutivederivations on 1198601
By Theorem 3 in [1] there is a basis 1198901 sdot sdot sdot 1198902119904+2 of 1198601such that[1198901 sdot sdot sdot 119890119895 sdot sdot sdot 1198902119904+2] = 0 119905 lt 119895 le 2119904 + 3 minus 119905[1198901 sdot sdot sdot 119890119894 sdot sdot sdot 1198902119904+2] = (minus1)119894 1198902119904+3minus1198941 le 119894 le 119905 or 2119904 + 3 minus 119905 lt 119894 le 2119904 + 2 (32)
If 119903 is even then 119903 = 2119905 ge 4 By Theorem 8 and (32)endomorphism 1198631 of 1198601 defined by
1198631 (119890119894) = 119890119894 119894 = 1 sdot sdot sdot 119904 + 1minus119890119894 119894 = 119904 + 2 sdot sdot sdot 2119904 + 2 (33)
is an involutive derivation on 1198601 Therefore the endomor-phism 119863 of 119860 defined by119863 (119890119894) = 119890119894 1 le 119894 le 119904 + 1119863 (119890119895) = minus119890119895 119904 + 2 le 119895 le 2119904 + 2119863 (1198902119904+3) = plusmn1198902119904+3
(34)
is involutive derivation on 119860If dim1198601 = 119903 is odd and endomorphism 119863 of 119860 is an
involutive derivation on 119860 then 119903 = 2119905 + 1 gt 4 Suppose119863(119890119894) = sum2119904+3119895=1 119886119894119895119890119895 1 le 119894 le 2119904 + 3 Then[119863 (1198902119904+3) 1198901198941 sdot sdot sdot 1198901198942119904]= 119863 [1198902119904+3 1198901198941 sdot sdot sdot 1198901198942119904]minus 2119904sum119895=1
[1198902119904+3 sdot sdot sdot 119863119890119894119895 sdot sdot sdot 1198901198942119904] = 0 (35)
We get 119863(1198902119904+3) isin 119885(119860) = F1198902119904+3 Since 1198632 = 119868119889 119863(1198902119904+3) =plusmn1198902119904+3 By (32) 1198601 = F1198901 + sdot sdot sdot + F119890119905 + F1198902119904+1 + sdot sdot sdot + F1198902119904+3minus119905and (minus1)119894119863 (1198902119904+3minus119894)
= 2119904+2sum119896=1119896=119894
[1198901 sdot sdot sdot 119890119894 sdot sdot sdot 119863 (119890119896) sdot sdot sdot 1198902119904+2] (36)
where 1 le 119894 le 119905 and 2119904 + 3 minus 119905 lt 119894 le 2119904 + 2 Then 1198861198942119904+3 = 0for 1 le 119894 le 119905 or 2119904 + 3 minus 119905 lt 119894 le 2119904 + 2 and 1198631198601 sube 1198601 Thenthe endomorphism 1198632 of 1198601 defined by1198632 (119890119894) = 119863 (119890119894) 1 le 119894 le 119905 or 2119904 + 3 minus 119905 lt 119894 le 2119904 + 2
1198632 (119890119895) = 119863(119890119895) minus 1198861198952119904+31198902119904+3 = 2119904+2sum119895=1
119886119894119895119890119895119905 lt 119895 le 2119904 + 3 minus 119905
(37)
is an involutive derivation on the (2119904+2)-dimensional (2119904+1)-Lie algebra 1198601 contradiction (Theorem 8) Therefore theredoes not exist involutive derivation on 119860
5 Structure of 3-Lie Algebras withInvolutive Derivations
Let (119871 [ ]) be a Lie algebra over F and 119901 be an element whichis not contained in 119871 Then 119860 = 119871 + F119901 is a 3-Lie algebra inthe multiplication[119909 119910 119911] = 0[119901 119909 119910] = [119909 119910]
for all 119909 119910 119911 isin 119871 (38)
And the 3-Lie algebra (119860 [ ]) is called one-dimensionalextension of 119871Theorem 15 Let 119860 be a 3-Lie algebra then 119860 is one-dimensional extension of a Lie algebra if and only if there existsan involutive derivation 119863 on 119860 such that dim1198601 = 1 ordim119860minus1 = 1Proof If 119860 is an one-dimensional extension of a Lie algebra119871 then 119860 = 119871 + F119901 Define the endomorphism 119863 of 119860 by119863(119901) = minus119901 (or 119901) and 119863(119909) = 119909 (or minus119909) forall119909 isin 119871 Thanks to(38)1198632 = 119868119889 and 119863([119909 119910 119911]) = 0 = [119863119909 119910 119911] + [119909119863119910 119911] +[119909 119910119863119911]119863([119901 119909 119910]) = [119901 119909 119910] = [119863119901 119909 119910] + [119901119863119909 119910] +[119901 119909 119863119910] for all 119909 119910 119911 isin 119871 Therefore 119863 is an involutivederivation on 119860 and dim119860minus1 = 1 (or dim1198601 = 1)
Conversely let 119863 be an involutive derivation on a 3-Liealgebra 119860 and dim119860minus1 = 1 (or dim1198601 = 1) Let 119860minus1 = F119901and 1198601 = 119871 (or 119860minus1 = 1198711198601 = F119901) where 119901 isin 119860minus119871 Thanksto Theorem 3 119871 is a Lie algebra with the multiplication[119909 119910] = [119901 119909 119910] for all 119909 119910 isin 119871 and 119860 is one-dimensionalextension of 119871
Let (119871 [ ]1) and (119871 [ ]2) be Lie algebras and 1199091 sdot sdot sdot 119909119898be a basis of 119871 For convenience denote Lie algebras (119871 [ ]119896)by 119871119896 119896 = 1 2 respectively Suppose 1199011 and 1199012 are twodistinct elements which are not contained in 119871 and 3-Lie algebras (119861 [ ]1) and (119862 [ ]2) are one-dimensionalextensions of Lie algebras 1198711 and 1198712 respectively where119861 = 119871 + F1199011 119862 = 119871 + F1199012 Then 119863119890119903(1198711) and 119863119890119903(1198712) aresubalgebras of 119892119897(119871)Definition 16 Let 1198711 = (119871 [ ]1) and 1198712 = (119871 [ ]2) be twoLie algebras and 1199011 1199012 be two distinct elements which arenot contained in 119871 and 119860 = 119871 + F1199011 + F1199012 Then 3-algebra(119860 [ ]) is called a two-dimensional extension of Lie algebras119871119896 119896 = 1 2 where [ ] 119860 and 119860 and 119860 997888rarr 119860 defined by[119909 119910 1199011] = [119909 119910]1 [119909 119910 1199012] = [119909 119910]2 [119909 119910 119911] = 0[1199011 1199012 119909] = 1205821199091199011 + 1205831199091199012 forall119909 119910 119911 isin 119871 120582119909 120583119909 isin F
(39)
If 119860 is a 3-Lie algebra then 119860 is called a two-dimensionalextension 3-Lie algebra of Lie algebras 119871119896 119896 = 1 2
8 International Journal of Mathematics and Mathematical Sciences
Let 119860 = 119871 + 119882 be a two-dimensional extension of Liealgebras 119871119896 119896 = 1 2 where 119882 = F1199011 + F1199012 Define linearmappings 1198631 1198632 119871 997888rarr 119864119899119889(119871) and 119863 119871 997888rarr 119882 by1198631 (119909) = 119886119889 (1199011 119909) 1198632 (119909) = 119886119889 (1199012 119909) 119863 (119909) = 119886119889 (1199011 1199012) (119909) forall119909 isin 119871
(40)
that is for all 119910 isin 119871 1198631(119909)(119910) = [1199011 119909 119910] = [119909 119910]11198632(119909)(119910) = [1199012 119909 119910] = [119909 119910]2 119863(119909) = [1199011 1199012 119909] We havethe following result
Theorem 17 Let 3-algebra 119860 be a two-dimensional extensionof Lie algebras 1198711 and 1198712 Then 119860 is a 3-Lie algebra if and onlyif linear mappings 1198631 1198632 and 119863 satisfy that 1198631 1198711 997888rarr119863119890119903(1198711) 1198632 1198712 997888rarr 119863119890119903(1198712) are Lie homomorphisms and1198631 (1199093) ([1199091 1199092]2) = [1198631 (1199093) (1199091) 1199092]2+ [1199091 1198631 (1199093) (1199092)]2minus 1205821199093 [1199091 1199092]1 minus 1205831199093 [1199091 1199092]2
(41)
1198632 (1199093) ([1199091 1199092]1) = [1198632 (1199093) (1199091) 1199092]1+ [1199091 1198632 (1199093) (1199092)]1+ 1205821199093 [1199091 1199092]1 + 1205831199093 [1199091 1199092]2 (42)
119863 ([1199091 1199092]1) = (12058311990911205821199092 minus 12058211990911205831199092) 1199011119863 ([1199091 1199092]2) = (12058311990911205821199092 minus 12058211990911205831199092) 1199012 (43)
120582[11990911199092]1 = 120583[11990911199092]2 = 12058311990911205821199092 minus 12058211990911205831199092 120583[11990911199092]1 = 120582[11990911199092]2 = 0 (44)
119863119896 (1199091) (1199092) = minus119863119896 (1199092) (1199091) 119891119900119903 119886119897119897 1199091 1199092 isin 119871 119896 = 1 2 (45)
where 1199091 1199092 1199093 isin 119871 119863(119909119894) = 1205821199091198941199011 + 1205831199091198941199012 119894 = 1 2 3Proof If 119860 is a two-dimensional extension 3-Lie algebrathen by Definition 16 linear mappings 119863119896 satisfy that119863119896(119871119896) sube 119863119890119903(119871119896) and119863119896 are Lie homomorphisms 119896 = 1 2Thanks to (39)1198631 (1199093) ([1199091 1199092]2) = [1199012 [1199011 1199093 1199091] 1199092]+ [1199012 1199091 [1199011 1199093 1199092]]+ [[1199011 1199093 1199012] 1199091 1199092]= [1198631 (1199093) (1199091) 1199092]2+ [1199091 1198631 (1199093) (1199092)]2minus 1205821199093 [1199091 1199092]1 minus 1205831199093 [1199091 1199092]2
(46)
for all 1199091 1199092 1199093 isin 119871 (41) holds Similarly we have (42)Thanks to (39) and (40)119863([1199091 1199092]1) = 119886119889 (1199011 1199012) [1199091 1199092]1= 120582[11990911199092]11199011 + 120583[11990911199092]11199012= (12058311990911205821199092 minus 12058211990911205831199092) 1199011119863 ([1199091 1199092]2) = 119886119889 (1199011 1199012) [1199091 1199092]2= 120582[11990911199092]21199011 + 120583[11990911199092]21199012= (12058311990911205821199092 minus 12058211990911205831199092) 1199012
(47)
Equations (43) and (44) hold Equation (45) follows from (39)and (40) directly
Conversely by (39) forall1199091 1199092 1199093 119909 isin 119871[1199091 1199092 1199093] = 0[1199011 1199091 1199092] = 1198631 (1199091) (1199092) = [1199091 1199092]1 [1199012 1199091 1199092] = 1198632 (1199091) (1199092) = [1199091 1199092]2 [1199011 1199012 119909] = 119863 (119909) = 120582 (119909) 1199011 + 1205831199091199012(48)
Since119863119896(119871) sube 119863119890119903(119871119896) and119863119896 are Lie homomorphisms 119861 =119871 + F1199011 and 119862 = 119871 + F1199012 are 3-Lie algebras which are one-dimensional extension 3-Lie algebras of Lie algebras 119871119896 119896 =1 2 respectively
Next we only need to prove that the multiplication on119860 defined by (39) satisfies (1) For all 119909119894 isin 119871 1 le 119894 le5 that products [[1199091 1199092 1199093] 1199094 1199095] [[119901119895 1199092 1199093] 1199094 1199095][[1199091 1199092 1199093] 1199094 119901119895] and [[1199091 1199092 119901119895] 1199094 119901119895] satisfy (1) 119895 =1 2 follow from that 119861 and 119862 are one-dimensional extension3-Lie algebras of 119871119896 and (39) directlyFrom (41) and (42) it follows that products[[119901119894 1199091 1199092] 119901119895 1199093] 1 le 119894 = 119895 le 2 satisfy (1) It
follows from (43)ndash(45) that products [1199011 1199012 [119901119894 1199091 1199092]][1199091 1199092 [119901119894 1199012 1199093]] and [119901119894 1199091 [1199011 1199012 1199092] 119894 = 1 2 satisfy(1) We omit the computation process
Theorem 18 Let (119860 [ ]) be a 3-Lie algebra Then 119860 is a two-dimensional extension 3-Lie algebra of Lie algebras if and onlyif there is an involutive derivation119879 on119860 such that dim1198601 = 2or dim119860minus1 = 2Proof If 119860 is a two-dimensional extension 3-Lie algebra ofLie algebras Then byTheorem 15 there are Lie algebras 1198711 =(119871 [ ]1) and 1198712 = (119871 [ ]2) such that 119860 = 119871 + 119882 and themultiplication of 119860 is defined by (39) where119882 = F1199011 + F1199012
Define the endomorphism 119879 of 119860 by 119879(119909) = 119909 119879(1199011) =minus1199011 119879(1199012) = minus1199012 or 119879(119909) = minus119909 119879(1199011) = 1199011 119879(1199012) =1199012 forall119909 isin 119871Then1198792 = 119868119889 and1198601 = 119871119860minus1 = 119882 or119860minus1 = 1198711198601 = 119882Thanks to (38) and (41)-(45) 119879 is a derivation of 119860Conversely if there is an involutive derivation 119879 on the3-Lie algebra 119860 such that dim119860minus1 = 2 (or dim1198601 = 2)
By Theorem 4 [1198601 1198601 1198601] = 0 [1198601 1198601 119860minus1] sube 1198601[1198601 119860minus1 119860minus1] sube 119860minus1 Let 119871 = 1198601 and 119860minus1 = F1199011 + F1199012Then [119871 119871 1199011] sube 119871 [119871 119871 1199012] sube 119871 and (119871 [ ]1) and
International Journal of Mathematics and Mathematical Sciences 9
(119871 [ ]2) are Lie algebras where [119909 119910]1 = [119909 119910 1199011] [119909 119910]2 =[119909 119910 1199012] forall119909 119910 isin 119871Thanks toTheorem 17 the 3-Lie algebra119860 is a two-dimensional extension 3-Lie algebra of Lie algebras1198711 and 1198712Data Availability
No data were used to support this study
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
The first named author was supported in part by the NaturalScience Foundation (11371245) and theNatural Science Foun-dation of Hebei Province (A2018201126)
References
[1] V Filippov ldquon-Lie algebrasrdquo SiberianMathematical Journal vol26 no 6 pp 126ndash140 1985
[2] R P Bai and P P Jia ldquoThe real compact n-Lie algebras andinvariant bilinear formsrdquo Acta Mathematica Scientia Series AShuxueWuli Xuebao Chinese Edition vol 27A no 6 pp 1074ndash1081 2007
[3] S Kasymov ldquoOn a theory of n-Lie algebrasrdquo Algebra and Logicvol 26 no 3 pp 277ndash297 1987
[4] C Bai L Guo and Y Sheng ldquoBialgebras the classical Yang-Baxter equation andmain triples for 3-Lie algebrasrdquoMathemat-ical Physics 2016
[5] R Bai and G Song ldquoThe classification of six-dimensional 4-Liealgebrasrdquo Journal of Physics A Mathematical and General vol42 no 3 035207 17 pages 2009
[6] R Bai G Song and Y Zhang ldquoOn classification of n-Liealgebrasrdquo Frontiers of Mathematics in China vol 6 no 4 pp581ndash606 2011
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
8 International Journal of Mathematics and Mathematical Sciences
Let 119860 = 119871 + 119882 be a two-dimensional extension of Liealgebras 119871119896 119896 = 1 2 where 119882 = F1199011 + F1199012 Define linearmappings 1198631 1198632 119871 997888rarr 119864119899119889(119871) and 119863 119871 997888rarr 119882 by1198631 (119909) = 119886119889 (1199011 119909) 1198632 (119909) = 119886119889 (1199012 119909) 119863 (119909) = 119886119889 (1199011 1199012) (119909) forall119909 isin 119871
(40)
that is for all 119910 isin 119871 1198631(119909)(119910) = [1199011 119909 119910] = [119909 119910]11198632(119909)(119910) = [1199012 119909 119910] = [119909 119910]2 119863(119909) = [1199011 1199012 119909] We havethe following result
Theorem 17 Let 3-algebra 119860 be a two-dimensional extensionof Lie algebras 1198711 and 1198712 Then 119860 is a 3-Lie algebra if and onlyif linear mappings 1198631 1198632 and 119863 satisfy that 1198631 1198711 997888rarr119863119890119903(1198711) 1198632 1198712 997888rarr 119863119890119903(1198712) are Lie homomorphisms and1198631 (1199093) ([1199091 1199092]2) = [1198631 (1199093) (1199091) 1199092]2+ [1199091 1198631 (1199093) (1199092)]2minus 1205821199093 [1199091 1199092]1 minus 1205831199093 [1199091 1199092]2
(41)
1198632 (1199093) ([1199091 1199092]1) = [1198632 (1199093) (1199091) 1199092]1+ [1199091 1198632 (1199093) (1199092)]1+ 1205821199093 [1199091 1199092]1 + 1205831199093 [1199091 1199092]2 (42)
119863 ([1199091 1199092]1) = (12058311990911205821199092 minus 12058211990911205831199092) 1199011119863 ([1199091 1199092]2) = (12058311990911205821199092 minus 12058211990911205831199092) 1199012 (43)
120582[11990911199092]1 = 120583[11990911199092]2 = 12058311990911205821199092 minus 12058211990911205831199092 120583[11990911199092]1 = 120582[11990911199092]2 = 0 (44)
119863119896 (1199091) (1199092) = minus119863119896 (1199092) (1199091) 119891119900119903 119886119897119897 1199091 1199092 isin 119871 119896 = 1 2 (45)
where 1199091 1199092 1199093 isin 119871 119863(119909119894) = 1205821199091198941199011 + 1205831199091198941199012 119894 = 1 2 3Proof If 119860 is a two-dimensional extension 3-Lie algebrathen by Definition 16 linear mappings 119863119896 satisfy that119863119896(119871119896) sube 119863119890119903(119871119896) and119863119896 are Lie homomorphisms 119896 = 1 2Thanks to (39)1198631 (1199093) ([1199091 1199092]2) = [1199012 [1199011 1199093 1199091] 1199092]+ [1199012 1199091 [1199011 1199093 1199092]]+ [[1199011 1199093 1199012] 1199091 1199092]= [1198631 (1199093) (1199091) 1199092]2+ [1199091 1198631 (1199093) (1199092)]2minus 1205821199093 [1199091 1199092]1 minus 1205831199093 [1199091 1199092]2
(46)
for all 1199091 1199092 1199093 isin 119871 (41) holds Similarly we have (42)Thanks to (39) and (40)119863([1199091 1199092]1) = 119886119889 (1199011 1199012) [1199091 1199092]1= 120582[11990911199092]11199011 + 120583[11990911199092]11199012= (12058311990911205821199092 minus 12058211990911205831199092) 1199011119863 ([1199091 1199092]2) = 119886119889 (1199011 1199012) [1199091 1199092]2= 120582[11990911199092]21199011 + 120583[11990911199092]21199012= (12058311990911205821199092 minus 12058211990911205831199092) 1199012
(47)
Equations (43) and (44) hold Equation (45) follows from (39)and (40) directly
Conversely by (39) forall1199091 1199092 1199093 119909 isin 119871[1199091 1199092 1199093] = 0[1199011 1199091 1199092] = 1198631 (1199091) (1199092) = [1199091 1199092]1 [1199012 1199091 1199092] = 1198632 (1199091) (1199092) = [1199091 1199092]2 [1199011 1199012 119909] = 119863 (119909) = 120582 (119909) 1199011 + 1205831199091199012(48)
Since119863119896(119871) sube 119863119890119903(119871119896) and119863119896 are Lie homomorphisms 119861 =119871 + F1199011 and 119862 = 119871 + F1199012 are 3-Lie algebras which are one-dimensional extension 3-Lie algebras of Lie algebras 119871119896 119896 =1 2 respectively
Next we only need to prove that the multiplication on119860 defined by (39) satisfies (1) For all 119909119894 isin 119871 1 le 119894 le5 that products [[1199091 1199092 1199093] 1199094 1199095] [[119901119895 1199092 1199093] 1199094 1199095][[1199091 1199092 1199093] 1199094 119901119895] and [[1199091 1199092 119901119895] 1199094 119901119895] satisfy (1) 119895 =1 2 follow from that 119861 and 119862 are one-dimensional extension3-Lie algebras of 119871119896 and (39) directlyFrom (41) and (42) it follows that products[[119901119894 1199091 1199092] 119901119895 1199093] 1 le 119894 = 119895 le 2 satisfy (1) It
follows from (43)ndash(45) that products [1199011 1199012 [119901119894 1199091 1199092]][1199091 1199092 [119901119894 1199012 1199093]] and [119901119894 1199091 [1199011 1199012 1199092] 119894 = 1 2 satisfy(1) We omit the computation process
Theorem 18 Let (119860 [ ]) be a 3-Lie algebra Then 119860 is a two-dimensional extension 3-Lie algebra of Lie algebras if and onlyif there is an involutive derivation119879 on119860 such that dim1198601 = 2or dim119860minus1 = 2Proof If 119860 is a two-dimensional extension 3-Lie algebra ofLie algebras Then byTheorem 15 there are Lie algebras 1198711 =(119871 [ ]1) and 1198712 = (119871 [ ]2) such that 119860 = 119871 + 119882 and themultiplication of 119860 is defined by (39) where119882 = F1199011 + F1199012
Define the endomorphism 119879 of 119860 by 119879(119909) = 119909 119879(1199011) =minus1199011 119879(1199012) = minus1199012 or 119879(119909) = minus119909 119879(1199011) = 1199011 119879(1199012) =1199012 forall119909 isin 119871Then1198792 = 119868119889 and1198601 = 119871119860minus1 = 119882 or119860minus1 = 1198711198601 = 119882Thanks to (38) and (41)-(45) 119879 is a derivation of 119860Conversely if there is an involutive derivation 119879 on the3-Lie algebra 119860 such that dim119860minus1 = 2 (or dim1198601 = 2)
By Theorem 4 [1198601 1198601 1198601] = 0 [1198601 1198601 119860minus1] sube 1198601[1198601 119860minus1 119860minus1] sube 119860minus1 Let 119871 = 1198601 and 119860minus1 = F1199011 + F1199012Then [119871 119871 1199011] sube 119871 [119871 119871 1199012] sube 119871 and (119871 [ ]1) and
International Journal of Mathematics and Mathematical Sciences 9
(119871 [ ]2) are Lie algebras where [119909 119910]1 = [119909 119910 1199011] [119909 119910]2 =[119909 119910 1199012] forall119909 119910 isin 119871Thanks toTheorem 17 the 3-Lie algebra119860 is a two-dimensional extension 3-Lie algebra of Lie algebras1198711 and 1198712Data Availability
No data were used to support this study
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
The first named author was supported in part by the NaturalScience Foundation (11371245) and theNatural Science Foun-dation of Hebei Province (A2018201126)
References
[1] V Filippov ldquon-Lie algebrasrdquo SiberianMathematical Journal vol26 no 6 pp 126ndash140 1985
[2] R P Bai and P P Jia ldquoThe real compact n-Lie algebras andinvariant bilinear formsrdquo Acta Mathematica Scientia Series AShuxueWuli Xuebao Chinese Edition vol 27A no 6 pp 1074ndash1081 2007
[3] S Kasymov ldquoOn a theory of n-Lie algebrasrdquo Algebra and Logicvol 26 no 3 pp 277ndash297 1987
[4] C Bai L Guo and Y Sheng ldquoBialgebras the classical Yang-Baxter equation andmain triples for 3-Lie algebrasrdquoMathemat-ical Physics 2016
[5] R Bai and G Song ldquoThe classification of six-dimensional 4-Liealgebrasrdquo Journal of Physics A Mathematical and General vol42 no 3 035207 17 pages 2009
[6] R Bai G Song and Y Zhang ldquoOn classification of n-Liealgebrasrdquo Frontiers of Mathematics in China vol 6 no 4 pp581ndash606 2011
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
International Journal of Mathematics and Mathematical Sciences 9
(119871 [ ]2) are Lie algebras where [119909 119910]1 = [119909 119910 1199011] [119909 119910]2 =[119909 119910 1199012] forall119909 119910 isin 119871Thanks toTheorem 17 the 3-Lie algebra119860 is a two-dimensional extension 3-Lie algebra of Lie algebras1198711 and 1198712Data Availability
No data were used to support this study
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
The first named author was supported in part by the NaturalScience Foundation (11371245) and theNatural Science Foun-dation of Hebei Province (A2018201126)
References
[1] V Filippov ldquon-Lie algebrasrdquo SiberianMathematical Journal vol26 no 6 pp 126ndash140 1985
[2] R P Bai and P P Jia ldquoThe real compact n-Lie algebras andinvariant bilinear formsrdquo Acta Mathematica Scientia Series AShuxueWuli Xuebao Chinese Edition vol 27A no 6 pp 1074ndash1081 2007
[3] S Kasymov ldquoOn a theory of n-Lie algebrasrdquo Algebra and Logicvol 26 no 3 pp 277ndash297 1987
[4] C Bai L Guo and Y Sheng ldquoBialgebras the classical Yang-Baxter equation andmain triples for 3-Lie algebrasrdquoMathemat-ical Physics 2016
[5] R Bai and G Song ldquoThe classification of six-dimensional 4-Liealgebrasrdquo Journal of Physics A Mathematical and General vol42 no 3 035207 17 pages 2009
[6] R Bai G Song and Y Zhang ldquoOn classification of n-Liealgebrasrdquo Frontiers of Mathematics in China vol 6 no 4 pp581ndash606 2011
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
