Shouchuan Zhang and Bizhong Yang- An Example of Double Cross Coproducts with Non-trivial Left...

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arXiv:math/0

410130v2

[math.Q

A]26Apr2005

An Example of Double Cross Coproducts with Non-trivial Left

Coaction and Right Coaction in Strictly Braided Tensor

Categories

Shouchuan Zhang Bizhong Yang

Department of Mathematics, Hunan University

Changsha 410082, P.R.China. E-mail:[email protected]

Beishang Ren

Department of Mathematics, Guangxi Normal College

Nanning 530001, P.R.China.

Abstract

An example of double cross coproducts with both non-trivial left coaction and non-trivialright coaction in strictly braided tensor categories is given.2000 Mathematics subject Classification: 16w30.Keywords: Hopf algebra, braided tensor category, double cross coproduct.

0 Introduction and Preliminaries

The double cross coproducts in braided tensor categories have been studied by Y.Bespalov,B.Drabant and author in [2] [12]. However, hitherto any examples of double cross coproducts

with both non-trivial left coaction and non-trivial right coaction in strictly braided tensor cat-egories (i.e. the braiding is not symmetric ) have not been found. Therefore Professor S.Majidasked if there is such example.

In this paper we first give the cofactorisation theorem of Hopf algebras in braided tensorcategories. Using the cofactorisation theorem and Sweedler four dimensional Hopf algebra, weconstruct such example.

We denote the multiplication, comutiplication, evaluation d, coevaluation b, braiding andinverse braiding by

,

,

,

, and ,

respectively. For convenience, we denote the inverse of morphism f by f if f has an inverse.

This work was supported by the National Natural Science Foundation (No. 19971074)

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Since every braided tensor category is always equivalent to a strict braided tensor categoryby [12, Theorem 0.1], we can view every braided tensor category as a strict braided tensorcategory and use braiding diagrams freely.

1 The cofactorisation theorem of bialgebras in braided tensor

categories

Throughout this section, we work in braided tensor category (C, C) and assume that all Hopfalgebras and bialgebras are living in (C, C) unless otherwise stated. We give the cofactorisationtheorem of bialgebras in braided tensor categories in this section.

We first recall the double bicrossproducts in [12]. Let H and A be two bialgebras in braidedtensor categories and

: HA A , : HA H,

: A H A , : H HA

morphisms in C.

D =:

A H

A H A H

, mD =:

A H A H

A H

and D = A H , D = A H. We denote (AH, mD, D, D, D) by

A H,

which is called the double bicrossproduct of A and H.When and are trivial, we denote A

H by A H. When and are trivial,

we denote A H by A

H. We call A H a double cross product and denote it by

AH in short. We call A H a double cross coproduct.

Theorem 1.1 (Factorisation theorem) (See [9, Theorem 7.2.3]) Let X , A and H be bialge-bras or Hopf algebras. Assume thatjA and jH are bialgebra or Hopf algebra morphisms from Ato X and H to X respectively. If =: mX(jA jH) is an isomorphism from AH onto X asobjects inC, then there exist morphisms

: HA A and : HA H

such that A H becomes a bialgebra or Hopf algebra and is a bialgebra or Hopf algebraisomorphism from A H onto X.

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Let us recall Sweedlers four dimensional Hopf algebra H4. That is, H4 is a Hopf algebragenerated g and x with relations

g2 = 1, x2 = 0, xg = gx

and (x) = x1+gx, (g) = gg, (x) = 0, (g) = 1, S(x) = xg,S(g) = g. Let {e1, eg , ex, egx}

denote the dual basis of{1,g,x,gx}.

Example 2.2 LetH be Sweedlers four dimensional Hopf algebra over field k with char k =2. Let D = D(H). Thus B =: D [b] D is quasitriangular, but it is not triangular by Lemma2.1. Considering [3, Theorem 2.5], B has a quasitriangular structure RB, defined in precedingTheorem 1.4 with U = V = 1 1, and RB never is triangular . Thus (BM, C

RB) is a strictlybraided tensor category by [10, Theorem 10.4.2 (3)]. It follows from Theorem 1.4 (ii) thatD [b] D = D D for some and . Furthermore, D D is a double cross coproduct.We shall show that both left coaction and right coaction are non-trivial.

Proof.D

D D

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Let u and v denote the first term and the second term, respectively.

u =

g x egx

S

S S

S

gx ex

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=

H A H

, but

eg g D

x id D

= 0 .

Thus is not trivial.

Acknowledgement This work was supported by the National Natural Science Foundation(No. 19971074) and the fund of Hunan education committee. Authors thank the editors forvaluable suggestion and help.

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References

[1] Y. Bespalov, Cross modules and quantum groups in braided categories. Applied categoricalstructures, 5 (1997), 155204.

[2] Y. Bespalov and B.Drabant, Cross product bialgebras I, J. algebra, 219 (1999), 466505.

[3] H.X.Chen, Quasitriangular structures of bicrossed coproducts, J. Algebra, 204 (1998)504531.

[4] Y. Doi. Braided bialgebras and quadratic bialgebras. Communications in algebra, 5(1993)21, 17311749.

[5] Y. Doi and M.Takeuchi, Multiplication algebra by two-cocycle - the quantum version ,Communications in algebra, 14 (1994)22, 57155731.

[6] V. G. Drinfeld, Quantum groups, in Proceedings International Congress of Mathemati-cians, August 3-11, 1986, Berkeley, CA pp. 798820, Amer. Math. Soc., Providence, RI,1987.

[7] C. Kassel. Quantum Groups. Graduate Texts in Mathematics 155, Springer-Verlag, 1995.

[8] S. Majid, Algebras and Hopf algebras in braided categories, Lecture notes in pure andapplied mathematics advances in Hopf algebras, Vol. 158, edited by J. Bergen and S.Montgomery, Marcel Dekker, New York, 1994, 55105.

[9] S. Majid, Foundations of Quantum Group Theory, Cambridge University Press, Cam-bridge, 1995.

[10] S. Montgomery, Hopf Algebras and Their Actions on Rings. CBMS Number 82, AMS,Providence, RI, 1993.

[11] M.E.Sweedler, Hopf Algebras, Benjamin, New York, 1969.

[12] Shouchuan Zhang, Hui-Xiang Chen, The double bicrossproducts in braided tensor cate-gories, Communications in Algebra, 29(2001)1, p3166.

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