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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:z9491@yahoo.com.cn
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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