CHEN Quan-guo,TANG Jian-gang

(School of Mathematics and Statistics,Yili Normal University,Yining 835000,China)

Group Twisted Tensor Biproducts over Hopf Group Coalgebras

CHEN Quan-guo,TANG Jian-gang

(School of Mathematics and Statistics,Yili Normal University,Yining 835000,China)

In this paper,we introduce the concept of a group twisted tensor biproduct and give the necessary and sufficient conditions for the new object to be a Hopf group coalgebra.

Hopf group coalgebra;twisted tensor biproduct;comodule coalgebra

§1.Introduction

As a generalization of ordinary Hopf algebras,Hopf group-algebras appeared in the work of Turaev[1]on homotopy quantum f i eld theories.A purely algebraic study of Hopf groupcoalgebras can be found in the references[29].

As a generalization of Majid’s double cross products,a general product AT?RB between two bialgebras A and B connected via a twisted map R:B?A→A?B and a cotwisted map T:A?B→B?A was introduced in Caenepeel et al[10].The product construction AT?RB is equipped with both smash product constructionand smash coproduct construction A×TB on A?B.In particular,the authors derived in Caenepeel et al[10,Theorem4.5]necessary and sufficient conditions for ATRB to be a bialgebra.Recently,by weakening the condition that A is a bialgebra replaced by that A is both an algebra and a coalgebra(but not necessarily bialgebra),Ma Tian shui and Wang Shuan hong[11]introduced a twisted tensor biproduct denoted by ATRB,generalizing Radford’s bismash products in Radford[12].

The aim of this paper is to give a general version of the twisted tensor biproducts(called group twisted tensor biproduct).We give the necessary and sufficient conditions for the new object to be a Hopf group coalgebra.

The article is organized as follows.

In Section 2,we recall the def i nitions and some of the basic properties of Hopf group coalgebras,group(co)module(co)algebras,respectively.

In Section 3,we introduce the group twisted tensor biproduct ARTB={ARTBα}α∈π, where π is a discrete group and B is a semi-Hopf π-coalgebra,A is only an algebra and a coalgebra connected by a twisted map R={Rα:Bα?A→A?Bα}α∈πand a cotwisted map T={Tα:A?Bα→Bα?A}.We f i nd the necessary and sufficient conditions for the new object ARTB to be a Hopf π-coalgebra(see Theorem 3.1).

§2.Preliminaries

Throughout,k is a f i xed f i eld and π is a discrete group with unit e.Unless otherwise stated, all vector spaces are over k and all maps are k-linear.

2.1π-Coalgebras

A π-coalgebra is a family of k-spaces B={Bα}α∈πtogether with a family of k-linear maps Δ={Δα,β:Bαβ→Bα?Bβ}α,β∈π(called a comultiplication)and a k-linear map ε:Be→k (called a counit)such that Δ is coassociative in the sense that

for any α,β,γ∈π and

for all α∈π.

Following the Sweedler’s notation for π-coalgebras,for any α,β∈π and b∈Bαβ,one write

2.2Hopf π-Coalgebras

A semi-Hopf π-coalgebra is a π-coalgebra B=({Bα}α∈π,Δ={Δα,β},ε)such that the following datas hold.

Each Bαis an algebra with multiplication mαand unit 1α∈Bα,for all α,β∈π,Δα,βand ε:Be→k are algebra maps.

A semi-Hopf π-coalgebra B=({Bα,mα,1α}α∈π,Δ={Δα,β},ε)is called a Hopf π-coalgebra, if there exists a family of k-linear maps S={Sα:Bα→Bα?1}α∈π(called an antipode)such that

2.3π-B-Comodule Coalgebras

Let B=({Bα}α∈π,Δ={Δα,β},ε)be a π-coalgebra and(V,ΔV,εV)a coalgebra.V is called a left π-comodule coalgebra,if there exists a family of maps ρV={:V→Bα?V}α∈π, which will be called a comodulelike structure and denoted by(v)=v(?1,α)?v(0,α),satisfying the following conditions

(i)For any α,β∈π and v∈V,we have

(ii)For any α∈π and v∈V,

(ii)V is counitary in the sense that,for any α∈π and v∈V,

2.4π-B-Module Algebras

Let B=({Bα,mα,1α}α∈π,Δ={Δα,β},ε)be a semi-Hopf π-coalgebra and A an algebra with the unit 1A.A is called a left π-B-module algebra,if the following conditions hold

(1)A is a left Bα-module,for each α∈π;

(2)b·(aa′)=(b(1,α)·a)(b(2,β)·a′),for all b∈Bαβand a,a′∈A;

(3)b·1A=ε(b)1A,for all b∈Be.

§3.Group Twisted Tensor Biproducts

In this section,we shall introduce the concept of a group twisted tensor biproducts and give the necessary and sufficient conditions for the new object to be a Hopf π-coalgebra.

3.1Group Twisted Tensor Products

Let A be an algebra with unit 1Aand B={Bα,mα,1α}α∈πa family of algebras.Suppose that R={Rα:Bα?A→A?Bα}α∈πis a family of linear maps.The A#RB is def i ned to be a family of vector spaces{A?Bα}α∈πwith the product given by

where aRα?bRα=arα?brα=Rα(b?a),for all b,b′∈Bαand a,a′∈A.We say that A#RB is a π-twisted tensor product,if each A#RBαis an associative algebra with the unit 1A#1α.In the case,the map R is called a twisted map.

Proposition 3.1With the notation as above.Then A#RB is a π-twisted tensor product if and only if the following conditions hold

for all a,a′∈A,b,b′∈Bα.

ProofStraightforward.

3.2Group Twisted Tensor Coproducts

Let(A,ΔA,εA)be a coalgebra and B=({Bα}α∈π,Δ={Δα,β},ε)a π-coalgebra.Given a family of linear maps T={Tα:A?Bα→Bα?A}.Then the π-twisted tensor coproduct A×TB={A?Bα}α∈πhas the coproduct given by

where bTα?aTα=btα?atα=Tα(a?b),for all b∈Bαand a∈A.We say that A×TB is a π-twisted tensor coproduct,if A×TB is a π-coalgebra with the counit εA?ε.The map T is called a cotwisted map.

Proposition 3.2With the notation as above.Then A×TB is a π-twisted tensor coproduct if and only if the following conditions hold,for all α,β∈π

3.3Group Twisted Tensor Biproducts

Theorem 3.1Let B=({Bα,mα,1α}α∈π,Δ={Δα,β},ε,S={Sα})be a Hopf πcoalgebra.Let A be both an algebra and a coalgebra(but not necessarily bialgebra)so that there exists a linear map SA:A→A satisfying SA(a1)a2=1AεA(a)and a1SA(a2)=1AεA(a). Then the following statements are equivalent

1)The conditions(C1)～(C8)such that,for all a,a′∈A

(C7)aRαβ1?bRαβ(1,α)Tα?aRαβ2Tα?bRαβ(2,β)=a1Rα?b(1,α)TαRα1αtα?1ATαa2tαrβ?b(2,β)rβ,for all a∈A,b∈Bαβ;

(C8)εRe=ε?εAand εAis an algebra map. 2)(ATRB={ATRBα}α∈πεˉ,Sˉ)is a Hopf π-coalgebra,where the multiplication,εˉeandSˉ are given as

In this case,we call ATRB a π-twisted tensor biproduct.

Proof1)?2)By(C1),we have that ATRB is a family of associative algebras.By (C2),we know thatˉΔ=ˉΔα,βis coassociative andˉε is the counit.

In what follows,we prove thatis an algebra homomorphism.Indeed,let T=t=U= u=s and R=r=P=p,we compute,for all a,a′∈A,b,b′∈Bαβ,

From the condition(C8),we get thatˉε is an algebra homomorphism.For all a∈A,b∈Beand α∈π,we compute

So we prove

Similarly,it follows that

Therefore,we conclude that(ATRB={ATRBα}α∈π,ˉΔ,ˉε,ˉS)is a Hopf π-coalgebra.

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2)?1)Straightforward.

Corollary 3.1Let B be a Hopf π-coalgebra.Let A be both an algebra and a coalgebra (but not necessarily bialgebra).Suppose that A is a left π-B-comodule coalgebra.Then the following are equivalent

1)For all a,a′∈A and b∈Bαβ,the conditions(A1)～(A5)below hold

2)(A#RB,ˉΔ,ˉε,ˉS)is a Hopf π-coalgebra with the multiplication,the coproductˉΔ,the counit ˉε and the antipodeˉS given by

ProofLet the cotwisted map T be de fi ned by Tα(a?b)=a(?1,α)b?a(0,α),for any a∈A and b∈Bα.By Theorem 3.1,we can get the corollary.

Corollary 3.2Let B be a Hopf π-coalgebra.Let A be both an algebra and a coalgebra (but not necessarily bialgebra).Suppose that A is a left π-B-module algebra.Then the following are equivalent

1)(A×TB={A×TBα},={Δˉα,β},εˉ,Sˉ={Sˉα})is a Hopf π-coalgebra,where the multiplication,,εˉeandSˉ are given as

2)The conditions(B1)～(B7)hold

(B1)A×TB is a π-twisted tensor coproduct;

(B2)T(1A?1α)=1α?1A,Δ(1A)=1A?1A;

(B3)(aa′)1?1αTα?(aa′)2Tα=a1(1αTα(1.e)·a′1)?1αTα(2,α)1αtα?a2Tαa′2tα;

(B4)bTα?aTα=1αTαbtα?aTα1Atα,for all a∈A,b∈Bα;

(B5)(b(1,α)b′(1,α))Tα?1ATα?b(2,β)b′(2,β)=b(1,α)Tαb′(1,α)tα?1ATα(b(2,β)(1,e)·1Atα)?b(2,β)(2,β)b′(2,β),for all b,b′∈Bαβ;

(B6)(b(1,e)·a)1?b(2,αβ)(1,α)Tα?(b(1,e)·a)2Tα?b(2,αβ)(2,β)=(b(1,α)Tα(1,e)·a1)?b(1,α)Tα(2,α)1αtα?1ATα(b(2,β)(1,e)·a2tα)?b(2,β)(2,β),for all a∈A and b∈Bαβ;

(B7)(εA?ε)Re=ε?εAand εAis an algebra map. 3)The conditions(C1)～(C5)hold

(C1)A×TB is a π-twisted tensor coproduct;

(C2)T(1A?1α)=1α?1A,Δ(1A)=1A?1A;

(C3)ΔA(a(b·a′))=a1(b(1,e)Te·a′1)?a2Te(b(2,e)·a′2),for all a,a′∈A and b∈Be;

(C4)(b(2,α)b′)Tα?(a(b(1,e)·a′))Tα=b(1,α)Tαb′tα?aTα(b(2,e)·a′tα),for all a,a′∈A and b,b′∈Bα;

(C5)For all a∈A and b∈Be,εA(b·a)=ε(b)εA(a)and εAis an algebra map.

ProofLet the twisted map R be de fi ned by Rα(b?a)=b(1,e)·a?b(2,α)for all a∈A and b∈Bα.By Theorem 3.1,it is obvious that 1)?2).

2)?3)We observe that,from(B3)and(B5),we have the following equations

Now we compute

and so the condition(C3)is proved.Also,

and so we get the condition(C4).

3)?2)It is easy to have equations(B4),(B5),Eqs(3.10),(3.11),(3.13)and

Next,we check(B6)as follows,for all b∈Bαβand a∈A,

This completes the proof of the corollary.

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tion:16W30

CLC number:O153.3Document code:A

1002–0462(2014)02–0274–09

date:2012-11-30

Supported by the Fund of the Key Disciplines of Xinjiang Uygur Autonomous Region (2012ZDXK03)

Biography:CHEN Quan-guo(1980-),male,native of Shangqiu,Henan,an associate professor of Yili Normal University,Ph.D.,engages in Hopf algebras.