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Abstract: The antipode of a Yetter-Drinfeld Hopf algebra is an anti-algebra and anti-coalgebra map is proved.It is also proved that the tensor algebra of Yetter-Drinfeld Hopf module is a Yetter-Drinfeld Hopf algebra.
Key words: Hopf algebra; antipode; comodule; Yetter-Drinfeld Hopf algebras
CLC number: O 154.1 Document code: A Article ID: 1000-5137(2014)05-0523-06
Corresponding author: Yanhua Wang,Associate professor,E-mail: yhw@mail.shufe.edu.cn.1 Introduction
Let H be a Hopf algebra (bialgebra),a left-left Yetter-Drinfeld module over Hopf algebra (bialgebra) H is a k-linear space V which is a left H-module,a left H-comodule and satisfies a certain compatibility condition.Yetter-Drinfeld modules were introduced by Yetter in ［1］ under the name of "crossed bimodule".Radford proved that pointed Hopf algebras can be decomposed into two tensor factors,one factor of the two factors is no longer a Hopf algebra,but a rather a Yetter-Drinfeld Hopf algebra over the other factor ［2］.Subsequently,Schauenburg proved that the category of Yetter-Drinfeld module over H was equivalent to the category of left module over Drinfeld double,and also to the category of Hopf module over H ［3］,and Sommerhauser studied Yetter-Drinfeld Hopf algebra over groups of prime order ［4］.
Some conclusions of Hopf algebras can be applied to Yetter-Drinfeld Hopf algebras.For example: Doi considered the Hopf module theory of Yetter-Drinfeld Hopf algebras in ［5］,Scharfschwerdt proved the Nichols Zoeller theorem for Yetter-Drinfeld Hopf algebras in ［6］,and Andruskiewitsch and Schneider gave the trace formula for Yetter-Drinfeld Hopf algebras in ［7］.
In this paper,we generalized the antipode properties of Hopf algebras to Yetter-Drinfeld Hopf algebras.We proved the antipode of a Yetter-Drinfeld Hopf algebra is an anti-algebra and anti-coalgebra map,see Proposition 1 and Proposition 2.We study the tensor algebra of Yetter-Drinfeld module,and show that the tensor algebra of Yetter-Drinfeld module is a Yetter-Drinfeld Hopf algebra under a tensor multiplication and a "twisted" comultiplication,see Theorem 4.
In the following,k will be a field.All algebras and coalgebras are over k.All unadorned are taken over k.
2 Preliminaries of Yetter-Drinfeld Hopf algebras
The tensor algebras of Yetter-Drinfeld module for all h∈H,v∈V.The category of left Yetter-Drinfeld module is denoted by HHYD.
We begin by recalling the notion of Yetter-Drinfeld Hopf algebras.A is a Yetter-Drinfeld bialgebra in HHYD if A is a k-algebra and a k-coalgebra with comultiplication △ and counit and the following (a1)-(a5) hold, One easily see that S is both H-linear and H-colinear.In general,Yetter-Drinfeld Hopf algebras are not ordinary Hopf algebras because the bialgebra axiom asserts that they obey (a5).However,it may happen that Yetter-Drinfeld Hopf algebras are ordinary Hopf algebras when the pre-braiding is trivial,for details see ［4］. Next,we give a basic property of Yetter-Drinfeld Hopf algebra.It is well know that the antipode of a Hopf algebra is an anti-algebra and anti-coalgebra map,see ［8-10］.This is also true for Yetter-Drinfeld Hopf algebra.The following lemma give the character.
Proposition 1 A is a Yetter-Drinfeld Hopf algebra with antipode S: A A,then S is an anti-algebra and anti-coalgebra automorphism,a,b∈A,that is The proof of S is an anti-coalgebra automorphism is similar to the proof of S is an anti-algebra automorphism.
In general,it is not always easy to verify a given map S: A A is the antipode for a Yetter-Drinfeld Hopf algebra A,but it should be simpler to check (a6) only on generators of A.Thus,it is convenient to have the following
Proposition 2 Let A be a bialgebra in HHYD and S: A A be an algebra anti-automorphism.Assume that A is generated as an algebra by a subset X of A,such that (Sid)(a)=u(a)=(idS)(a) for all a∈X.Then S is the antipode of Yetter-Drinfeld Hopf algebra A.
References:
［1］ D.N.Yetter.Quantum groups and representation of monoidal categories ［J］.Math.Proc.Cambridge Philos.Soc.,1990,108:261-290.
［2］ D.Radford.The structure of Hopf algebras with a projection ［J］.J.Algebra,1985,92:322-347.
［3］ P.Schauenburg.Hopf modules and Yetter-Drinfeld modules ［J］.J.Algebra,1994,169:874-890.
［4］ Y.Sommerhauser.Yetter-Drinfeld Hopf algebras over groups of prime order ［C］//Lecture Notes in Mathemtics,Vol.1789,Springer,2002.
［5］ Y.Doi.Hopf module in Yetter-Drinfeld categories ［J］.Comm.Algebra,1998,26(9):3057-3070.
［6］ Scharfschwerdt.The Nichols-Zoeller theorem for Hopf algebras in the category of Yetter-Drinfeld modules ［J］.Comm.Algebra,2001,29(6):2481-2487.
［7］ Y.Doi.The trace formula for braided Hopf algebras ［J］.Comm.Algebra,2000,28(4):1881-1895.
［8］ S.Dascalescu,C.Nastasescu,S.Raianu.Hopf algebra; An introduction ［M］.New York:Marcel Dekker,Inc.,2001.
［9］ S.Montgomery.Hopf algebras and their actions on rings ［C］.CBMS Regional Conf.Series in Math.82,Amer.Math.Soc.,Providence,RI 1993.
［10］ M.E.Sweedler.Hopf algebras ［M］.New York:Benjamin,1969.
(Zhenzhen Feng,Hui Yu)
Key words: Hopf algebra; antipode; comodule; Yetter-Drinfeld Hopf algebras
CLC number: O 154.1 Document code: A Article ID: 1000-5137(2014)05-0523-06
Corresponding author: Yanhua Wang,Associate professor,E-mail: yhw@mail.shufe.edu.cn.1 Introduction
Let H be a Hopf algebra (bialgebra),a left-left Yetter-Drinfeld module over Hopf algebra (bialgebra) H is a k-linear space V which is a left H-module,a left H-comodule and satisfies a certain compatibility condition.Yetter-Drinfeld modules were introduced by Yetter in ［1］ under the name of "crossed bimodule".Radford proved that pointed Hopf algebras can be decomposed into two tensor factors,one factor of the two factors is no longer a Hopf algebra,but a rather a Yetter-Drinfeld Hopf algebra over the other factor ［2］.Subsequently,Schauenburg proved that the category of Yetter-Drinfeld module over H was equivalent to the category of left module over Drinfeld double,and also to the category of Hopf module over H ［3］,and Sommerhauser studied Yetter-Drinfeld Hopf algebra over groups of prime order ［4］.
Some conclusions of Hopf algebras can be applied to Yetter-Drinfeld Hopf algebras.For example: Doi considered the Hopf module theory of Yetter-Drinfeld Hopf algebras in ［5］,Scharfschwerdt proved the Nichols Zoeller theorem for Yetter-Drinfeld Hopf algebras in ［6］,and Andruskiewitsch and Schneider gave the trace formula for Yetter-Drinfeld Hopf algebras in ［7］.
In this paper,we generalized the antipode properties of Hopf algebras to Yetter-Drinfeld Hopf algebras.We proved the antipode of a Yetter-Drinfeld Hopf algebra is an anti-algebra and anti-coalgebra map,see Proposition 1 and Proposition 2.We study the tensor algebra of Yetter-Drinfeld module,and show that the tensor algebra of Yetter-Drinfeld module is a Yetter-Drinfeld Hopf algebra under a tensor multiplication and a "twisted" comultiplication,see Theorem 4.
In the following,k will be a field.All algebras and coalgebras are over k.All unadorned are taken over k.
2 Preliminaries of Yetter-Drinfeld Hopf algebras
The tensor algebras of Yetter-Drinfeld module for all h∈H,v∈V.The category of left Yetter-Drinfeld module is denoted by HHYD.
We begin by recalling the notion of Yetter-Drinfeld Hopf algebras.A is a Yetter-Drinfeld bialgebra in HHYD if A is a k-algebra and a k-coalgebra with comultiplication △ and counit and the following (a1)-(a5) hold, One easily see that S is both H-linear and H-colinear.In general,Yetter-Drinfeld Hopf algebras are not ordinary Hopf algebras because the bialgebra axiom asserts that they obey (a5).However,it may happen that Yetter-Drinfeld Hopf algebras are ordinary Hopf algebras when the pre-braiding is trivial,for details see ［4］. Next,we give a basic property of Yetter-Drinfeld Hopf algebra.It is well know that the antipode of a Hopf algebra is an anti-algebra and anti-coalgebra map,see ［8-10］.This is also true for Yetter-Drinfeld Hopf algebra.The following lemma give the character.
Proposition 1 A is a Yetter-Drinfeld Hopf algebra with antipode S: A A,then S is an anti-algebra and anti-coalgebra automorphism,a,b∈A,that is The proof of S is an anti-coalgebra automorphism is similar to the proof of S is an anti-algebra automorphism.
In general,it is not always easy to verify a given map S: A A is the antipode for a Yetter-Drinfeld Hopf algebra A,but it should be simpler to check (a6) only on generators of A.Thus,it is convenient to have the following
Proposition 2 Let A be a bialgebra in HHYD and S: A A be an algebra anti-automorphism.Assume that A is generated as an algebra by a subset X of A,such that (Sid)(a)=u(a)=(idS)(a) for all a∈X.Then S is the antipode of Yetter-Drinfeld Hopf algebra A.
References:
［1］ D.N.Yetter.Quantum groups and representation of monoidal categories ［J］.Math.Proc.Cambridge Philos.Soc.,1990,108:261-290.
［2］ D.Radford.The structure of Hopf algebras with a projection ［J］.J.Algebra,1985,92:322-347.
［3］ P.Schauenburg.Hopf modules and Yetter-Drinfeld modules ［J］.J.Algebra,1994,169:874-890.
［4］ Y.Sommerhauser.Yetter-Drinfeld Hopf algebras over groups of prime order ［C］//Lecture Notes in Mathemtics,Vol.1789,Springer,2002.
［5］ Y.Doi.Hopf module in Yetter-Drinfeld categories ［J］.Comm.Algebra,1998,26(9):3057-3070.
［6］ Scharfschwerdt.The Nichols-Zoeller theorem for Hopf algebras in the category of Yetter-Drinfeld modules ［J］.Comm.Algebra,2001,29(6):2481-2487.
［7］ Y.Doi.The trace formula for braided Hopf algebras ［J］.Comm.Algebra,2000,28(4):1881-1895.
［8］ S.Dascalescu,C.Nastasescu,S.Raianu.Hopf algebra; An introduction ［M］.New York:Marcel Dekker,Inc.,2001.
［9］ S.Montgomery.Hopf algebras and their actions on rings ［C］.CBMS Regional Conf.Series in Math.82,Amer.Math.Soc.,Providence,RI 1993.
［10］ M.E.Sweedler.Hopf algebras ［M］.New York:Benjamin,1969.
(Zhenzhen Feng,Hui Yu)