A Note of Yetter-Drinfeld Hopf Algebras

Journal of Modern Mathematics Frontier Volume 3 Issue 3, September 2014 doi: 10.14355/jmmf.2014.0303.02

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A Note of Yetter-Drinfeld Hopf Algebras Yanhua Wang School of Mathematics, Shanghai University of Finacce and Economics Shanghai, 200433, China yhw@mail.shufe.edu.cn Abstract

in Theorem 4.

This note gives a property of cohomology of Yetter-Drinfeld Hopf algebras.

In the following, k will be a field. All algebras and coalgebras are over k. All unadorned ⊗ are taken over k. For modules, comodules and Hopf algebras see [9], [10] and [11].

Keywords Hopf Algebras; Yetter-Drinfeld Module; Yetter-Drinfeld Hopf Algebras; Cohomology

Introduction Yetter-Drinfeld modules were introduced by Yetter in [1] under the name of "crossed bimodule". A YetterDrinfeld module over Hopf algebra H is a k-linear space V which is a left H-module, a left H-comodule and satisfies a certain compatibility condition. In general, a Yetter-Drinfeld Hopf algebra is a Hopf algebra in the category of Yetter-Drinfeld module over H. In some sense, Yetter-Drinfeld Hopf algebras are generalizations of Hopf algebras.

Yetter-Drinfeld Hopf algebra plays an important role in the classification of Hopf algebras. 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-Drinfel'd Hopf algebra over the other factors [2]. Subsequently, Schauenburg proved that the category of YetterDrinfel'd module over H was equivalent to the category of left module over Drinfel'd double, and also to the category of Hopf module over H [3]; Sommerhauser studied Yetter-Drinfel'd Hopf algebra over groups of prime order [4].

On the other hand, some conclusions of Hopf algebras can be applied to Yetter-Drinfeld Hopf algebras. For example: Doi considered the Hopf module theory and the trace of Yetter-Drinfeld Hopf algebras in [5] and [6] respectively; Scharfschwerdt proved the Nichols Zoeller theorem for Yetter-Drinfeld Hopf algebras in [7], and Wang gave the freedom of Yetter-Drinfeld Hopf algebras in [8], and so on.

In this note, we give a property about cohomology of Yetter-Drinfeld Hopf algebra, i.e., n

HH Ae (A, M)  Ext nA (k, G(M)), n ≥ 0,

Preliminaries Let (A, m, u) be an algebra. The arrow “ → ” denotes left module action. Let (C, ∆, ε ) be a coalgebra, the left C-comodule map is denoted as

ρ: V → C ⊗ V: v  ∑ v-1 ⊗ v0 .

Let ( H , m, u , ∆, ε , S ) be a Hopf algebra with antipode S. Recall the definition of Yetter-Drinfeld Hopf module and Yetter-Drinfeld Hopf algebras from [10].

Definition1. A left Yetter-Drinfel'd module over H is a k-vector space V which is both a left H-module and left H-comodule and satisfies the compatibility condition -1 0 -1 0 ∑ (h → v) ⊗ (h → v) = ∑ h1 v Sh 3 ⊗ h 2 → v

for all h ∈ H, v ∈ V. The category of left Yetter-Drinfeld module over H is denoted by

H HYD

.

Definition2. We say A is a Yetter-Drinfeld Hopf algebra or Hopf algebra in HHYD if A is an algebra and coalgebra satisfying (1)—(6) (1) A is a left H-module algebra, i.e.,

h → (ab)= ∑ (h1 → a)(h 2 → b), h → 1=ε (h)1.

(2) A is a left H-comodule algebra, i.e.,

ρ (ab)= ∑ (ab)-1 ⊗ (ab)0 = ∑ a -1b-1 ⊗ a 0 b0 , ρ (1)=1 ⊗ 1.

(3) A is a left H-module coalgebra, i.e.,

∆(h → a)= ∑ (h1 → a1 ) ⊗ (h 2 → a 2 ),

ε (h → a)=ε (h)ε (a).

(4) A is a left H-comodule coalgebra, i.e.,

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Journal of Modern Mathematics Frontier Volume 3 Issue 3, September 2014

that we have a morphism of left (A ⊗ A)τ -module

-1 0 0 -1 -1 0 0 ∑ a ⊗ (a )1 ⊗ (a ) 2 = ∑ a1 a 2 ⊗ a1 ⊗ a 2 , -1 0 ∑ a ε (a )=ε (a)1H .

g: (A ⊗ A)τ ⊗ N → I, (a ⊗ b)n  (a ⊗ b)f(n),

(5) ∆, ε are algebra maps in HHYD i.e.,

a, b ∈ A, n ∈ N. Since I is an injective object in M , there exists a morphism (A ⊗ A)τ

for

∆(ab)= ∑ a1 (a 2-1 → b1 ) ⊗ a 20 b 2 ∆(1)=1 ⊗ 1, ε (ab)=ε (a)ε (b), ε (1)=1 ⊗ 1 (6) There exists a linear map S: A → A in

H HYD

that ∑ S(a1 )a2 =uε (a )= ∑ a1S(a2 ) .

such that g ′  j=g. Define

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].

Let A be a Yetter-Drinfeld Hopf algebra over Hopf H. Then (A ⊗ A)τ becomes an algebra with multiplication

(a ⊗ b)(c ⊗ d)= ∑ a(b-1 → c) ⊗ b0 d. It is easy to know that (A ⊗ A)τ is free over A, see [11].

τ

(A ⊗ A)

AM

f ′: N ′ → Res(I), n ′  g ′((1 ⊗ 1) ⊗ n ′).

Now it is easy to check f ′  i=f and this shows that Res(I) is an injective object in A M . This complete the poof.

Define a functor G to be

Main Results

Denote by

g ′: (A ⊗ A)τ ⊗ N ′ → I

such

(A ⊗ A)τ

M → A M.

Explicitly, let M be an A-bimodule, then G(M)=M as kspace and the left A-module structure is given by

am= ∑ a1 (a 2-1 → m)S(a 20 ) , a ∈ A, m ∈ M.

the category of left A-module and

M the category of left (A ⊗ A)τ -module. Note

that (A ⊗ A)τ is considered as a braided algebra as above. Since ∆: A → (A ⊗ A)τ is an algebra map, there is a restriction functor

Res:

G=Res  F: A M A →

(7)

The main theorem in this section is as follows: Theorem4. Let (A, m, u, ∆, ε , S) be a Yetter-Drinfeld Hopf algebra and M be a A-bimodule. Then there exists an isomorphism n

(A ⊗ A)τ

HH A (A, M)  Ext nA (k, G(M)), n ≥ 0, e

M → A M.

More precisely, if M is a (A ⊗ A)τ -module, then Res(M)=M as space, and its left A-action is given by

am= ∑ (a1 ⊗ a 2 )m , a ∈ A, m ∈ M. We have the following conclusion.

Proposition 3. The functor Res:

where k is viewed as the trivial A-module via the counit ε .

Proof: First we show that there exists a natural isomorphism

Hom Ae (A, _)  Hom A (k, G(_)) (A ⊗ A)τ

M → A M is

In fact, for each A-module M, we have

exact and it preserves injective objects.

Hom Ae (A, M)={m ∈ M| am=ma, for all a ∈ A}

and

Proof: The exactness of Res is obvious by the definition of left A-module. Let I be an injective object in (A ⊗ A)τ M .

Hom A (k, G(M))={m ∈ M| am=ε (a)m for all a ∈ A\}.

We

To show that Res(I) be an injective objective in A M , take any morphism i: N → N ′ and a morphism f: N → Res(I) in A M . We claim that there exists some morphism f ′: N → Res(I) such that f ′  i=f , then we are

deduce Hom Ae (A, _)  Hom A (k, G(_)) by (7). Thus, for n ≥ 1 , we have

HH nA (A,_)=Ext nA (A,_) e

e

n

=R Hom A (A,_) e

done. In fact, (A ⊗ A)τ is a free over A, then it is a left

=R n(Hom A (k,_)  G)

A-module. So we have a monomorphism of left (A ⊗ A)τ -module:

=R n n(Hom A (k,_))  G

j=(A ⊗ A)τ ⊗ i: (A ⊗ A)τ ⊗ N → (A ⊗ A)τ ⊗ N ′

=Ext nA (k, G(_)).

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Note

n

=Ext nA (k,_)  G

Journal of Modern Mathematics Frontier Volume 3 Issue 3, September 2014

This complete the proof.

groups of prime order, Springer, L.N.M. 1789, (2002).

Sweedler M. E., Hopf algebras, Benjamin, New York, (1969).

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www.sjmmf.org

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Yetter-Drinfel'd

modules, Journal of Algebra, 169, 874-890,(1994).

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108, 261-290,(1990).

Yanhua Wang was born in Jilin Province, Dec. 1975. She received the Ph.D degree in Pure mathematics from University of Science and Technology of China , Hefei, China, in 2004.

She is currently an associate Professor in the School of Mathematics, Shanghai University of Finance and Economics, Shanghai, China. She has been a visiting research scholar in the Department of Mathematics, University of Washington, USA from 2012 to 2013. Her current research interests include Hopf algebras, homological algebra and quantum group.

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