Guohua LIU Xiaofan ZHAO

1 Introduction

Let V be a finite-dimensional vector space over a field k of characteristic 0.Then for any positive integer m,the symmetric group Smacts on V?mvia the twist map and the Lie algebra gl(V)acts on V?mvia its comultiplication.Schur’s double centralizer theorem originally established a correspondence between the above representations,which stated that Smand U(gl(V))are mutual centralizers in EndkV?m.Berele and Regev[1]generalized this result to the Lie superalgebra pl(V),where V is a Z2-graded vector space,Jimbo[2]stated a similar result for Uq(sl(2)),and Kirillov and Reshetikhin[3]for Uq(su(2)).Fischman[4]used purely Hopf algebraic methods to give a short proof of both these situations.In 1994,Cohen,Fischman and Westreich[5]considered the situation of triangular Hopf algebras.

In[6],the authors introduced and studied almost-triangular Hopf algebras as a generalization of triangular Hopf algebras.Naturally,this paper is devoted to establishing the Cohen-Fischman-Westreich’s double centralizer theorem for triangular Hopf algebras(see[5])in the setting of almost-triangular Hopf algebras.

This paper is organized as follows.In Section 2,we recall some definitions and results about quasi-triangular Hopf algebras and R-Lie algebras.In Section 3,we introduce the definition of the R-universal enveloping algebra of an R-Lie algebra in the setting of almost-triangular Hopf algebras,which generalizes the corresponding results in the setting of triangular Hopf algebras.In the final section,we establish the Cohen-Fischman-Westreich’s double centralizer theorem for almost-triangular Hopf algebras(see Theorem 4.2).

2 Preliminaries

Throughout this paper,k is a fixed field.Unless otherwise stated,all vector spaces,algebras,coalgebras,maps and unadorned tensor products are over k.For a coalgebra C,we denote its comultiplication by Δ(c)=c(1)? c(2),?c ∈ C and for a left C-comodule(M,?),we denote its coaction by ?(m)=m[?1]? m[0],?m ∈ M,where the summation symbols are omitted.We refer to[7]for the Hopf algebras theory.

Let H be a bialgebra and A be a left H-module algebra.The smash product AH of A and H is defined as follows:For all a,b∈A and h,g∈H,

(i)as k-spaces,AH=A?H,

(ii)multiplication is given by

Note that AH is an algebra with the unit 1A1H.

Recall from[8]that a quasi-triangular Hopf algebra is a pair(H,R),where H is a Hopf algebra and R=R1?R2∈H?H(where the summation symbols are also omitted)satisfying the following conditions(with r=R):

(1)R is invertible,

(2)RΔ(h)=Δop(h)R for all h∈H,

(3)(Δ?id)(R)=R1?r1?R2r2,

(4)(id?Δ)(R)=R1r1?r2?R2.It is easy to get that

Remark 2.1(1)If the antipode of H is bijective,then(S?S)(R)=R.

(2)(H,R)is triangular if R1r2?R2r1=1?1.

(3)(H,R)is almost-triangular if R1r2?R2r1∈C(H?H)=C(H)?C(H),the center of H?H(see[6]).

LetHM denote the category of the left H module category.For each V∈HM,EndkV∈HM,where for each f∈EndkV and h∈H,

Moreover,if V,W∈HM,then V?W∈M,where for each v?w∈V?W and h∈H,

The tensor algebra of V,T(V)is an H module algebra.ThenHM is a monoidal category.When(H,R)is quasi-triangular,the categoryHM is a braided category with the braiding ψV,W:V?W→W?V given by for any V,W∈HM,and v∈V,w∈W.

Let(H,R)be a quasi-triangular Hopf algebra.Recall from[5]that an R-Lie algebra is an object L∈HM together with an H-morphism[,]R:L?L→L satisfying

(i)R-anticommutativity:[l1,l2]R= ?[R2·l2,R1·l1]R;

(ii)R-Jacobi identity:

for all l1,l2,l3∈ L,where{l1?l2?l3}R=[l1,[l2,l3]R]R,S312= ψ12?ψ23,S231= ψ23?ψ12,ψ23(l1?l2?l3)=l1?(R2·l3)?(R1·l2)and ψ12(l1?l2?l3)=R2·l2?R1·l1?l3.

Note that[,]Rbeing an H-module homomorphism means that for all h∈H and l1,l2∈L,

3 R-universal Enveloping Hopf Algebras

In this section,we introduce the definition of the R-universal enveloping algebra of an RLie algebra(see[5])in the setting of almost-triangular Hopf algebras,which generalizes the corresponding results in the setting of triangular Hopf algebras.

Let(H,R)be a triangular Hopf algebra and A be any left H-module algebra.In[5],the authors derived an R-Lie algebra denoted by A?from A by defining an inner R-Lie product

for any a,b∈A.

However,if(H,R)is an almost-triangular Hopf algebra,A?is not necessarily an R-Lie algebra.In the following,we will discuss the condition under which A?is an R-Lie algebra.Unless otherwise stated,we always let(H,R)denote an almost-triangular Hopf algebra.

Proposition 3.1Let(A,·)be a leftH-module algebra satisfyingR1r2·a?R2r1·b=a?bfor alla,b∈A.Then(A,[,]R)is anR-Lie algebra.

ProofIt is easy to get that[,]Rsatisfies the R-anticommutativity.Indeed,for any a,b∈A,

In order to check the R-Jacobi identity,we have the following computations:For any a,b,c∈A,

and

Hence

Example 3.1Let(H,R)be a triangular Hopf algebra,and then any left H-module algebra A satisfies R1r2·a?R2r1·b=a?b for any a,b∈ A.So A?is an R-Lie algebra with[,]R.

Example 3.2For any Hopf algebra H,H is a left H-module algebra via the adjoint action,i.e.,h?g=h(1)gS(h(2))for all h,g∈H.If(H,R)is almost-triangular,then H?is an R-Lie algebra with[,]R.

ProofBy Proposition 3.1,we just need to show that

for all h,g∈H.For this,we compute

Example 3.3Let(H,R)be an almost-triangular Hopf algebra with a bijective antipode.If V is a finite-dimensional left H-module satisfying R1r2·v1?R2r1·v2=v1?v2for any v1,v2∈V,then EndkV is a left H-module algebra satisfying

for any f1,f2∈ EndkV,where(h·f)(v)=h(1)·f(S(h(2))·v)for any h ∈ H,f ∈ EndkV and v∈V.Therefore,EndkV?is an R-Lie algebra with[,]R.

ProofIt is easy to check that EndkV is a left H-module algebra.We just prove the identity

for any f1,f2∈EndkV.Indeed,for any v1,v2∈V,we have the following computations:

where the fourth identity holds because(S?S)(R)=R and the fifth holds because S(R1)·v1?R2·v2=R2·v1?R1·v2.So from Proposition 3.1,EndkV?is an R-Lie algebra with[,]R.

Example 3.4Let(H,R)be an almost-triangular Hopf algebra and V be a finite-dimensional left H-module such that the representation of H on V,

is a surjection.Then EndkV is a left H-module algebra satisfying

where h·f= πV(h(1))fπV(S(h(2))),πV:H → EndkV,defined by

is an algebra homomorphism.Hence EndkV?is an R-Lie algebra with[,]R.

ProofFor any f,f'∈ EndkV,since πVis a surjection,there exist h,h' ∈ H such that πV(h)=f and πV(h')=f'.Then we have

Hence EndkV?is an R-Lie algebra with[,]R.

Remark 3.1(i)If V is a simple H-module,then πVis a surjection.

(ii)Let V be a semi-simple H-module,i.e.,V=⊕···⊕,where for any i,j=1,···,s,ViVjas H-modules when ij,and Viare simple H-modules.If k1= ···=ks=1,then πVis a surjection.

Definition 3.1Let(L,[,]R)be anR-Lie algebra satisfying

AnR-universal enveloping algebra ofLis a pair(U,u),whereUis an associative leftH-module algebra such that

u:L→ U?is anR-Lie homomorphism,and the following holds:For any associativeH-module algebraAsatisfyingR1r2·a?R2r1·b=a?b?a,b∈ A,and anyR-Lie homomorphismf:L→ A?,there exists a uniqueH-module algebra homomorphismg:U→ A,such thatg?u=f.

Proposition 3.2LetAbe a leftH-module algebra such that for alla,b∈ A,R1r2·a?R2r1·b=a?b.Then the mapu:A?→U(A?)is an injection.

ProofClearly the identity map from A?to A?is an R-Lie map.By the universality of U(A?),there exists a unique H-module algebra homomorphism g:U(A?)→ A?such that g?u=id.Hence u is an injection.

Proposition 3.3Let(L,[,]R)be anR-Lie algebra satisfyingR1r2·l1?R2r1·l2=l1?l2for alll1,l2∈L.ThenLhas anR-universal enveloping algebra(U(L)=T(L)/I,u),whereIis the ideal ofT(L)generated by

andu:L → T(L)/Iis the canonical map:l→ l+I=l.

ProofThe proof is similar to the one in the setting of triangular Hopf algebras in[5].

From now on,we write[,]for[,]R.

Proposition 3.4LetLbe anR-Lie algebra satisfyingR1r2·l1?R2r1·l2=l1?l2for alll1,l2∈L.Then there exists anH-module algebra homomorphism

ProofDefine f:L⊕L→U(L)?U(L)by

Next we show that f is an R-Lie homomorphism.Obviously,f is an H-module homomorphism.It suffices to show that

Recall that the multiplication in U(L)?U(L)is

Then we have

So f is an R-Lie map.

Now by the universal property of U(L⊕L),there exists an H-module algebra homomorphism g:U(L⊕L)→U(L)?U(L).

Theorem 3.1LetLbe anR-Lie algebra satisfyingR1r2·l1?R2r1·l2=l1?l2for alll1,l2∈L.ThenU(L)in Proposition3.4is a Hopf algebra in the categoryHMwith

for alll∈Land,t∈ U(L).

ProofAnalogous to the proof in the case of triangular Hopf algebras([5,Theorem 2.6]).

Next we give an application of Theorem 3.1.Let V be a finite-dimensional left H-module such that for any v1,v2∈ V,R1r2·v1?R2r1·v2=v1?v2.So from Example 3.4 and Theorem 3.1,U(EndkV?)is a Hopf algebra in the categoryHM,which implies that U(EndkV?)H is a Radford’s biproduct.In the following,we will discuss when U(EndkV?)H is an almosttriangular Hopf algebra.

Theorem 3.2Let(H,R)be an almost-triangular Hopf algebra andVbe a finite-dimensional leftH-module such that for anyv1,v2∈ V,R1r2·v1?R2r1·v2=v1?v2.Thenis an almost-triangular Hopf algebra if and only ifR1r2?

ProofDenote U(EndkV?)H by B and(1R1)? (1R2)byIt is easy to get that

(ΔB?id)()=andNext we show that(B,R)is almost cocommutative if and only if R1r2?R2r1·f=1?f for any f∈EndkV.For this,on the one hand,for any f?h∈B,we have

On the other hand,

Then from the above computations,we get that(B,R)is almost cocommutative if and only if

which is equivalent to R1r2?R2r1·f=1?f.

Finally,we check thatbelongs to the center of B?B if and only if R1r2?R2r1·f=1?f,and R1r2·f?R2r1=f?1 for any f∈EndkV.Indeed,we have the following computations:

and

Thus it is not hard to get the conclusion.So we complete the proof.

Remark 3.2If V is a finite-dimensional left H-module such that the representation of H on V is a surjection,then R1r2?R2r1·f=1?f,and R1r2·f?R2r1=f?1 for any f ∈ EndkV.

4 Cohen-Fischman-Westreich’s Double Centralizer Theorem in the Setting of Almost-Triangular Hopf Algebras

In this section,we always let(H,R)be an almost-triangular Hopf algebra and V be a finitedimensional left H-module such that for any v1,v2∈ V,R1r2·v1?R2r1·v2=v1?v2.In Section 3,we have already showed that U(EndkV?)H is a Radford’s biproduct.

In the following,we always denote U(EndkV?)H by B.Obviously,V is a left B-module via(fh)·v=f(h·v)for any f∈ EndkV,h∈H and v∈V.So we have a representation of B on V ρ :B → EndkV given by

Clearly,ρ is a surjection.The representation ρ induces a representation ρmon Vmas follows:

for any b∈B.

Notation 4.1(i)For any b ∈ B,denote ρ(b)by b.So

(ii)For any h ∈ H,it is easy to get thatSo denoteandby

Now we consider the symmetric group Sm.Define a representation φ :kSm→ EndkV?mby

The action of kSmon EndkV?mis given by

In the following lemma,we have repeated occurrences of R denoted by R1,···,Rj,where R=Rifor all i.For convenience,we shall writefor an empty word andfor 1.

Lemma 4.1Let(H,R)be an almost-triangular Hopf algebra andVbe a finite-dimensional leftH-module such that for anyv1,v2∈ V,R1r2·v1?R2r1·v2=v1?v2.Then for anyf ∈ (EndkV)?,we have

(i)

(ii)

Proof

Since

it suffices to check that·f.Indeed,we have

Hence with the same idea of Lemma 3.7 in[5],we can obtain our lemma.

Theorem 4.1Let(H,R)be an almost Hopf algebra andVa finite-dimensional vector space over a fieldkof characteristic0.IfVis a leftH-module such that for anyv1,v2∈Vandf ∈ EndkV,R1r2·v1?R2r1·v2=v1?v2,R1r2?R2r1·f=1?f,andR1r2·f ?R2r1=f?1,then we have

(i)Endφ(kSm)V?m= ρm(B);

(ii)Endρm(B)V?m= φ(kSm).

Proof(i)First we show that ρm(B)? (EndkV?m)kSm.Indeed,for any b∈ B,we have

Next we claim that Endφ(kSm)V?m=(EndkV?m)kSm.On the one hand,for any f1? ···?fm∈ Endφ(kSm)V?mand v1,···,vm∈ V,we have

which means f1? ···? fm∈ (EndkV?m)kSm.So Endφ(kSm)V?m? (EndkV?m)kSm.On the other hand,for any f1? ···?fm∈ (EndkV?m)kSm,we compute

which means f1? ···?fm∈ Endφ(kSm)V?m.So(EndkV?m)kSm? Endφ(kSm)V?m.Therefore Endφ(kSm)V?m=(EndkV?m)kSm.

Since there exists a trace 1 element in EndkV?m,we have(EndkV?m)kSm=t·(EndkV?m),whereThus,to show(i),it suffices to show that ρm(B) ? t·(EndkV?m)which follows as in[5].

(ii)Follows,as in[4].

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