THE MILITARU-STEFAN LIFTING THEOREM OVER WEAK HOPF ALGEBRAS

WANG Yong

(College of Information Engineering,Nanjing Xiaozhuang University,Nanjing 211171,China)

THE MILITARU-STEFAN LIFTING THEOREM OVER WEAK HOPF ALGEBRAS

WANG Yong

(College of Information Engineering,Nanjing Xiaozhuang University,Nanjing 211171,China)

The paper is concerned with extension modules for weak Hopf-Galois extensions. By using faithfully flat weak Hopf-Galois extension theory,we investigative the Militaru-Stefan lifting theorem over weak Hopf algebras,which extends the corresponding result given in[10]. Moreover,we characterizer weak stable modules by a weak cleft extension of endomorphism rings of induced modules.

weak Hopf algebra;weak Hopf-Galois extension;weak cleft extension

1 Introduction and Preliminaries

Let H be a Hopf algebra,A a faithfully flat Hopf-Galois extension over its subalgebra of coinvariants B and M a B-module.Generalizing a result due to Dade[7]on strongly graded rings,Militaru and Stefan checked the following classical result:the B-action on M can be extended to an A-action if and only if there exists a total integral and algebra map φ :H → ENDA(M ⊗BA),where ENDA(M ⊗BA),consisting of the rational space of EndA(M ⊗BA),was introduced by Ulbrich[17].Moreover,Caenepeel also studied and obtained this result using isomorphisms of small categories in[4].

The purpose of the present paper is to investigate the above result in the case of weak Hopfalgebras.But this is not a direct promotion,we give a new simple proof.

Weak bialgebras(or weak Hopfalgebras),as a generalization ofordinary bialgebras(or Hopf algebras)and groupoid algebras,were introduced by B¨ohm and Szlach´anyi in[3](see also their joint work with Nillin[2]).The main difference between ordinary and weak Hopf algebras comes from the fact that the comultiplication of the latter is no longer required to preserve the unit(equivalently,the counit is not required to be an algebra homomorphism). Consequently,there are two canonical subalgebras(HLand HR)playing the role of “noncommutative bases” in a weak Hopf algebra H.Moreover,the wellknown examples of weakHopfalgebras are groupoid algebras,face algebras and generalized Kac algebras(see[8,20]). The main motivation for studying weak Hopfalgebras comes from quantum field theory and operator algebras.It turned out that many results of classical Hopf algebra theory can be generalized to weak Hopf algebras.

This paper is organized as follows.In Section 1,we recall some basic definitions and give a summary of the fundamental properties concerning weak Hopf algebras.In Section 2,based on the work of[19],we obtain the main result of this paper by a new method,that is,the Militaru-Stefan lifting theorem over weak Hopf algebras.As an application,we check that if A/B is a weak right H-Galois extension,then the weak smash product EndB(M)#H is isomorphic to ENDA(M ⊗BA)as an algebra for any M ∈ MA,which extends Theorem 2.3 in[18],given for a finite dimensional Hopf algebra.Moreover,for any B-module M, we prove that there exists a one-to-one correspondence between all A-isomorphism classes of extensions of M to a right A-module and the conjugation classes of total integrals and algebra maps t:H → ENDA(M ⊗BA).In Section 3,under the condition “faithfully flat weak Hopf-Galois extensions”,we mainly prove that a right B-module M is weak H-stable if and only if ENDA(M ⊗BA)/EndB(M)is a weak cleft extension,which generalizes Theorem 3.6 in[15].

We always work over a fixed field k and follow Montgomery’s book[11]for terminologies on algebras,coalgebras and comodules,butomitthe usualsummation indicesand summation symbols.

In what follows,we recallsome concepts and results used in this paper.

Defi nition 1.1[2]Let H be both an algebra and a coalgebra.If H satisfies conditions (1.1)–(1.3)below,then it is called a weak bialgebra.If it satisfies conditions(1.1)–(1.4) below,then it is called a weak Hopf algebra with antipode S.

For any x,y,z ∈ H,

where Δ2=(Δ ⊗ id)Δ.

where Δ(1)=11⊗ 12.

For any weak bialgebra H,define the mapsL,R:H → H by the formulas

We have that HL=Im(L)and HR=Im(R)(see[2,5]).

By[2],the antipode S of a weak Hopf algebra H is anti-multiplicative and anticomultiplicative,that is,for any h,g ∈ H,

The unit and counit are S-invariants,that is,S(1H)=1H,ε ◦ S= ε.

H is always considered as a weak Hopf algebra.The following results(W1)− (W9)are given in[2].For any x ∈ HL,y ∈ HRand h,g ∈ H,

Let H be a weak Hopf algebra with bijective antipode S.Then it is clear that S−1is anti-multiplicative and anti-comultiplicative such that

The following results(W13)− (W14)are given in[12].

If the antipode S is bijective,then for any h ∈ H,

Defi nition 1.2[5]Let H be a weak bialgebra,and A a right H-comodule,which is also an algebra,such that

for any a,b ∈ A.In this case we call A a weak right H-comodule algebra.

Defi nition 1.3[5]Let H be a weak Hopf algebra and A a weak right H-comodule algebra.If M is both a right A-module and a right H-comodule such that for any m ∈M,a ∈ A,then M is called a weak right(A,H)-Hopf module.

Similarly,we can define the weak left right(A,H)-Hopf modules.We denote by MHAthe category of weak right(A,H)-Hopf modules,and right A-linear H-colinear maps,andAMHthe category ofweak left right(A,H)-Hopfmodules,and left A-linear right H-colinear maps.

Defi nition 1.4[12] Let H be a weak bialgebra.The algebra A is called a weak left H-module algebra if A is a left H-module via h ⊗ a → h ·a such that for any a,b ∈ A and h∈H,

Defi nition 1.5[12]Let H be a weak Hopfalgebra and A a weak left H-module algebra. A weak smash product A#H of A with H is defined on a k-vector space A ⊗HLH,where H is a left HL-module via its multiplication and A is a right HL-module via

Its multiplication is given by the familiar formula:for any a,b ∈ A and h,g ∈ H,

Then by[12],A#H is an associative algebra with unit 1A#1H.

Defi nition 1.6[1]Let H be a weak Hopf algebra and A a weak right H-comodule algebra.A map φ :H → A is called a total integral if φ is a right H-comodule map and φ(1H)=1A.

2 The Militaru-Stefan Lifting Theorem

In this section,we always assume that H is a weak Hopfalgebra with bijective antipode S and A a weak right H-comodule algebra.

Denote B=AcoH={a ∈ A|ρ(a)=a(0)⊗L(a(1))}.Then by[9,23],we know that B is a subalgebra of A,McoH={m ∈ M|ρ(m)=m(0)⊗L(m(1))}is a right B-submodule of M for any M ∈ MHA.Set

for any N ∈ MA.Then by[19],N ☒ H ∈ MHA,whose action and coaction are given by

Defi nition 2.1[9]If the given map

is a bijection,we say that A/B is a weak right H-Galois extension,where A is a left and right B-module via its multiplication.

We will write for any h ∈ H, β−1(1A☒ h)=h[1]⊗Bh[2]∈ A ⊗BA.

Lemma 2.2Let N ∈ MA.If A/B is a weak right H-Galois extension,then N ☒ H ～= N ⊗BA as weak right(A,H)-Hopfmodules,where the A-action and H-coaction on N ⊗BA are given by

for any a,b ∈ A,n ∈ N.

ProofDefine a map ϕ to be the composite

that is, ϕ(n ☒ h)=n ·h[1]⊗Bh[2].This implies ϕ is a bijection.Additionally,by Lemma 2.2 in[13],we can easily check that ϕ is both a right A-module map and a right H-comodule map.Thus N ☒ H ～=N ⊗BA as weak right(A,H)-Hopf modules.

Lemma 2.3The following assertions are equivalent.

(1)There exists a totalintegral and algebra map φ :H → A.

(2)B#H ～=A as weak right H-comodule algebras.

If these assertions hold then B is a weak left H-module algebra via the adjoint action h ·b= φ(h1)bφ(S(h2)).

ProofDefine a map τ:H → B#H,h → 1#h.For any h,g ∈ H,(1#h)(1#g)=1#hg. This implies that τ is an algebra map.Obviously, τis a total integral.Hence the map φ = λ ◦ τ:H → A is also a totalintegraland algebra map,where the map λ :B#H → A is an isomorphism of right H-comodule algebras.

Conversely,assume that there exists a totalintegraland algebra map φ :H → A.Then B is a weak left H-module algebra via the adjoint action h ·b= φ(h1)bφ(S(h2)).

In fact,since φ is a right H-comodule map,(φ ⊗ idH)Δ(1H)= ρAφ(1H),that is,φ(11)⊗12=1(0)⊗ 1(1).Hence

In view of Theorem 3.3 in[22],we know that the rest is true.

Take M,N ∈ MHA.Consider ρ(f) ∈ HomA(M,N ⊗ H)as

for any f ∈ HomA(M,N),m ∈ M,where the A-action on N ⊗ H is induced by the A-action on N.Then by[19], ρ(f)is right A-linear.In addition,by[19],we know that HomA(M,N) becomes a right HR-module via

for any f ∈ HomA(M,N)and y ∈ HR,where M is a left HR-module via

Recall from[19],we say that a map f ∈ HomA(M,N)is rationalif there is an element fi⊗ fj∈ HomA(M,N)⊗ H such that

for any m ∈ M,where Δ(1H)=11⊗ 12.Set HOMA(M,N)={f ∈ HomA(M,N)|f is rational}.Then by(2.3)and(2.6),for any f ∈ HOMA(M,N),

By[19],we know that HOMA(M,N)is a right H-comodule via(2.7),ENDA(M)= HOMA(M,M)is a weak right H-comodule algebra,ENDA(M)coH=EndHA(M),and(2.6) is equivalent to that

for any m ∈ M and f ∈ HOMA(M,N).

From(2.8),for any M ∈ MHA,we can easily check that M ∈ENDA(M)MHthe category of weak left right(ENDA(M),H)-Hopf modules,and left ENDA(M)-linear right H-colinear maps,where M is a left ENDA(M)-module via f ·m=f(m)for any f ∈ E N DA(M),m ∈ M.

Let M ∈ MHA.Consider the induction functor − ⊗HRM and the functor HOMA(M,−) between MHand MHA:

where for a right H-comodule P,it is a right HR-module via p ·y=p(0)ε(p(1)y)for any p ∈ P,y ∈ HR,M is a left HR-module via(2.5),and the A-action and H-coaction on P ⊗HRM are given by

With notation as above,the following assertion holds.

Lemma 2.4Let M ∈ MHA.Then(− ⊗HRM,HOMA(M,−))is an adjoint pair.

ProofTo show that(− ⊗HRM,HOMA(M,−))is an adjoint pair,it suffi ces to prove that HomH(P,HOMA(M,N)) ～=HomHA(P ⊗HRM,N)for any P ∈ MH,M,N ∈ MHA.

Define a map F:HomHA(P ⊗HRM,N) → HomH(P,HOMA(M,N))by

The map F is welldefined.In fact,for any f ∈ HomHA(P ⊗HRM,N),p ∈ P,m ∈ M,

that is, ρ(F(f)(p))=F(f)(p(0)) ← 11⊗ p(1)12.The right A-linearity of f implies that F(f)(p)is also a right A-linear map.Hence F(f)(p) ∈ HOMA(M,N).Moreover,in the light ofthe right H-colinearity of f,we can easily show that F(f)is also a right H-colinear map.

Now,we define a map G:HomH(P,HOMA(M,N)) → HomHA(P ⊗HRM,N)by

Obviously,G is welldefined,and F is a bijection with inverse G.Hence HomH(P,HOMA(M,N))～=HomHA(P ⊗HRM,N).

Consider H as a right H-comodule via its comultiplication,hence by(2.9),H ⊗HRM ∈MHA.Then the following assertion holds.

Lemma 2.5Let M ∈ MHA.Then H ⊗HRM ～=M ☒ H as weak right(A,H)-Hopf modules,where M ☒ H is a weak right(A,H)-Hopf module via(2.1).

ProofDefine a map

Using(W2),we can check that δis well defined.It is easy to see that δis both a right A-module map and a right H-comodule map.

In what follows,we show that δis a bijection with inverse

The map γ is well defined,since for any m ∈ M,h ∈ H,y ∈ HR,

that is,Imγ ⊆ H ⊗HRM.

Now we calculate that

for any h ⊗HRm ∈ H ⊗HRM,and

where the fifth equality follows by the fact that m(0)⊗R(m(1))=m ·1(0)⊗ S(1(1))for any m ∈ M(see[19]).Therefore H ⊗HRM ～=M ☒ H as weak right(A,H)-Hopf modules.

In what follows,we obtain the Militaru-Stefan lifting theorem over weak Hopfalgebras,

which extends Theorem 2.3 in[10].

Theorem 2.6Let A/B be a weak right H-Galois extension and A faithfully flat as a

left B-module.Assume that(M,≺)is a right B-module.Then the following assertions are equivalent.

(1)M can be extended to a right A-module.

(2)There exists a total integral and algebra map φ :H → ENDA(M ⊗BA),where

M ⊗BA is a weak right(A,H)-Hopf module via(2.2).

(3)There is a weak left H-module algebra structure on EndB(M)such that

as weak right H-comodule algebras.

Proof(1) ⇔ (2)Since A/B is a weak right H-Galois extension and A is faithfully flat as a left B-module,the functor − ⊗BA is an equivalence between MBand MHAaccording to[6].Hence we have a sequence of isomorphisms:

where the first isomorphism follows by Lemma 2.4,the second one by Lemma 2.5 and the third one by Lemma 2.2.This resulting isomorphism relates the desired A-action ↼ on M to the multiplicative total integralφ on ENDA(M ⊗BA).

In fact,the associativity and unitality of the action ↼ are equivalent to the multiplicativity and unitality of φ,respectively.Indeed,there are further similar isomorphisms:

and

They relate,respectively,

with

and

with

while

with(−) ↼ 1A:M → M;furthermore the unit of ENDA(M ⊗BA)with the identity map on M.So(1) ⇔ (2)holds.

(2)⇔ (3)Since A/B is a weak right H-Galois extension and A is faithfully flat as a left B-module,the functor − ⊗BA is an equivalence between MBand MHAaccording to[6], hence

So by Lemma 2.3,(2)⇔ (3)holds.

The following conclusion extends Theorem 3.5 in[16].

Proposition 2.7Let A/B be a weak right H-Galois extension and A faithfully flat as a left B-module.Assume that(M,≺)is a right B-module.Then the following assertions are equivalent.

(1) ι:M → M ⊗BA,m → m ⊗B1Ais a B-split monomorphism.

(2)ENDA(M ⊗BA)is a relative injective H-comodule.

ProofWe only sketch the proof.This result can be derived from the isomorphism HomH(H,ENDA(M ⊗BA)) ～=HomB(M ⊗BA,M)together with Theorem 1.7 in[1]and the observation in the proofof Theorem 2.6 about the simultaneous unitality of the corresponding morphisms κ ∈ HomB(M ⊗BA,M)and φ ∈ HomH(H,ENDA(M ⊗BA)).

Remark(1)Let A/B be a weak right H-Galois extension and A faithfully flat as a left B-module.Assume that(M,↼)is a right A-module.Then(M,↼)is also a right B-module, which can be extended to a right A-module.Therefore,by Theorem 2.6,EndB(M)#H ～= ENDA(M ⊗BA)as weak right H-comodule algebras,which extends Theorem 2.3 in[18], given for a finite dimensional Hopf algebra.

(2)By[6,21],we know that H is a weak right H-Galois extension of HL,hence,by(1), EndHL(H)#H ～=ENDH(H ⊗HLH)as algebras.In particular,if H is a finite dimensional

weak Hopfalgebra,then by Corollary 3.4 in[12],we have H#H∗～=E ndHL(H)as algebras. Then there exists an algebra isomorphism(H#H∗)#H ～=ENDH(H ⊗HLH).

Set

For any φ1,φ2∈ ΩE,if there exists ψ ∈ AutB(M)such that

for any h ∈ H,we say that φ1,φ2are conjugate,denoted by φ1～ φ2.It is obvious that ～is an equivalence relation on ΩE.We denote by ΩEthe quotient set of ΩErelative to this equivalence relation ～.

With notation as above,the following assertion holds.

Theorem 2.8Let A/B be a weak right H-Galois extension and A faithfully flat as a left B-module.Consider M as a right B-module.Then there is a bijection between all A-isomorphism classes of extensions of M to a right A-module and ΩE.

ProofBy the proof of Theorem 2.6,we know that HomH(H,ENDA(M ⊗BA)) ～= HomB(M ⊗BA,M).This isomorphism relates

with the map

where ψ ∈ AutB(M).Therefore,the bijection between ΩEand the set of extensions of M, induces a bijection between ΩEand the set of A-isomorphism classes of extensions of M.

Recall from Remark 2.8(1)in[19],we know that ENDA(A) ～=A as weak right H-comodule algebras.Hence ENDA(B ⊗BA) ～=ENDA(A) ～=A as weak right H-comodule algebras.Let M=B,then ΩE= ΩA={φ ∈ HomH(H,A)|φ is an algebra map}.At the same time,it is easy to see that equation(2.10)is replaced by the equation

where b ∈ U(B)={b ∈ B|b is invertible}.That is,for any φ1,φ2∈ ΩA,φ1,φ2are conjugate if there exists b ∈ U(B)such that for any h ∈ H,(2.11)holds.Denote by ΩAthe quotient set of ΩArelative to this conjugate relation.Then by Theorem 2.6 and Theorem 2.8,the following assertion holds.

Corollary 2.9Let A/B be a weak right H-Galois extension and A faithfully flat as a left B-module.Consider M as a right B-module.Then the following assertions are equivalent.

(1)B can be extended to a right A-module.

(2) ΩA/= ∅.

(3)There exists a weak left H-module algebra structure on B such that B#H ～=A as weak right H-comodule algebras.

Furthermore,there exists a one-to-one correspondence between the set of isomorphism classes of extensions of B and ΩA.

3 Weak Stable Modules

In this section,we always assume that H is a weak Hopfalgebra with bijective antipode S,A a weak right H-comodule algebra and B=AcoH.

Defi nition 3.1If there exists a right H-comodule map φ :H → A,called a weak cleaving map,and a map ψ :H → A that satisfy the following conditions

for any h ∈ H.Then we say that A/B is a weak cleft extension(see[14]).

Defi nition 3.2Let M be both a right B-module and a left HL-module.M is called weak H-stable if M ⊗BA and H ⊗HLM are isomorphic as right H-comodules and right B-modules,where H is a right HL-module via

for any h ∈ H,x ∈ HL,and the actions and coactions are given by

for any b ∈ B,m ⊗Ba ∈ M ⊗BA,h ⊗HLm ∈ H ⊗HLM.

Lemma 3.3Let M ∈ MHA.Then H ⊗HLM is a weak right(A,H)-Hopfmodule,where H is a right HL-module as in(3.1),M is a left HL-module via x ·m=m(0)ε(m(1)S(x))for any x ∈ HL,m ∈ M,and the A-action and H-coaction on H ⊗HLM are given by

for any h ⊗HLm ∈ H ⊗HLM,a ∈ A.

ProofThe A-action on H ⊗HLM is welldefined,since for any x ∈ HL,a ∈ A,h⊗HLm ∈H ⊗HLM,

where the third equality follows by the fact that m(0)⊗R(m(1))=m ·1(0)⊗ S(1(1))for any m ∈ M.Using(W2)and the fact that S(HL) ⊆ HR,we can easily show that the H-coaction on H ⊗HLM is also well defined.What is more,it is easy to see that H ⊗HLM is a weak right(A,H)-Hopf module.

Let M ∈ MHA.By Lemma 2.5,we know that H ⊗HRM is a weak right(A,H)-Hopf module.In view of Lemma 3.3,we obtain the following result.

Lemma 3.4Let M ∈ MHA.Then H ⊗HRM ～=H ⊗HLM as weak right(A,H)-Hopf modules.

ProofWe first have a welldefined map

In fact,for any h ⊗HLm ∈ H ⊗HLM,x ∈ HL,

where the fourth equality follows by(W3)and the fact that S(HL) ⊆ HR.And for any h ∈ H,m ∈ M,y ∈ HR,

that is,Imθ ⊆ H⊗HRM.Moreover,from(1.6),we can easily show thatθ is a right A-module map,and θ is a right H-comodule map,because

Next,we show that θis a bijection with inverse

The map ϑ is well defi ned,since for any y ∈ HR,h ⊗HLm ∈ H ⊗HLM,

and for any h ∈ H,m ∈ M,x ∈ HL,

where the second equality follows by(W9)and the fact that S(HL) ⊆ HR.This implies Imϑ ⊆ H ⊗HLM.

Now we calculate that

that is,θ is a bijection with inverse ϑ.

Therefore,H ⊗HRM ～=H ⊗HLM as weak right(A,H)-Hopf modules.

With notation as above,we obtain the following result which extends Theorem 3.6 in [15].

Theorem 3.5Let A/B be a weak right H-Galois extension and A faithfully flat as a left B-module.Let M be both a right B-module and a left HL-module.Then the following assertions are equivalent.

(1)M is weak H-stable.

(2)ENDA(M ⊗BA)/EndB(M)is a weak cleft extension.

ProofBy the proof of Theorem 2.6,we have a sequence of isomorphisms

where the second isomorphism is given by

for any f ∈ HomB(M ⊗BA,M),g ∈ HomHB(M ⊗BA,H ⊗HLM).This resulting isomorphism relates Φ ∈ HomHB(M ⊗BA,H ⊗HLM)with φ ∈ HomH(H,ENDA(M ⊗BA))which is given by

Moreover,by Theorem 2.6 and Lemma 3.4,we have the following sequence of isomorphisms

This resulting isomorphism relates Ψ ∈ HomHB(H ⊗HLM,M ⊗BA)with ψ ◦ S−1∈ HomH(H, ENDA(M ⊗BA))which is given by

Therefore, φ and ψ ◦ S−1satisfy conditions(1)and(2)in Definition 3.1 if and only if Φ is a bijection with inverse Ψ,that is,ENDA(M ⊗BA)/EndB(M)is a weak cleft extension if and only if M is weak H-stable.

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弱Hopf代数上的Militaru-Stefan提升定理

王勇
(南京晓庄学院信息工程学院,江苏 南京 211171)

本 文 研 究 了 弱Hopf-Galois扩 张 的 扩 张 模. 利 用 忠 实 平 坦 的 弱Hopf-Galois扩 张 理 论, 研 究 了弱Hopf代数上 的Militaru-Stefan提 升定理, 推广了文献[10]中的相应结果. 进一步地, 通过诱导模的自同态环的cleft扩张刻画了弱稳定模.

弱Hopf代数; 弱Hopf-Galois扩张; 弱cleft扩张

:16T05;16T15

O153.3

tion:16T05;16T15

A < class="emphasis_bold">Article ID:0255-7797(2017)02-0325-15

0255-7797(2017)02-0325-15

∗Received date:2015-09-14 Accepted date:2016-03-04

Foundation item:Supported by National Natural Science Foundation of China(11401522);Natural Science Foundation for Colleges and Universities in Jiangsu Province(13KJD110008);Postdoctoral Research Program of Zhejiang Province(BSH1402029);Qing Lan Project.

Biography:Wang Yong(1982–),male,born at Xuzhou,Jiangsu,associate professor,major in Hopf algebras.