Abstract：Some sufficient and necessary conditions are given for the equivalence between two crossed product actions of Hopf algebra H on the same linear category, and the Maschke theorem is generalized. Based on the result of the crossed product in the classic Hopf algebra theory, first, let A be a k-linear category and H be a Hopf algebra, and the two crossed products A#σH and 
σ′H are isomorphic under some conditions. Then, let A#σH be a crossed product category for a finite dimensional and semisimple Hopf algebra H. If V is a left A#σH-module and W⊆V is a submodule such that W has a complement as a left A-module, then W has a complement as a A#σH-module.
Key words：linear category; inner action; crossed product; generalized Maschke theorem
Group-graded rings and algebras have been the study focus and several fundamental results have been obtained. Cohen and Montgomery[1]obtained the duality theorem for graded algebras. In this paper, the graded algebras are treated from the point of view of Hopf algebras. A G-graded algebra A can be viewed as a kG-comodule algebra, and thus can be viewed as a kG-module algebra when G is a finite group. In fact, the concept of graded algebras can be extended to linear categories, leading to group-graded k-linear categories. Similarly, if a k-linear category is graded by a finite group G, then it can be called a kG-module category. More generally, it is natural to define the concept of the module category for any Hopf algebra H, as done in Refs.[2-4].
As a continuation of the work in Ref.[5], we provide some sufficient and necessary conditions for the equivalence between two crossed product actions of Hopf algebra H on the same linear category,and generalize the Maschke theorem.
Throughout this paper, we work over field k, and all the vector spaces, algebras and tensor product are over k.
In this section, we recall the concept of the crossed product of a linear category with a Hopf algebra H. For the crossed product action of a Hopf algebra H on an algebra, one can refer to Refs.[6-7]. If A is a linear category and x,y are objects of A, we denote yAx the space Hom(x,y), and denote A0 the class of objects.
Definition 1 The weak action of a Hopf algebra H on a linear category A is defined by the map H⊗yAx→yAx given by h⊗y fx|→h·y fx, for any x,y∈A0 and h∈H such that
h·(zgy∘y fx)=∑(h1·zgy)∘(h2·y fx)
h·1x=ε(h)1x, 1H·y fx=y fx
where y fx∈yAx,zgy∈zAy.
Definition 2 Let H be a Hopf algebra and A be a k-linear category. Assume the weak action of H on A and σ={σx}x∈A0∈Hom(H⊗H,A) with σx∈Hom(H⊗H,xAx) convolution invertible. The crossed product A#σH of A with H is a category such that (A#σH)0=A0, and for any objects x, y and z, y(A#σH)x=yAx⊗H as a vector space. Therefore, the composition of morphisms is given by
(zgy#h)∘(y fx#k)=∑zgy∘(h1·y fx)∘σx(h2,k1)#h3k2
for any h, k∈H, zgy∈zAy,y fx∈yAx.
In what follows, we assume that the morphisms of all crossed product categories are associative with identity morphisms {1x#1H}x∈A0.
Definition 3 Let H be a Hopf algebra and A be a k-linear category. Consider the crossed product A#σH. We call the action of H on A inner if there is a collection of maps u={ux}x∈A0 with ux∈Hom(H,xAx) convolution invertible such that for any y fx∈yAx and h∈H,
(1)
The above equation has an equivalent form:
∑(h1·y fx)∘ux(h2)=∑uy(h)∘y fx
(2)
Proposition 1 Let A be a k-linear category and A#σH be the crossed product such that the action of H on A is inner, via some invertible u∈Hom(H, A). Define τ={τx}x∈A0∈Hom(H⊗H, A) by
(3)
Then τ is a cocycle and A#σH≅Aτ[H], a twisted product with trivial action and a category isomorphism which is also a left A-module, right H-comodule map.
Proof Define φ:A#σH→Aτ[H]by y fx#h→∑y fxux(h1)⊗h2. It is easy to confirm that φ is a left A-module and a right H-comodule map. Then define ψ:Aτ[H]→A#σH by y fx⊗
#h2.
It is straightforward to confirm that φ and φ are inverses. We show that φ is an algebra map. For any z fy,ygx in A, and h, k∈H,
φ[(z fy#h)(ygx#
#
∑z fy(h1·ygx)σx(h2,k1)ux(h3k2)⊗h4k3=
⊗h5k3=
∑z fyuy(h1)ygxux(k1)τx(h2,k2)⊗h3k3=
⊗
⊗
φ(z fy#h)φ(ygx#k)
Since Aτ[H]≅A#σH as categories, the composition in Aτ[H] is associative and τ is a cocycle.
Now we give some necessary and sufficient conditions for the two crossed products to be isomorphic.
Theorem 1 Let A be a k-linear category and H be a Hopf algebra with two crossed product actions h⊗y fx|→h·y fx and h⊗y fx|→h▷y fx with respect to two cocycles σ, σ′:H⊗H→A, respectively. Assume that
is an isomorphism of linear category, which is also a left A-module and a right H-comodule map. Then there is a collection of invertible maps u={ux}x∈A0∈Hom(H, A) such that for any y fx∈yAx, h, k∈H,
1) φ(y fx#h)=∑y fxux(h1)#′h2;
3) 
).
Conversely given a collection of maps u={ux}x∈A0∈Hom(H, A) such that 2) and 3) hold, then the map φ in 1) is an isomorphism.
Proof Define ux∈Hom(H, xAx) by ux(h)=(id⊗ε) φ(1x⊗h) for any h∈H. Then
(id⊗ε)φ(y fx#h)= (id⊗
){(y fx⊗1)[φ(1x#h)]}=
y fx∘ux(h)
as φ is a left A-module map. Since φ is a right H-comodule map, we have
(id⊗Δ)∘φ=(φ⊗id)∘(id⊗Δ)
Apply id⊗ε⊗id to both sides of the equation. The left side becomes φ, and the right side becomes [(id⊗ε)∘φ⊗id]∘(id⊗Δ), which evaluated at y fx#h is
∑(id⊗ε)∘φ(y fx#h1)⊗h2=∑y fxux(h1)⊗h2
This proves 1).
#σ′H is an isomorphism satisfying the same hypotheses of φ, we may set vx(h)=(id⊗ε)φ-1(1x⊗h) and conclude as above that φ-1(y fx#′h)=∑y fxvx(h1)#h2. We claim that v=u-1.
1x#h= φ-1φ(1x#
#
∑ux(h1)v(h2)#h3
Applying id⊗ε to both sides, we obtain ∑ux(h1)vx(h2)=ε(h)1x. Similarly we obtain ∑vx(h1)ux(h2)=ε(h)1x, and thus v=u-1.
Now the equation φ-1[(zgy#′h)(y fx#′k)]=φ-1·(zgy#′h)φ-1(y fx#′k) becomes
#h4k3=
∑zgyvy[(h1)(h2·y fxvx(k1)]σx(h3,k2)#h4k3
Set x=y=z and f=g=1x, and apply id⊗ε to both sides, we obtain
This proves 3) after inverting vx(hk) and using v=u-1.
Again using the above equation with y=z, g=1z and k=1, and applying id⊗ε to both sides, we have
∑(h1▷y fx)vx(h2)=∑vy(h1)(h2·y fx)
Inverting v, we obtain 2).
The converse follows as in the proof of Proposition 1.
Recall from Ref.[5] that for a crossed product category A#σH, the family of maps γx: H→xAx#σxH given by h|→1x#h is invertible in Hom(H,xAx#σxH). Then by the equation
(1y#h)(y fx#1)=∑(h1·f)#h2
denoting f=f#1, we have
(4)
for any h∈H and f∈yAx.
Recall Ref.[8], let A be a k-linear category. A left module over A is a family of linear spaces M={xM}x∈A0 together with the structure maps ·:yAx⊗xM→yM satisfying
z fy·(ygx·m)=fg·m 1x·m=m
A morphism of the left A-module is a family of linear maps u={xu:xM→xN}x∈A0 satisfying
y u(f·m)=f·xu(m)
for an y f∈yAx and m∈xM.
Proposition 2 Let A#σH be a crossed product category for a finite dimensional, semisimple Hopf algebra H. If V∈A#σHM and W⊆V is a submodule such that W has a complement in AM, then W has a complement in A#σHM.
Proof Let π:V→W be an A-projection with πx:xV→xW.
for any x∈A0 and v∈xV.
First, for any f∈yAx and v∈xV
where the second identity uses Eq.(4), and the third identity is obtained by π being A-linear.
h⊗Δ(t)=∑h1⊗Δ[tε(h2)]=∑h1⊗t1h2⊗t2h3
Then for any h∈H and v∈xV.
#σH-map.
w. The proof is completed.
References：
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摘要：给出了Hopf代数与线性范畴2个不同交叉积之间等价的充要条件,并推广了Maschke定理．基于经典Hopf代数的方法,首先设A为k-线性范畴且H为Hopf代数,则2个交叉积A#σH与
σ′H在某些条件下是同构的.其次设A#σH为有限维半单Hopf代数H的交叉积范畴.若V为左A#σH-模且W⊆V为V的子模,W作为左A-模在V中有补,则W作为左A#σH-模在V中有补.
关键词：线性范畴;内作用;交叉积;广义Maschke定理
中图分类号：O153.3
Received：2014-05-05.
Biographies：Lu Daowei (1987—), male, graduate; Wang Shuanhong (corresponding < class="emphasis_italic">author
Foundation item：s：The National Natural Science Foundation of China (No.11371088), the Natural Science Foundation of Jiangsu Province (No.BK2012736), the Fundamental Research Funds for the Central Universities, the Research Innovation Program for College Graduates of Jiangsu Province (No.KYLX15_0109).
Citation：Lu Daowei, Wang Shuanhong. Equivalence of crossed product of linear categories and generalized Maschke theorem[J].Journal of Southeast University (English Edition),2016,32(2):258-260．< class="emphasis_italic">doi
doi：:10.3969/j.issn.1003-7985.2016.02.020.
doi：10.3969/j.issn.1003-7985.2016.02.020