Hopf crossed products were introduced independently by Yukio et al. [1] and Blattner et al. [2] as a Hopf algebraic generalization of group crossed products. In particular, a Hopf crossed product is, in fact, always a Hopf cleft extension, provided that the cocycle that appeared in a Hopf crossed product is convolution-invertible [3-5].

Hopf group-algebras were related to homotopy quantum field theories, which are generalizations of ordinary topological quantum field theories[3,6-8]. In 2007, Wang et al. [9-11] introduced group smash products of Hopf group-algebras. Group crossed products of Hopf group-coalgebras were introduced [12-13]. Other related works can be found in Refs.[14-17].

In this article, we introduce and study the notions of a group crossed product and a group cleft extension. We then characterize group crossed products by the group cleft extension. Finally, we prove the equivalences of the group crossed products for the Hopf group-algebras.

1 Group Cleft Extensions and Existence of Group Crossed Products

Definition 1 Let A=({Aα}α∈π,Δ,ε) be a Hopf π-algebra with the bijective antipode S and J as algebra. We say that A acts weakly on J if there exists a family of maps:

a⊗xa⇀αx, ∀α∈π, a∈Aα,x∈J, such that

1) 1α⇀x=x, ∀x∈J; α∈π;

2) a⇀α(xy)=(a(1,α)⇀αx)(a(2,α)⇀αy), ∀a∈Aα, x,y∈J；

3) a⇀α1J=εα(a)1J, ∀x∈J.

Furthermore, if J is an Aα module for each α∈π and satisfies 2) and 3), we call J a left π-A-module-like algebra.

Definition 2 Let A=({Aα}α∈π,Δ,ε) be a Hopf π-algebra and J a left π-A-module-like algebra. Let χα,β:Aα#Aβ→J be a family of k-linear maps and suppose that χ is an invertible map. Suppose that J acts weakly on each Aα with α∈π. For any α∈π, there is a π-crossed product J#χAα with the multiplication given by (x#αa)(y#βb)=x(a(1,α)⇀αy)χα,β(a(2,α),b(1,β))#αβa(3,α)b(2,β), for all a,b∈Aα, Aβ, x,y∈J, α,β∈π, and the unit is 1J#1α.

Proposition 1 With the above notations, J#χAα is a Hopf π-crossed product if and only if the following conditions hold: ∀a∈Aα,b∈Aβ,c∈Aγ,∀α,β∈π, and x,y∈J.

χα,β(a,1β)=χα,γ(a,1γ)=εα(a)1J

(1)

χα,β(a(1,α),b(1,β))χαβ,γ(a(2,α)b(2,β),c)=

(a(1,α)⇀αχβ,γ(b(1,β),c(1,γ)))χα,βγ(a(2,α),b(2,β)c(2,γ))

(2)

χα,β(a(1,α),b(1,β))(a(2,α)b(2,β)⇀αβx)=

a(1,α)⇀α(b(1,β)⇀βy)χα,β(a(2,α),b(2,β))

(3)

Proposition 2 If is a family of associative algebras.

Remark 1 1) If π=1, the Hopf π-crossed product is then the ordinary Hopf crossed product.

2) If we take χα,β(a,b)=εα(a)εβ(b)1J, ∀α,β∈π, a∈Aα, b∈Aβ, the Hopf π-crossed product becomes the Hopf π-smash product.

Let A be a Hopf algebra. For any α∈π, denote δα as the one-dimensional linear space generated by α. Then we have a Hopf group algebra H={Hα=A⊗δα}α∈π with the structure (a⊗α)(1,α)⊗(a⊗α)(2,α)=a1⊗α⊗a2⊗α, εα(a⊗α)=ε(a), Sα(a⊗α)=S(a)⊗α-1.

If J#σA is a crossed product with σ: A⊗A→J. Define

χα: Hα⊗Hβ→A, χα(a⊗α,b⊗β)=σ(a,b) for all x∈J, we then have the crossed product

Definition 3 Let J be a left π-Aα-module-like algebra.

1) We say that H⊂J is a π-Aα-extension if J is a right π-Aα-comodule algebra with a family of k-linear maps ρ={ρα:J→J⊗Aα},

JcoAα={x∈J|ρα(x)=x⊗1α∈J⊗Aα, ∃x∈J, ∀α∈π}

which is called a π-subalgebra of the right π-co-invariants.

2) A π-Aα-extension H⊂J is a π-Aα-cleft if there exists a family of right π-Aα-comodule maps γ={γα:Aα→J}α∈π such that γ is convolution-invertible in the sense that there exists a family of maps satisfying

εα(a)1J ∀a∈Aα, α∈π

Lemma 1 Let H⊂J be a π-Aα-cleft extension with a right π-Aα-comodule structure map: ρ={ρα:J→J⊗Aα} via xx(0,0)⊗x(0,α) for α∈π and a π-Aα-cleft structure map: γ={γα:Aα→J}α∈π such that γα(1Aα)=1J with We then have

Proof First, observe that since ρ is an algebra map, ρα°γ(α)-1 is the inverse of ρα°γα=(γα⊗id)°Δα. Let ∀x∈J. Then, [(ρα°γα)*θ](x)=1J⊗1Aα. Thus, θ is a right inverse of ρα°γα, and so, by the uniqueness of the inverse.

As for (L2), we compute This finishes the proof.

Proposition 3 Let H⊂J be a π-Aα-cleft via γ={γα:Aα→J}α∈π such that γα(1Aα)=1J with α∈π. Then, there is a Hopf π-crossed product with a weak action of Aα on J given by

and a family of convolution-invertible maps χ={χα,β:Aα⊗Aβ→J}α,β∈π given by

∀a∈Aα, ∀b∈Aβ

Furthermore, there is an algebra isomorphism given by x⊗axγα(a) with α∈π such that Φ={Φα}α∈π is both a left π-J-module and a right π-Aα-comodule map, where the right π-Aα-comodule structure map of is given by x#a→x#a(1,α)⊗a(2,α).

Proof First, we compute for x∈J, a∈Aα,

⊗Sα-1(a(3,α-1)))=a⇀αx⊗1Aα∈J⊗Aα

and thus, a⇀αx∈H=JcoAα. Furthermore, it is easy to see that Definition 1 2) and 3) hold.

Similarly, we can prove that χ={χα}α∈π has values in A. In fact, ∀a,b∈Aα,Aβ,

ρα(χα,β(a,b))=ραγα(a(1,α))ραγβ(b(1,β))·

Now, for α∈π, we define by h

It is easy to show that Ψα is the inverse of Φα with α∈π. Furthermore, Φ is an algebra-like map: Φα(x#a)Φβ(y#b)=Φα,β((x#a)(y#b)). Therefore, we have

Finally, it is easy to check that Φ={Φα}α∈π is a left π-J-module-like map and is a right π-Aα-comodule map.

Proposition 4 Let be a Hopf π-crossed product and define γ={γα:Aα→J#Aα}α∈π by γα(a)=1J#a. Then γ={γα}α∈π is a family of convolution invertible maps with the inverse

Proof Let It is then straightforward to verify that ν is a left inverse for γ, and now we have να-1(a(1,α-1))γα-1(a(2,α-1))=εα-1(a)1J#1Aα.

To check that ν is a right inverse for γ is more complicated. By a computation similar to the above, we have

a(5,α))]χα(α(2,α),S(a(3,α)))#1Aα

(4)

and hence, ν is a right inverse for γ if and only if

(5)

Since χ={χα,β:Aα⊗Aβ→J} is invertible, Eq. (2) gives

·

(a(3,α),b(3,β)c(3,γ))=a⇀αχβ,γ(b,c)

(6)

for any a∈Aα,b∈Aβ,c∈Aγ.

Let a∈Aα act on the identity We have

εα(a)εβ(b)εγ(c)1J

(7)

Hence, from Eq. (7), we obtain

(8)

We may now verify Eq. (6) using Eq. (8):

By Proposition 3 and Proposition 4, we can now get the main result of this section as follows.

Theorem 1 With the above notations, a π-Aα-extension H⊂J={Hα⊂Jα}α∈π is a π-Aα-cleft if and only if

2 Equivalences of Group Crossed Products

In this section, we will study the equivalences of group crossed products in the setting of Hopf group-coalgebras. Let J be a Hopf π-algebra, Aα a family of coalgebras A={Aα,mα,1Aα}α∈π over k, and γ={γα:Aα→J}α∈π a family of convolution-invertible linear maps. Define and the weak action of Aα on J by for any x∈J and ∀a,b∈Aα,Aβ.

Lemma 2 Let be a Hopf π-crossed algebra. Then, χγαμα=(χγα)μα and ⇀γαμα=(⇀γα)μα where γ={γα:Aα→J}α∈π and μ={μα:Aα→J}α∈π are a family of convolution-invertible linear maps.

The proof is clear.

Theorem 2 Let J be a Hopf π-algebra, Aα a family of coalgebras A={Aα,mα,1Aα}α∈π, and γ={γα:Aα→J}α∈π a family of convolution-invertible linear maps. If χ={χα:Aα⊗Aα→J}α∈π is a family of k-linear maps, we then have the following assertions with the above notations χγα for any α,β∈π:

2) χ satisfies Eq. (1) if and only if χγ satisfies Eq. (1);

3) (χ,⇀) satisfies Eq. (2) if and only if (χγ,⇀γ) satisfies Eq. (2);

4) If(χ,⇀) satisfies Eq. (2), (χ,⇀) satisfies Eq. (3) if and only if (χγ,⇀γ) satisfies Eq. (3);

5) is a Hopf π-crossed algebra, and they are isomorphic.

Proof 1) Define by x⊗a→xγα(a(1,α))⊗a(2,α), ∀x,y∈J, ∀a, b∈Aα,Aβ, Φα,β((x⊗a)(y⊗b))=x(a(1,α)⇀γαy)χγα(a(2,α),b(1,β))γα,β(a(3,α)b(2,β))⊗a(4,α)b(3,β)=Φα(x⊗a)Φβ(y⊗b). Φ is clearly bijective, since

3) If (χ,⇀) satisfies Eq. (3), then

(a(1,α))⇀γα(b(1,β)⇀γβx))χγαα,β(a(2,α),b(2,β))=

γα(a(1,α))(a(2,α)⇀γβ(b(1,β)))χα,β(a(3,α),b(2,β))

·

Conversely, we get it from Lemma 2.

4) If (χ,⇀) satisfies Eq. (2) and Eq. (3), then, for a∈Aα, b∈Aβ, c∈Aγ，

γα(a(1,α))(a(2,α)⇀[γβ(b(1,β))(b(2,β)⇀γγ(c(1,γ)))·

·

γα(a(4,α))(a(5,α)⇀γβγ(b(5,β)c(4,γ)))χα,βγ(a(5,α),b(6,β)c(5,γ))·

2) and 5) of Theorem 2 are clearly proved.

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Hopf群代数上的交叉积

游弥漫1 鹿道伟2 王栓宏3

(1华北水利水电大学数学与统计学院, 郑州 450045 )

(2济宁学院数学系, 曲阜 273155 )

(3东南大学数学学院, 南京 211189 )

摘要：首先给出了Hopf群代数的群交叉积定义，并给出了群交叉积是群代数的充分必要条件.引入了Hopf群代数的cleft扩张理论，并证明了Hopf群代数的交叉积与cleft扩张等价.然后，给出了2个Hopf群交叉积等价的充分必要条件.最后,结合Hopf群交叉积与cleft扩张的等价理论得到,群文叉积一般由2-余循环构造，作为代数同构于带有卷积可逆映射的2-余循环的群交叉积.一般2-余循环的余单位性质等价于带有卷积可逆映射的2-余循环余单位性质, 通常意义下的2-余循环和弱作用结合条件等价于带有卷积可逆映射的2-余循环及其弱作用结合条件；同时得到,由一般2-余循环构造的Hopf π-交叉积代数同构于带有卷积可逆映射的2-余循环构造的Hopf π-交叉积代数.

关键词：Hopf π-代数; cleft 扩张理论; π-余模像代数; 群交叉积

DOI:10.3969/j.issn.1003-7985.2021.03.015

Received 2021-05-10，Revised 2021-07-02.

Biographies：You Miman(1984—)，female，doctor, lecturer; Lu Daowei(corresponding author), male, doctor, associate professor, ludaowei620@126.com.

Foundation item：The National Natural Science Foundation of China (No. 11871144, 11901240).

Citation：You Miman, Lu Daowei, Wang Shuanhong.Crossed products for Hopf group-algebras[J]．Journal of Southeast University (English Edition),2021,37(3):339-342．DOI:10.3969/j.issn.1003-7985.2021.03.015.

中图分类号：O153