Let H be a Hopf algebra. Schauenburg[1] obtained a braided monoidal category equivalence between the category of right-right Yetter-Drinfeld modules over H and the category of two-sided two-cosided Hopf modules over H under some suitable assumption. This yields new sources of braiding by which one can obtain the solutions to the Yang-Baxter equation, which plays a fundamental role in various areas of mathematics[23].

In 1997, Doi[4] studied the duality of any finite-dimensional Hopf modules in the left-left Yetter-Drinfeld where L denotes any ordinary Hopf algebra over the ground field k with a bijective antipode.

The most well-known examples of Hopf algebras are the linear spans of (arbitrary) groups. Dually, also the vector space of linear functionals on a finite group carries the structure of a Hopf algebra. In the case of quasigroups (nonassociative groups), however, it is no longer a Hopf algebra, but more generally, a Hopf quasigroup[510], which is a specific case of the notion of unital coassociative bialgebra[11].

Motivated by these notions and structures, this paper aims to construct Sweedler’s dual of infinite-dimensional Hopf algebras in

Throughout this paper, let k be a fixed field. We will work over k. Let C be a coalgebra with a coproduct Δ. We will use Heyneman-Sweedler’s notation[12], Δ(c)=∑c1⊗c2 for all c∈C, for coproduct.

1 Preliminaries

Recall from Ref.[5] that a Hopf coquasigroup is a unital associative algebra H，armed with three linear maps: Δ:H→H⊗H，ε:H→K and S:H→H satisfying the following equations for all a,b∈H:

Δ(ab)=Δ(a)Δ(b)

ε(ab)=ε(a)ε(b)

(id⊗ε)Δ(a)=a=(ε⊗id)Δ(a)

∑S(a1)a21⊗a22=1⊗a=∑a1S(a21)⊗a22

∑a11⊗S(a12)a2=a⊗1=∑a11⊗a12S(a2)

Recall from Ref.[6], the authors gave the notion of a left H-quasimodule over a Hopf quasigroup H. Duality, a left H-quasicomodule over a Hopf coquasigroup H is a vector space M with a linear map ρ:M→H⊗M, where ρ(m)=∑m-1⊗m0such that ∑ε(m-1)m0=m and

∑S(m-1)m0-1⊗m00=∑m-1S(m0-1)⊗m00=1⊗m

for all m∈M.

Moreover, the authors studied the notion of the left-left Yetter-Drinfeld quasimodule M=(M,·,ρ) over a Hopf quasigroup H.

Duality, a left-left Yetter-Drinfeld quasicomodule M=(M,·,ρ) over a Hopf coquasigroup H is a left H-module (M,·) and a left H-quasicomodule (M,·) satisfying the following equations:

∑(a1·m)-1a2⊗(a1·m)0=∑a1m-1⊗a2·m0

∑a1·m⊗a21⊗a22=∑a11·m⊗a12⊗a2

∑a1⊗a21·m⊗a22=∑a11⊗a12·m⊗a2

for all a∈H,m∈M.

Remark that the first equation is equivalent to the following formula:

ρ(a·m)=∑a11m-1S(a2)⊗a12·m0

YDQCM to denote the category of the left-left Yetter-Drinfeld quasicomodules over a Hopf coquasigroup H. Moreover, if we assume that M is an ordinary left H-comodule, we say that M is a left-left Yetter-Drinfeld module over H. Obviously, the left-left Yetter-Drinfeld modules with the obvious morphisms is a subcategory of HHYDQCM. We denote it by HHYDCM.

Note that if the antipode S of Hopf quasigroup H is bijective, then the is a braided monoidal category with a “pre-braiding” defined as

τ:M⊗N→N⊗M, τ(m⊗n)=∑m-1·n⊗m0

τ-1:N⊗M→M⊗N, τ-1(n⊗m)=∑m0⊗S-1(m-1)·n

for any m∈M and n∈N.

One can check the following lemmas and Corollary 1.

Lemma 1 Let H be a Hopf coquasigroup. Then, is a monoidal category.

Lemma 2 Let H be a Hopf coquasigroup with a bijective antipode S. Then, the monoidal category with the pre-braiding defined above is a braided monoidal category if and only if the following identity holds:

∑m-11·n⊗m-12·p⊗m0=∑m-1·n⊗m0-1·p⊗m00

YDQCM, m∈M, n∈N and p∈P, this category will be denoted as

Corollary 1 Let H be a Hopf coquasigroup with a bijective antipode S. If the following equations hold:

∑m-11·n⊗m-12⊗m0=∑m-1·n⊗m0-1⊗m00

∑m-11⊗m-12·n⊗m0=∑m-1⊗m0-1·n⊗m00

YDQCM, m∈M,n∈N, then (HHYDQCM,⊗,k) is a braided monoidal category, and we denote it as

Let H be a Hopf coquasigroup with a bijective antipode S. Under the hypotheses of the above results, we have the relationship:

In what follows, let L denote a Hopf coquasigroup with a bijective antipode SL. Let H be a Hopf algebra in i.e., explicitly, it is both a L-algebra and a L-coalgebra with comultiplication Δ and counit ε, and the following identities hold:

Δ(xy)=∑x1(x2-1·y1)⊗x20y2,Δ(1)=1⊗1

ε(xy)=ε(x)ε(y),ε(1H)=1

ρH(xy)=∑(xy)-1⊗(xy)0=

∑x-1y-1⊗x0y0,ρH(1H)=1L⊗1H

∑x-1⊗x01⊗x02=∑x1-1x2-1⊗x10⊗x20

∑x-1εH(x0)=εH(x)1

l·(xy)=∑(l1·x)(l2·y),l·1H=ε(l)1H

Δ(l·x)=∑(l1·x1)⊗(l2·x2),ε(l·x)=ε(l)ε(x)

SH(xy)=∑((S(x))-1·SH(y))(S(x))0=

∑(x-1·S(y))S(x0),S(1)=1

SH(xy)=∑((S(x))-1·SH(y))(S(x))0=

∑(x-1·S(y))S(x0),S(1)=1

for any x,y∈H and l∈L.

2 A Generalization of Sweedler’s Dual of Hopf Algebras

In this section, let H be a Hopf coquasigroup with a bijective antipode S, and B an infinite-dimensional Hopf algebra in

Let (A,mA,μA) be an associative algebra. Then, we have coalgebra A0 given in Ref.[13] as

A0={f∈A*|Kerf⊃an ideal of A of cofinite dimension}

Let (B,mB,μB,DB,εB) be a bialgebra in Recall that B0 is the subspace of all b*∈B* vanishing on some cofinite ideal I of B. Let i:B*⊗B*→(B⊗B)* be the natural embedding, defined as (i(f⊗g))(a⊗b)=f(a)g(b) for f,g∈B* and a,b∈B. For all f∈B*, the following statements are equivalent:

dim(fB)<∞,dim(B⇀fB)<∞

For any f∈B* and a,b∈B, we define (a⇀f)(b)=f(ba) and (fa)(b)=f(ab). This defines a B-B bimodule structure on B*.

We consider the action of H on B* given by (h·f)(b)=f(S(h) ·b) and the quasicoaction of H on B* defined by ρ(f)(b)=S-1(b(-1))⊗f(b0) for all h∈H, b∈B and f∈B*.

Let A, B be algebras in Then, we have the braided tensor product algebra A⊗B with the product (x⊗y)(a⊗b)=∑x(y(-1)·a)⊗y0b for all x,a∈A and y,b∈B.

It is not difficult for one to check the following two lemmas.

Lemma 3 The action :B*⊗B→B is a left H-linear and the action ⇀:B⊗B*→B is a left Hcop-linear.

Lemma 4 B0 is an H-submodule of (B*,

Proposition 1 B0 is a subalgebra of (B*,

Proof By Lemma 3, for any f, g∈B* and a, b∈B, we obtain

((fg)a)(b)=(fg)(ab)=(f⊗g)Δ(ab)=

f(a1(a2(-1)·b1)g(a20b2)=

f(a2(-1)2·[(S-1(a2(-1)1)·a1)b1])g(a20b2)=

(S-1(a2(-1)2)·f)[(S-1(a2(-1)1)·a1)b1]×

g(a20b2)=[(S-1(a2(-1)2)·f)

(S-1(a2(-1)1)·a1)](b1)(ga20)(b2)=

Δ*[(S-1(a2(-1))·(fa1)⊗ga20)](b)

Thus,

(fg)B⊆Δ*[H·(fB)⊗gB]⊆Δ*[(H·f)B)⊗gB]

Combining f∈B0 with Lemma 4, we can conclude that H·f∈B0. Moreover, since f,g∈B0, the left-hand side of the above containment is finite dimensional. Hence, fg∈B0. Finally, it is easy to check that

Lemma 5 We have that i○ τ:(B*)op⊗(B*)op→(B⊗B)*op is an algebra map in

Proof Applying the quasicoaction of H on B*, the proof is complete.

Now, we obtain the main result of this paper which gives a characterization of Sweedler’s dual of Hopf algebras

Theorem 1 Let H be any Hopf coquasigroup with a bijective antipode S. If (B,mB,μB, ΔB, εB) be a Hopf algebra in with antipode SB, then

Proof According to Ref.[13], we check that

1) B0 is an H-subquasicomodule of B*.

2) Observe that ○ i○ τ:B0⊗B0→B*. It is morphism is. Thus, (B0,(mB0)op, is an algebra in HM.

3) Observe that (ΔB0)op is the composite map is a coalgebra in HM.

4) (ΔB0)op:(B0)op→(B0)op⊗(B0)op is an algebra map.

6)

In the setting of Hopf coquasigroups, the notion of the left H-module is exactly the same as that for ordinary Hopf algebras since it only depends on the algebra structure of H. Thus, the proof of these assertions is either trivial or will become trivial after acquainting the Hopf coquasigroup calculus developed above.

References

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范畴上Hopf代数的Sweedler对偶

张 涛 王栓宏

(东南大学数学学院, 南京 211189)

摘要：首先，给出了Hopf余拟群H上的左-左Yetter-Drinfeld拟余模M=(M,·,ρ)的概念，其为Hopf代数上的左-左Yetter-Drinfeld模结构的推广.其次，介绍了辫子张量范畴的定义并且给出其具体的结构映射.最后，讨论辫子张量范畴上的无限维Hopf代数Sweedler的对偶问题. 证明了如果(B,mB,μB, ΔB, εB)是上有对极SB的Hopf代数， 那么是上有对极的Hopf代数，从而推广了Hopf代数上的相应结果.

关键词：Hopf(余)拟群；Yetter-Drinfeld拟(余)模；辫子张量范畴；对偶

DOI:10.3969/j.issn.1003-7985.2020.03.016

Received 2019-06-21，Revised 2020-01-20.

Biographies：Zhang Tao (1990—), male, Ph.D. candidate; Wang Shuanhong(corresponding author), male, doctor, professor, Shuanwang@seu.edu.cn.

Foundation item：The National Natural Science Foundation of China (No.11371088, 11571173, 11871144).

Citation：Zhang Tao, Wang Shuanhong. Sweedler’s dual of Hopf algebras in of Southeast University (English Edition),2020,36(3):364366．DOI:10.3969/j.issn.1003-7985.2020.03.016.

中图分类号：O153.5