# A Class of Weak Hopf Algebras

Authors: Dongming Cheng
Journal: International Journal of Mathematics and Mathematical Sciences
 (2010)

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## Abstract

We introduce a class of noncommutative and noncocommutative
weak Hopf algebras with infinite Ext quivers and study their structure. We decompose
them into a direct sum of two algebras. The coalgebra structures of these
weak Hopf algebras are described by their Ext quiver. The weak Hopf extension
of Hopf algebraHn
has a quotient Hopf algebra and a sub-Hopf algebra which are
isomorphic to Hn.


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## Body

## 1. Introduction

Weak Hopf algebra was introduced by Li in 1998 as a generalization of Hopf algebras [1]. It had been proved in [1, 2]; for some sorts of finite dimensional weak Hopf algebras H, the quantum quasidouble D(H) of H is quasibraided equipped with some quasi-R-matrix R. Hence R is a solution of the Quantum Yang-Baxter Equation.First two examples of noncommutative and noncocommutative weak Hopf algebras were given in [3]. Up to now, many examples of weak Hopf algebras have been found [2, 4–7]. So far, all examples of weak Hopf algebras were based on some Hopf algebras and were constructed by weak extension.In this paper, we first give a Hopf algebra, denoted byHn. By weak extension, we construct a weak Hopf algebra W(n1,n2,n3) corresponding to Hn and study their structure. W(n1,n2,n3) has a quotient Hopf algebra and a sub-Hopf algebra which are isomorphic to Hn. And as an algebra, W(n1,n2,n3) can be decomposed into a direct sum of two algebras, one of which is Hn. The coalgebra structures of these weak Hopf algebras are described by their Ext quiver [8, 9].We organize our paper as follows. In Section2, we introduce the Hopf algebra Hn. In Section 3, we define a class of weak Hopf algebras W(n1,n2,n3). In Section 4, we study the structure of W(n1,n2,n3) and decompose W(n1,n2,n3) into a direct sum of Hn and some algebra of polynomials as an algebra. We give the Ext-quiver of coalgebra of W(n1,n2,n3) and prove that W(n1,n2,n3) has a quotient Hopf algebra and a sub-Hopf algebra which are isomorphic to Hn.


## 2. A Quiver Hopf Algebra

The Hopf AlgebraF(q) is defined in [10]. Let q∈k∖0. As a k-algebra F(q) is generated by a,b, and x subject to the relations(2.1)ab=1,ba=1,xa=qax,xb=q-1bx.
The coalgebra structure of F(q) is determined by(2.2)Δ(a)=a⊗a,Δ(b)=b⊗b,Δ(x)=x⊗a+1⊗x.ε(1)=ε(a)=ε(b)=1,ε(x)=0.We generalizeF(q) to Hn, which is defined as follows. Let k be a field, q∈k∖0, i=1,2,…,n. As a k-algebra Hn is generated by K,K-1, and Xi,i=1,2,…,n subject to the relations(2.3)KK-1=1,K-1K=1,XiK=qKXi,XiK-1=q-1K-1Xi.
The coalgebra structure of Hn is determined by(2.4)Δ(K)=K⊗K,Δ(K-1)=K-1⊗K-1,Δ(Xi)=Xi⊗K+1⊗Xi,ε(K)=ε(K-1)=1,ε(Xi)=0.
The antipode S is induced by(2.5)S(K)=K-1,S(K-1)=K,S(Xi)=-K-1Xi.


## 3. A Class of Weak Hopf Algebras

In this section, we construct a class of weak Hopf algebra corresponding toHn.First recall the definition of weak Hopf algebra [1].Definition 3.1.
Ak-bialgebra H=(H,μ,η,Δ,ε) is called a weak Hopf algebra if there exists T∈Homk(H,H) such that id*T*id=id and T*id*T=T where T is called a weak antipode of H.A weak Hopf algebra is called pointed if it is pointed as a coalgebra. If a weak Hopf algebraH is pointed, then the set of all group-like elements G(H) is a regular monoid [6].Now we construct weak Hopf algebraW corresponding to Hn. The set G(W) of group-like elements of weak Hopf algebra W is a regular monoid which has generators g, g¯, 1, subject to gg¯=gg¯,g2g¯=g,g¯2g=g¯.To construct all possible weak extension we need the following discussion.Recall, for any coalgebraC, that the group-like elements in C are the set G(C)={a∈C∣a≠0andΔ(a)=a⊗a}; necessarily ε(a)=1 for a∈G(C). Note that a simple subcoalgebra D of C is one-dimensional ⇔D=ka for some a∈G(C). A coalgebra is pointed if all of its simple subcoalgebras are one-dimensional. For a,b∈G(C), the a,b-primitive elements in C are the set Pa,b(C)={c∈C∣Δ(c)=c⊗a+b⊗c}; necessarily ε(c)=0 for c∈Pa,b(C). Note that k(a-b)={l(a-b)∣l∈k}⊂Pa,b(C); an a,b-primitive element c is nontrivial if c∉k(a-b)={l(a-b)∣l∈k}. If a=b=1, the 1,1-primitives are simply called primitive; otherwise they are called skew primitive.The following result is a generalization of [11].Lemma 3.2.
LetW be the weak Hopf algebra defined above. One has
(3.1)gPa,b(W)⊆Pga,gb(W),g¯Pa,b(W)⊆Pg¯a,g¯b(W).Proof.
Letu∈Pa,b(W), then Δ(u)=u⊗a+b⊗u. Hence,
(3.2)Δ(gu)=Δ(g)Δ(u)=(g⊗g)(u⊗a+b⊗u)=gu⊗ga+gb⊗gu∈Pga,gb(W).
The second inclusion is proved similarly.Corollary.
ForW, one has
(3.3)dimPgi+1,gi(W)=dimPgi,gi-1(W),i≥2,dimPg¯i,g¯i+1(W)=dimPg¯i-1,g¯i(W),i≥2,dimPg¯,g¯2(W)=dimPgg¯,g¯(W)=dimPg,gg¯(W)=dimPg2,g(W).Proof.
We only prove the first equation. In fact, the mapφ:Pgi,gi-1(W)→Pgi+1,gi(W), u↦gu is a linear map with inverse ψ:Pgi+1,gi(W)↦Pgi,gi-1(W), v↦g¯v. Hence, Pgi,gi-1(W) and Pgi+1,gi(W) are isomorphic as vector spaces.Since all the dimensions in Corollary3.3 are same, we have the following corollary.Corollary 3.4.
One has(3.4)dimP1,g¯(W)≤dimPg,gg¯(W),dimPg,1(W)≤dimPg,gg¯(W).Proof.
The mapφ:Pg,1(W)→Pg,gg¯(W), u↦gg¯u is a linear map. If φ(u)=gg¯u=l(g-gg¯), for some l∈k, then u∈kG(W), the vector space spanned by all group-like elements, because W is graded. Hence, u=l(g-1). Therefore, the linear map φ is an injection. Consequently,
(3.5)dimP1,g¯(W)≤dimPg,gg¯(W).
The proof of the second inequality is similar.By the above discussion we know that weak Hopf algebraW is determined by P1,g¯(W), Pg,1(W), and Pg,gg¯(W). Take x1,…,xn1 to be linearly independent nontrivial elements in P1,g¯(W), and y1,…,yn2 linearly independent nontrivial elements in Pg,1(W). Let(3.6)Pg,gg¯(W)=(gg¯P1,g¯(W)+gPg,1(W))⊕V,
and z1,…,zn3 a basis of V. Then W is determined by x1,…,xn1, y1,…,yn2, z1,…,zn3.To summarize, we define weak Hopf algebraW(n1,n2,n3) corresponding to Hn as follows.Definition 3.5.
Letk be a field. For any positive integers n1,n2,n3, and nonzero element q∈k, we define W(n1,n2,n3) to be associative algebra over field k generated by 1,g,g¯,xi,yj,zk, i=1,2,…,n1,j=1,2,…,n2,k=1,2,…,n3, subject to
(3.7)gg¯=gg¯,gg¯2=g,g¯2g=g¯,(3.8)gxi=qxig,g¯xi=q-1xig¯,i=1,2,…,n1,(3.9)gyj=qyjg,g¯yj=q-1yjg¯,j=1,2,…,n2,(3.10)gzkg¯=qzk,k=1,2,…,n3.W(n1,n2,n3) can be endowed with coalgebra structure by(3.11)Δ(g)=g⊗g,(3.12)Δ(xi)=xi⊗g+1⊗xi,(3.13)Δ(yj)=yj⊗1+g¯⊗yj,(3.14)Δ(zk)=zk⊗g+gg¯⊗zk,(3.15)ε(1)=ε(g)=ε(g¯)=1,ε(xi)=0,ε(yj)=0,ε(zk)=0,
while the weak antipode T is induced by(3.16)T(1)=1,T(g)=g¯,T(g¯)=g,(3.17)T(xi)=-xig¯,T(yj)=-gyj,T(zk)=-zkg¯,Theorem 3.6.
For any positive integersn1,n2,n3, W(n1,n2,n3) is a weak Hopf algebra.Proof.
First we must check that the coproductΔ is an algebra map. It suffices to prove that Δ preserves the relations (3.7)–(3.10). It is easy to see that Δ preserves the relations (3.7). And
(3.18)Δ(gxi)=(g⊗g)(xi⊗g+1⊗xi)=gxi⊗g2+g⊗gxi=(qxig)⊗g2+g⊗(qxig)=q(xi⊗g+1⊗xi)(g⊗g)=Δ(qxig),Δ(gyj)=(g⊗g)(yj⊗1+g¯⊗yj)=gyj⊗g+gg¯⊗gyj=(qyjg)⊗g+gg¯⊗(qyjg)=q(yj⊗1+g¯⊗yj)(g⊗g)=Δ(qyjg),Δ(gzkg¯)=(g⊗g)(zk⊗g+gg¯⊗zk)(g¯⊗g¯)=gzkg¯⊗ggg¯+ggg¯g¯⊗gzkg¯=(qzk)⊗g+gg¯⊗(qzk)=Δ(qzk).
Next we prove that T is the weak antipode. It suffices to prove that for each generator g,g¯,xi,yj,zk, the action of T*id*T is the same as that of T, and the action of id*T*id is the same as that of id.
Since
(3.19)(Δ⊗id)Δ(xi)=(Δ⊗id)(xi⊗g+1⊗xi)=(xi⊗g+1⊗xi)⊗g+1⊗1⊗xi=xi⊗g⊗g+1⊗xi⊗g+1⊗1⊗xi,
we get
(3.20)(id*T*id)(xi)=xig¯g+(-xig¯)g+xi=xi=id(xi),(T*id*T)(xi)=(-xig¯)gg¯+xig¯+(-xig¯)=-xi(g¯gg¯)=-xig¯=T(xi).
Since
(3.21)(Δ⊗id)Δ(yj)=(Δ⊗id)(yj⊗1+g¯⊗yj)=(yj⊗1+g¯⊗yj)⊗1+g¯⊗g¯⊗yj=yj⊗1⊗1+g¯⊗yj⊗1+g¯⊗g¯⊗yj,
it follows that
(3.22)(id*T*id)(yj)=yj+g¯(-gyj)+g¯gyj=yj=id(yj),(T*id*T)(yj)=(-gyj)+gyj+gg¯(-gyj)=-gyj=T(yj).
Since
(3.23)(Δ⊗id)Δ(zk)=(Δ⊗id)(zk⊗g+gg¯⊗zk)=(zk⊗g+gg¯⊗zk)⊗g+gg¯⊗gg¯⊗zk=zk⊗g⊗g+gg¯⊗zk⊗g+gg¯⊗gg¯⊗zk,zkgg¯=q-1gzkg¯gg¯=q-1gzkg¯=zk,gg¯zk=q-1gg¯gzkg¯=q-1gzkg¯=zk,
we get
(3.24)(id*T*id)(zk)=zkgg¯+gg¯(-zkg¯)g+ggg¯g¯zk=zk-qg¯zkg+gg¯zk=zk-zk+zk=zk=id(zk),(T*id*T)(zk)=(-zkg¯)gg¯+gg¯zkg¯+ggg¯g¯(-zkg¯)=(-zkg¯)+zkg¯+(-zkg¯)=-zkg¯=T(zk).


## 4. The Structure ofW(n1,n2,n3)

In this section we study the algebra and coalgebra structure ofW(n1,n2,n3).It is easy to prove that the elementsgg¯ and 1-gg¯ are a pair of orthogonal central idempotents. Set W1=W(n1,n2,n3)gg¯, W2=W(n1,n2,n3)(1-gg¯). We have the following.Theorem 4.1.
W(n1,n2,n3) can be written as a direct sum of two-sided ideals W(n1,n2,n3)=W1⨁W2. And one has the following.(1)
As an algebra,W1 is isomorphic to Hn, where n=n1+n2+n3.(2)
As an algebra,W2 is isomorphic to the free associative algebra k〈Y1,…,Yt〉 of t generators, where t=n1+n2.Proof.
(1)  Since gg¯ and 1-gg¯ are a pair of orthogonal central idempotents,
(4.1)W(n1,n2,n3)=W(n1,n2,n3)gg¯⊕W(n1,n2,n3)(1-gg¯)=W1⊕W2.
The isomorphism W1→Hn is induced by xigg¯↦Xi, yjgg¯↦Xn1+j, zkgg¯↦Xn1+n2+k, gg¯↦1, g2g¯↦K.
(2)  Note that zk(1-gg¯)=0 and xi(1-gg¯)yj(1-gg¯)=yj(1-gg¯)xi(1-gg¯). Since xi(1-gg¯),yj(1-gg¯) are generators of W2, the isomorphism W2→k〈Y1,…,Yt〉 is defined by (1-g2)↦1, xi(1-g2)↦Yi, yj(1-g2)↦Yn1+j.A weak Hopf idealJ of a weak Hopf algebra H is a bi-ideal such that T(J)⊂J, where T is the weak antipode of H. It is easy to see that H/J has a natural structure of a weak Hopf algebra.Theorem 4.2.
The idealJ in W(n1,n2,n3) generated by 1-gg¯ is a weak Hopf ideal. And the quotient weak Hopf algebra W(n1,n2,n3)/J is a Hopf algebra, which is isomorphic to Hn, where n=n1+n2+n3.Proof.
Since(4.2)Δ(1-gg¯)=1⊗1-gg¯⊗gg¯=1⊗1-gg¯⊗1+gg¯⊗1-gg¯⊗gg¯=(1-gg¯)⊗1+gg¯⊗(1-gg¯),T(1-gg¯)=T(1)-T(g¯)T(g)=1-gg¯,J is a weak Hopf ideal in W(n1,n2,n3).
The isomorphismW(n1,n2,n3)/J→Hn is defined by g+J↦K, g¯+J↦K-1, xi+J↦Xi, gyj+J↦Xn1+j, zk+J↦Xn1+n2+k.Now we give the Ext quiver ofW(n1,n2,n3). For the definition and calculation of Ext quiver, we refer to [5, 8, 9, 12].Figure 1
Ext quiver ofW(n1,n2,n3).The Ext quiver ofW(n1,n2,n3) is shown in Figure 1. The multiplicity of arrow g·→·1 is n1. The multiplicity of arrow 1·→·g¯ is n2. The multiplicity of other arrows is all n.Theorem 4.3.
The sub-coalgebraH related to the subquiver in Figure 2 is isomorphic to Hn as coalgebra.Figure 2
A subquiver of Ext quiver ofW(n1,n2,n3).Proof.
The isomorphismH→Hn is induced by gg¯↦1, g↦K, g¯↦K-1, xi↦Xi, gyj↦Xn1+j, zk↦Xn1+n2+k.Remark 4.4.
The isomorphisms described in Theorem4.1 are not isomorphisms of bialgebras.Remark 4.5.
The weak Hopf algebras discussed in [4, 5] also have quotient Hopf algebras and sub-Hopf algebras which are isomorphic to the related Hopf algebras.



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