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  浙江大学学报(理学版)  2016, Vol. 43 Issue (4): 379-388,410  DOI:10.3785/j.issn.1008-9497.2016.04.001
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穆尼尔艾哈迈德 , 李方 . 2016. 基本弱Hopf代数和弱覆盖箭图[J]. 浙江大学学报(理学版), 43(4): 379-388,410. DOI: 10.3785/j.issn.1008-9497.2016.04.001.
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AHMED Munir , LI Fang . 2016. Basic weak Hopf algebra and weak covering quiver[J]. Journal of Zhejiang University(Science Edition) , 43(4): 379-388,410. DOI: 10.3785/j.issn.1008-9497.2016.04.001.
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基金项目

Supported by the National Natural Science Foundation of China(11271318, 11571173) and the Zhejiang Provincial Natural Science Foundation of China (LZ13A010001)

作者简介

AHMED M(1963-),ORCID:http://orcid.org/0000-0003-4335-6560, male, doctor, assistant professor, the field of interest is classification and representation of algebras, E-mail:irmunir@yahoo.com.

通信作者

ORCID:http://orcid.org/0000-0002-4627-1581,E-mail:fangli@zju.edu.cn..

文章历史

收稿日期:2015-09-21
基本弱Hopf代数和弱覆盖箭图
穆尼尔艾哈迈德1 , 李方2     
1. 伊斯兰堡大学 模范男子学院, 巴基斯坦 伊斯兰堡;
2. 浙江大学 数学科学学院, 浙江 杭州 310027
摘要: 研究了代数闭域K上具有强分次Jacobson根r的有限维基本可裂弱Hopf代数,并刻画了有限维基本可裂半格分次弱Hopf代数H,即存在有限Clifford半群S,使得$H/r\cong k{{S}^{*}}$.还引入了弱覆盖箭图的概念,其路代数具有半格分次弱Hopf代数的结构,其箭图作为弱覆盖箭图被刻画.进一步地,证明了对上述H存在弱覆盖箭图Г和由长度大于2的路生成的理想I,使得$k\Gamma /I\cong H$.
关键词: 弱Hopf代数    弱覆盖箭图    分枝数据    
Basic weak Hopf algebra and weak covering quiver
AHMED Munir1 , LI Fang2     
1. Islamabad Model College for Boys, F-10/4, Islamabad, Pakistan
2. School of Mathematical Sciences, Zhejiang University, Hangzhou 310027, China
Abstract: We introduce a finite-dimensional basic and split weak Hopf algebra H over an algebraically closed field k with strongly graded Jacobson radical r. We obtain some structures of a finite-dimensional basic and split semilattice graded weak Hopf algebra, and observe that there exists a finite Clifford monoid S which is isomorphic to the set of all the isomorphism classes of 1-dimensional H-modules such that $H/r\cong k{{S}^{*}}$. We also introduce the notion of weak covering quiver whose path algebra admits a structure of semilattice graded weak Hopf algebra, and classify the path algebra corresponding to the weak covering quiver. Furthermore, we prove that, for a finite-dimensional basic semilattice graded weak Hopf algebra H over an algebraically closed field k with strongly graded radical, there exists a weak covering quiver Γ such that $k\Gamma /I\cong H$, where the ideal I is generated by the paths of length l≥2.
Key words: weak Hopf algebra    weak covering quiver    ramification data    

A finite-dimensional algebra A over a field k is defined to be basic if $A\cong \underset{i=1}{\overset{n}{\mathop{\oplus }}}\,$ Pi for some indecomposable projective A-modules Pi,where Pi${{P}_{i}}\cong {{P}_{j}}$. If {e1,e2,…,en} is the set of orthogonal idempotents of A,then Pi=Aei for 1≤i≤n and $1=\sum\limits_{i=1}^{n}{{{e}_{i}}}$. For a quiver Γ=(Γ01),the path algebra kΓ is finite-dimensional if,and only if,Γ has no oriented cycles. Here Γ0={v1,v1,…,vn} represents the set of vertices of the quiver Γ. The trivial paths of Γ may be identified by the symbols e1,e2,…,en. Let J be an ideal of kΓ generated by all the arrows Γ1 in Γ,then J becomes a nilpotent ideal. If Λ denotes the quotient kΓ/J,then $\Lambda =k\Gamma /J\cong k{{e}_{1}}\times k{{e}_{2}}\times \cdots \times k{{e}_{n}}$ is a k-algebra,where Λ=kΓ/J is semisimple and J=rad(kΓ)[1]. Further,$k\Gamma =\underset{i=1}{\overset{n}{\mathop{\oplus }}}\,\left( k\Gamma \right)$ is a basic k-algebra,where (kΓ)ei is an indecomposable projective kΓ-module for all 1≤i≤n,since (kΓ)ei/Jei is 1-dimensional over k. The set of paths {p|s(p)=i} can be considered as the k-basis for the indecomposable projective kΓ-module (kΓ)ei. The number of arrows from the vertex vi to the vertex vj is the dimension of k-vector space j Vi = vj (rad Λ /(rad Λ)2)vi[1]. We assume that {ar}mr=1 is a basis for all such vector spaces j Vi ,which are arrows from vi to vj.

A bialgebra H over a field k is called a weak Hopf algebra(LI's) if there exists an element T in the convolution algebra Homk(H,H) such that id * T * id=id and T * id * T=T,where * is the convolution product in Homk(H,H) and T is a weak antipode of H. A weak Hopf algebra H with weak antipode T is called a semilattice graded weak Hopf algebra if $H=\underset{\alpha \in Y}{\mathop \oplus }\,$ is a semilattice graded sum,where Hα is sub-weak Hopf algebra and is a Hopf algebra with antipode THα for each αY. There are Hopf algebra homomorphisms φα,β from Hα to Hβ if αβ=β such that for all a∈Hα,bHβ,the multiplication a°b in H can be given by a°b=φα,αβ(a)φβ,αβ(b)[2-3].

CIBILLS et al[4] defined the notion of Hopf quiver and classified the graded Hopf algebras using such quivers. The authors defined earlier the so-called notion of weak Hopf quiver of some Clifford monoid,and classified the semilattice of graded weak Hopf algebra structures over path coalgebra using this type of quiver[5-6].

GREEN et al[7] investigated the structures of finite-dimensional basic Hopf algebra over an algebraically closed field k. They introduced the covering quiver ΓG(W) in terms of a finite group G with a weight sequence W, and obtained a condition for a finite-dimensional basic Hopf algebra with respect to the path algebra of the covering quiver.

We generalize partially a well-known classical result which states that,“H is a finite-dimensional commutative basic and split Hopf algebra over an algebraically closed field k if,and only if,$H\cong {{\left( kG \right)}^{*}}$ for some finite group G”,to the version,“if H is a finite-dimensional basic and split semilattice graded weak Hopf algebra which is with a strongly graded Jacobson radical r,then there exists a Clifford monoid S such that$H/r\cong {{\left( kS \right)}^{*}}$ for a finite Clifford monoid S”. It also turns out that the existence of such a Clifford monoid S is a common structure associated with the finite-dimensional basic and split semilattice graded weak Hopf algebra. We call the Clifford monoid S a structure monoid of H.

We also introduce the notion of weak covering quiver ΓW(S) corresponding to a Clifford monoid S and a weight sequence W. We discover the structures of basic weak Hopf algebra by introducing the so called notion of weak covering quiver ΓW(S) in terms of a finite semigroup with a weight sequence W.We obtain a condition for a finite-dimensional basic weak Hopf algebra with respect to the path algebra of the weak covering quiver,and classify the structure of a semilattice graded weak Hopf algebra obtained from the path algebra of the weak covering quiver. We define a finitely generated basic semilattice graded weak Hopf algebra with a strongly graded Jacobson radical,then observe that for any finite-dimensional basic semilattice graded weak Hopf algebra H with a strongly graded Jacobson radical,there exists a finite Clifford monoid S and a weight sequence W such that $H\cong k{{\Gamma }_{W}}\left( S \right)/I$ for the corresponding weak covering quiver ΓW(S) and for some ideal I in the path algebra kΓW(S),such that Jn I ⊆J2 for an integer n≥2,where J is an ideal generated by the arrows in ΓW(S).

1 Structures of basic weak hopf algebras

Suppose H=(H, Δ,ε,T) is a finite-dimensional basic weak Hopf algebra over an algebraically closed field k,where Δ,ε and T denote the comultiplication,the counit and the weak antipode of H,respectively. The comultiplication Δ:H→HⓧH is given by

$\Delta \left( h \right)=\sum{_{\left( h \right)}{{h}_{\left( 1 \right)}}\otimes {{h}_{\left( 2 \right)}}.}$

The counit is an algebra homomorphism ε:Hk satisfying

$h=\sum{_{\left( h \right)}\varepsilon \left( {{h}_{\left( 1 \right)}}{{h}_{\left( 2 \right)}} \right)=\sum{_{\left( h \right)}\varepsilon }}\left( {{h}_{\left( 1 \right)}}{{h}_{\left( 2 \right)}} \right),$

and the weak antipode T∈Homk(H,H) is such that T * idH* T=T and idH* T * idH=idH,i.e.,

$\begin{align} & \sum{_{\left( h \right)}T}\left( {{h}_{\left( 1 \right)}}{{h}_{\left( 2 \right)}} \right)T\left( {{h}_{\left( 3 \right)}} \right)=T\left( h \right), \\ & \sum{_{\left( h \right)}}{{h}_{\left( 1 \right)}}T\left( {{h}_{\left( 2 \right)}} \right)\left( {{h}_{\left( 3 \right)}} \right)=\varepsilon \left( h \right)1H. \\ \end{align}$

If U and V are H-modules,then UV is an H-module via the actions

$h\bullet \left( u\otimes v \right)=\sum{_{\left( h \right)}}\left( {{h}_{\left( 1 \right)}}\bullet u \right)\otimes \left( {{h}_{\left( 2 \right)}}\bullet u \right)$

and

$\left( u\otimes v \right)\bullet h=\sum{_{\left( h \right)}}\left( u\bullet {{h}_{\left( 2 \right)}} \right)\otimes \left( u\bullet {{h}_{\left( 1 \right)}} \right)$

for hH,uU,vV.

The field k=kε equips an H-module structure via ε and is a unit element in the class of H-modules. U*=Homk(U,k) is a left H-module and becomes a right H-module if the action is composed by the weak antipode T.

If each simple module of H has multiplicity one in the maximal semisimple quotient subalgebra of H,then H is basic algebra. If r=rad H is the Jacobson radical,then we obtain that r is also a weak Hopf ideal of H. In view of [8]cor.8 and [7]lemma1.1,we can give the following lemma in terms of finite-dimensional basic weak Hopf algebra.

Lemma 1 If H=(H,Δ,ε,T) is a finite-dimensional basic semilattice graded weak Hopf algebra over k with the Jacobson radical r,where the weak antipode T is the anti-morphism of algebras,then the following statements are true:

(i) The Jacobson radical rad(HkH) of HkH is equal to rkH+Hkr.

(ii) The Jacobson radical r of H is a weak Hopf ideal.

Proof See the proof of (i) lemma 1.1(a) in [7]. For the proof of (ii),by [7]lemma 1.1(b) and [8] cor.8,the comultiplication Δ:H→HⓧH is a morphisms of k-algebars,and

$\Delta \left( r \right)\subseteq rad\left( H\otimes H \right)=r{{\otimes }_{k}}H+H{{\otimes }_{k}}r.$

As ε:HHop is a morphism of k-algebras,so ε(r)=0. It suffices to prove that T(r)r for the weak antipode T. Since H and Hop are weak Hopf algebras. Further,since T:H→Hop is an anti-morphism of k-algebars with T(xy)=T(y)T(x) for any x,yH. Thus,image of radical r under T is contained in the radical of Hop,since r is also an ideal of the ring H. Then for the opposite ring Hop,r=rad H=rad Hop,and T(r)⊆rad Hop=r.

If Γ is a finite quiver and the corresponding path coalgebra is kΓ,then φ:kΓ→kΓ/I is a natural surjection,where kerφ=I is an ideal of kΓ. If H is a basic weak Hopf algebra,then for some finite quiver Γ and for an ideal I in kΓ,HkΓ/I,where I is generated by the paths of length l≥2. The Jacobson radical r=rad H can be generated as 〈φ(ai)|ai∈Γ1〉 in kΓ/I. Since H/r≌kn for some integer n,therefore,H/r is a finite-dimensional commutative split weak Hopf algebra over the field k and is semisimple.

A weak Hopf algebra H is split as an algebra if the endomorphism rings Endk(V,V) of the simple modules V over H are isomorphic to the ground field k. In particular,if k is algebraically closed,then H is always split as an algebra. Since $ is a finite-dimensional split weak Hopf algebra,a finite-dimensional basic and split weak Hopf algebra,which means that all its simple H-modules are 1-dimensional over the field k. The multiplication structure in H-modules is given by the tensor product of modules due to the comultiplication on H. In view of the so-called semilattice graded weak Hopf ideal[9],we have the following fact.

Lemma 2 If $H=\underset{\alpha \in Y}{\mathop{\oplus }}\,{{H}_{\alpha }}$ is a finite-dimensional basic and split semilattice graded weak Hopf algebra,and $r=\underset{\alpha \in Y}{\mathop{\oplus }}\,{{r}_{\alpha }}$ is the Jacobson radical of H,where rα=r∩Hα,then:

(i) rα is the Jacobson radical of Hα for some α in Y;

(ii) r=⊕αYrα is semilattice graded.

Proof (i) The Jacobson radical of H is $r=\underset{\alpha \in Y}{\mathop{\oplus }}\,{{r}_{\alpha }}$,where rα=rHα for each α in Y. Applying lemma 1,r becomes a weak Hopf ideal with each rα as a Hopf ideal. If D is a subcoalgebra of a coalgebra C,then the coradical of D is D0=C0∩D and D0=C0∩D*,where D0 is the Jacobson radical of the dual D* of D and C0 is the Jacobson radical of C* [10]. H is finite dimensional pointed semilattice graded weak Hopf algebra,and Hα is a sub-Hopf algebra of H for each α. Replacing C* by H and D* by Hα,we obtain that Jacobson radical of Hopf algebra Hα is equal to the intersection of the Jacobson radical r of the weak Hopf algebra H with Hα,i.e.,rα=r∩Hα,for each α in Y.

(ii) Again,in view of lemma 1,we have that Jacobson radical r is a weak Hopf ideal. To prove that $r=\underset{\alpha \in Y}{\mathop{\oplus }}\,{{r}_{\alpha }}$ is semilattice graded,we just prove Hαrβ rαβ,rαHβ rαβ and rαrβ rαβ for all α,β in Y[9].

Since rα=r∩Hα =rad Hα,xHα H and yrβ=rHβ,then y∈r. Therefore,xy∈r and xy∈HαβHβ Hαβ,since r is an ideal of H. Consequently,xy∈r∩Hαβ=rαβ,which implies that Hαrβ rαβ.

Further,if x∈rα=r∩Hα and y∈rβ Hβ H,we obtain xy∈r. As r is an ideal of H and xy∈HαHβ Hαβ,so xy∈r∩Hαβ=rαβ and we have rαHβ rαβ.

Finally,if x∈rα=r∩Hα and y∈rβ=r∩Hβ,then xy∈r,xy∈HαHβ Hαβ H and xy∈r∩Hαβ=rαβ. Thus,we have rαrβ rαβ for all α,β in Y,and the Jacobson radical $r=\underset{\alpha \in Y}{\mathop{\oplus }}\,{{r}_{\alpha }}$ is semilattice graded.

Obviously,$H/r=\underset{\alpha \in Y}{\mathop{\oplus }}\,{{H}_{\alpha }}/{{r}_{\alpha }}$ is a weak Hopf algebra. To prove that H/r is a semilattice graded weak Hopf algebra,we only need to prove that $r=\underset{\alpha \in Y}{\mathop{\oplus }}\,{{H}_{\alpha }}/{{r}_{\alpha }}$ is semilattice graded sum. For this,we need to show that the embedding (Hα/rα)(Hβ/ rβ)Hαβ/rαβholds for all α,β in Y. To hold this embedding,we need to fix an assumption that rαrβ=rαβ for all α,β in Y.

Lemma 3 If $h=\underset{\alpha \in Y}{\mathop{\oplus }}\,{{H}_{\alpha }}$ is a finite-dimensional basic and split semilattice graded weak Hopf algebra,and $r=\underset{\alpha \in Y}{\mathop{\oplus }}\,{{r}_{\alpha }}$ is the Jacobson radical of H,where rα=r∩Hα. Further,if rαrβ=rαβ for all α,β in Y,then$H/r=\underset{\alpha \in Y}{\mathop{\oplus }}\,{{H}_{\alpha }}/{{r}_{\alpha }}$ is a semilattice graded weak Hopf algebra.

Proof For the elements x+rαHα/rα and y+rβHβ/rβ,where x is in Hα and y is in Hβ,the product

$\left( x+{{r}_{\alpha }} \right)\left( y+{{r}_{\beta }} \right)=xy+x{{r}_{\beta }}+{{r}_{\alpha }}y+{{r}_{\alpha }}{{r}_{\beta }}=xy+x{{r}_{\beta }}\in {{H}_{\beta }}/{{r}_{\beta }}\left( {{r}_{\alpha }}{{r}_{\beta }}={{r}_{\alpha \beta }} \right),$

where,xy∈HαHβ Hαβ,xrβHαrβ rαβ and rαy∈rαHβ rαβ,hence (x+rα)(y+rβ)=xy+rαβHαβ/rαβ. Thus,we have (Hα/rα)(Hβ/rβ) Hαβ/rαβ for all α,β in Y.

Definition 1 If H is a finite-dimensional basic semilattice graded weak Hopf algebra with Jacobson radical $r=\underset{\alpha \in Y}{\mathop{\oplus }}\,{{r}_{\alpha }}$,where rα=r∩Hα such that rαrβ=rαβ for all α,β in Y,then we call that H is with a strongly graded radical r.

In case of a weak Hopf algebra H which is not a Hopf algebra,a group-like element may not be invertible in the group sense. However,it is invertible in the regular monoid sense. If H is a bialgebra and I is a bi-ideal of H,then the quotient H/I is bialgebra. If H is a weak Hopf algebra which is not a Hopf algebra,then I may not be Hopf ideal but be a weak Hopf ideal if T(I) I. Thus,the group-like elements of H/I may not form a group,instead they form a regular monoid[11].

The following result gives a relationship between a finite-dimensional basic semilattice graded weak Hopf algebra and a finite-dimensional split semilattice graded weak Hopf algebra. There is a Clifford monoid attached to such a semilattice graded weak Hopf algebra,which corresponds to the group-like elements of the dual of semilattice graded weak Hopf algebra,as considered in [8] for the dual of a finite-dimensional Hopf algebra.

Theorem 1 If H is a finite-dimensional basic and split semilattice graded weak Hopf algebra with a weak antipode T as an anti-morphism of algebras. Further,if H is with a strongly graded radical,then there exists a Clifford monoid S as the isomorphism classes of 1-dimensional H-modules,such that H/r=(kS)*.

Proof Let $H=\underset{\alpha \in Y}{\mathop{\oplus }}\,{{H}_{\alpha }}$ be a finite-dimensional basic and split semilattice graded weak Hopf algebra,and r=⊕α∈Yrα be the Jacobson radical of H,where rα=r∩Hα is the Jacobson radical of the Hopf algebra Hα for every α in Y. Then,r becomes a weak Hopf ideal by lemma 1. Since H is with a strongly graded radical $r=\underset{\alpha \in Y}{\mathop{\oplus }}\,{{r}_{\alpha }}$,we have rαrβ=rαβ for all α,β in Y. Thus,by lemma 2,r equips a semilattice graded structure. As H is basic and split,we have H/r=kn for some positive integer n. Since the multiplication and comultiplication are the induced morphisms for H/r,and the weak antipode is the induced anti-morphism of H/r,applying lemma 3 above,H/r induces a semilattice graded weak Hopf algebra structure. For if φ:H→H/r is a surjection such that φ=(φαβ,…,φδ)=α∈Yφα,where φα:HαHα/r is a surjection,which is a restriction φ|HαHα/r for each α in Y. Further,Ker φ=r=rad H and for any x,y in H such that either x or y is in r,then xy is in r and φ(xy)=0=φ(x)φ(y). If both x and y are non-zero in H,then x+r and y+r will be non-zero in H/r and φ(xy)=xy+r=(x+r)(y+r)=φ(x)φ(y). Therefore,φ is an algebra morphism. Further,with the assumption rαrβ=rαβ,in view of lemma 3,we obtain

$\left( {{H}_{\alpha }}/{{r}_{\alpha }} \right)\left( {{H}_{\beta }}/{{r}_{\beta }} \right)\to {{H}_{\alpha \beta }}/{{r}_{\alpha \beta }}$

for all α,β in Y,hence H/r is a semilattice graded weak Hopf algebra. Thus,the dual (H/r)* of H/r is also semilattice graded weak Hopf algebra.

Let E=ei,α|1≤i≤nα;α∈Y be a set of orthogonal idempotents in H,where ∑α∈Ynα=n is the dimension of H/r. Then,E is also basis of H/r. Let B=gi,α|1≤i≤nα;αY be the corresponding dual basis of (H/r)*. Since gi,α∈Alg(H/r,k)=G((H/r)*) (see[10]th.2.3.1 ).

Thus,gi,α|1≤i≤nα=G((Hα/rα)*)=Gα that is a group for each α in Y and $\left( H/{{r}^{*}} \right)=\underset{\alpha \in Y}{\mathop{\oplus }}\,{{\left( {{H}_{\alpha }}/{{r}_{\alpha }} \right)}^{*}}$ is generated by $\underset{\alpha \in Y}{\mathop{\cup }}\,\left\{ {{g}_{i,\alpha }}1\le i\le {{n}_{\alpha }} \right\}=\underset{\alpha \in Y}{\mathop{\cup }}\,{{G}_{\alpha }}=S$,which is a semilattice of groups,hence a Clifford monoid. S generates (H/r)* and (H/r)*=kS implies H/r=(kS)*. The comultiplication in (H/r)*=kS is defined by

$\Delta \left( gt,\gamma \right)=\sum\limits_{\alpha \beta =\gamma ,i+j=l}{{{g}_{i,a}}\otimes {{g}_{j,\beta }},}$

where αβ=γ and i+j=l if,and only if,gi,α.gj,β=gl,αβ=gl,γ is the multiplication in S. Then,there is a one-to-one correspondence between the comultiplication structures on (H/r)* and the multiplication structures in S. Let g∈(H/r)*. Then,g:H/rk such that g-1∈Homk(H/r,k)=(H/r)* given by g-1(h)=g(T(h)).

If V=km is a 1-dimensional H-module,then V is a 1-dimensional H/r-module. Since the Jacobson radical is annihilator of V,there exists an algebra morphism g:H/rk such that h·m=g(h)·m and T(h·m)=m·T(h) for all hH/r. If Ug is a 1-dimensional H-module generated by g as a vector space with h·g=ε(h)g and T(h·g)=g·T(h)=ε(T(h))g,then VUg is as an H-module.

Moreover,

${{\left( {{U}_{g}} \right)}_{g}}-1={{U}_{\varepsilon }}={{k}_{\varepsilon }}\cong k.$

The 1-dimensional simple H-modules are generated by the basis elements of (H/r)*. If Ugi,α is a 1-dimensional simple module generated by gi,α for some 1≤inα,α in Y,then

${{U}_{{{g}_{i,a}}}}\otimes {{U}_{{{g}_{j,\beta }}}}={{U}_{{{g}_{i}},{{a}^{g_{i,a}^{-1}}}}}={{U}_{{{g}_{i+j}},\alpha \beta }}={{U}_{{{g}_{l,y,}}}}$

where αβ=γ and i+j=l. In particular,

${{U}_{{{g}_{i,a}}}}\otimes {{U}_{{{g}_{j,\alpha }}}}-1={{U}_{{{g}_{i}},{{a}^{g_{i,a}^{-1}}}}}={{U}_{{{\varepsilon }_{a}}}}={{k}_{{{\varepsilon }_{\alpha }}}},$

where

${{U}_{{{g}_{i,a}}}}-1={{U}_{T\left( {{g}_{i,\alpha }} \right)}}=Hom=\left( {{U}_{{{g}_{i,\alpha }}}},k \right).$

Since S is a Clifford monoid,its elements are regular such that for any s in S,there exists an element t in S such that sts=s and tst=t. For any 1-dimensional H-module Us,there exists a 1-dimensional H-module Ut such that UsUtUs=Us and UtUsUt=Ut. The weak antipode T is in (H/r)* such that its restriction T/(Hα/rα)* is an antipode in the Hopf algebra (Hα/rα)* for every α in Y.

Example 1 Let H=kS be a semilattice graded weak Hopf algebra of functions on some finite Clifford monoid S,where H is the dual of the Clifford monoid algebra kS. If k is algebraically closed and S is commutative,then H=kS=kS≌(kS)*. The Clifford monoid S is called the structure Clifford monoid of H.

Remark 1 In views of theorem 1 and example 1 above,one can observe that,the set S of isomorphism classes of simple modules over a finite-dimensional basic and split semilattice graded weak Hopf algebra H with a strongly graded radical r (which is the Jacobson radical of H) over an algebraically closed field k are Clifford monoids if,and only if H/r≌(kS)* .

As we know that $\Lambda =H/r\cong {{\left( kS \right)}^{*}}$ for some finite-dimensional basic semilattice graded weak Hopf algebra H over an algebraically closed field k with a strongly graded radical r,and for some finite Clifford monoid $S=\underset{\alpha \in Y}{\mathop{\cup }}\,{{G}_{\alpha }}$ is a semilattice Y of subgroups Gα. Let the comultiplication,the counit and the weak antipode in Λ be denoted by Δ′,ε′ and T′,respectively,and give in term of dual basis $\left\{ {{e}_{{{g}_{\alpha }}}} \right\}\underset{\alpha \in Y}{\mathop{{{g}_{\alpha }}\in {{G}_{\alpha }}}}\,$ in (kS)* as follows:

$\eqalign{ & \Delta '\left( {{e_{{g_\alpha }}}} \right) = \sum\limits_{\beta \gamma = \alpha } {{e_{g\beta }} \otimes {e_{g\gamma }},} \cr & \varepsilon '\left( {{e_{{g_\alpha }}}} \right) = {\delta _{{g_\alpha },{e_{{1_\alpha }}}}} = \left\{ \matrix{ 1,if{g_{\alpha = {e_{{1_\alpha }}}}} \hfill \cr 0,otherwise, \hfill \cr} \right. \cr} $

and

$T'\left( {{e}_{{{g}_{\alpha }}}} \right)={{e}_{g_{\alpha }^{-1}}}$

for all gαGα,gβGβ ,gγGγ; α,β,γ∈Y,and 1α is the identity in Gα and an idempotent element of S.

The set of primitive orthogonal idempotents $\left\{ {{e}_{{{g}_{\alpha }}}} \right\}\underset{\alpha \in Y}{\mathop{{{g}_{\alpha }}\in {{G}_{\alpha }}}}\,$ in H/r can be lifted to the set of primitive orthogonal idempotents $\left\{ {{v}_{{{g}_{\alpha }}}} \right\}\underset{\alpha \in Y}{\mathop{{{g}_{\alpha }}\in {{G}_{\alpha }}}}\,$ in H. We rewrite the lemma 1.2 of ref.[7] as the following new version.

Lemma 4 Let H be a finite-dimensional basic semilattice graded weak Hopf algebra over an algebraically closed field k with Jacobson radical r=rad H such that H is with a strongly graded radical. Then,there exists a finite Clifford monoid S with primitive orthogonal idempotents {vs}s∈S in H such that the following conditions are true:

(i) The counit ε for H is given by

$\varepsilon \left( {{v}_{s}} \right)=\left\{ \begin{align} & 1,if{{s}^{2}}=s \\ & 0,otherwise, \\ \end{align} \right.$

(ii) The weak antipode T of H is the induced morphism T:HH/r is given by T(vs)=vt for some t in S such that vtvsvt=vt and vsvtvs=vs,for all sS.

(iii) The comultiplication in H is the induced morphism

$\Delta :H\to H\otimes H/\left( r{{\otimes }_{k}}H+H{{\otimes }_{k}}r \right),$

given by Δ(vs)=∑t∈Svssvt$H/r\cong {{\left( kS \right)}^{*}}$ for all s,s′∈S such that vsvtvs=vs and vtvsvt=vt. By the above results,there exists a finite regular monoid S with commuting idempotents such that H/r≌(kS)* and with the primitive orthogonal idempotents {vs}s∈S of H which satisfies the conditions of lemma 4. If H* is the dual of H,then H* acts on the left of H by f·h=∑(h)f(h(2))h(1) and on the right of H by h·f=∑(h)f(h(1))h(2).

There exists an induced inclusion kSH*,such that H is a kS-bimodule. The regular element s in S is identified with the dual basis element of vs. Then,actions of kS on H are given by

$s.{{v}_{t}}={{v}_{ts}}',{{v}_{t}}\bullet s={{v}_{s't}}\left( \bmod ulor \right),$

where vs-1=vs such that vsvsvs=vs and vsvsvs=vs (or equivalently s′ss′=s′ and sss=s′ for all t,s,s′∈S).

If H is a finite-dimensional basic semilattice graded weak Hopf algebra over an algebraically closed field k and the comultiplication Δ,the counit ε and the weak antipode T on H are as defined above,then there exists a finite quiver Γ and an ideal I with Jm I J2,for some integer m and for an ideal J generated by all the arrows in Γ such that $H\cong k\Gamma /I$.

We obtain some formulae to calculate the images of T modulo r2 (where r=rad H) and the images of Δ modulo rad2(HⓧH) for the generators of H corresponding to the arrows in the quiver Γ. In view of lemma 2.1[7],we have the following fact.

Lemma 5 If H is a finite-dimensional basic semilattice graded weak Hopf algebra over an algebraically closed field k,such that H/r≌(kS)* for some Clifford monoid S. If s Vt denotes the vector space vsr/r2vt over k,then the following conditions hold:

(i) For xs Vt , we have Δ(x)=∑s′∈Ss′.xⓧvs+vsⓧx.s′ modulo rad2(HH).

(ii) For all s′,t′ in S,we have dimk(sVt)=dimk(t-1s s-1Vt-1t s-1).

(iii) The set {st-1|sVt≠{0}} is a conjugate closed subset of each maximal subgroup G of S such that s,t∈G.

(iv) If {vi}mi=1 is a basis for ⊕sS(sVe),where each vi is in some vector space s Ve ,then the set s.vimi=1,s∈S becomes a basis for r/r2.

In the left and the right actions of S on s Vt = vsr/r2vt,consider the linear isomorphism

$_{s}{{L}_{t}}\left( t' \right){{:}_{s}}{{V}_{t}}\to t{{'}^{-1}}_{s}{{V}_{t{{'}^{-1}}t.}}$

Applying definition of Δ and ε,it is easy to see that Δ satisfies the counitary property that the image of (εⓧid)°Δ is equal to that of (idⓧε)°Δ modulo r for vertices and modulo r2 for the elements of s Vt ,which are arrows.

If Trr/r2 is an induced morphism,then in view of lemma 2.2 in ref. [7],we can describe T as follows.

Lemma 6 The weak antipode T is an induced morphism T:HH/r2 which satisfies T(x)=-(s·xt,T(y)=-s·(y·t) and T(s′·x)=T(x)·s′-1,T(x·s′)=s′-1·T(x), for all italic>x in s Vt and s′ in S.

2 Weak covering quiver

GREEN et al[7] introduced the covering quiver. We introduce the notion of weak covering quiver here.

Let $H=\underset{\alpha \in Y}{\mathop{\oplus }}\,{{H}_{\alpha }}$ be a finite-dimensional basic semilattice graded weak Hopf algebra over an algebraically closed field k. For a finite quiver Γ,$H\cong k\Gamma /I$,where I is an ideal such that Jn I J2 for n≥2,and J is an ideal generated by the arrows in Γ,and ideal $I=\underset{\alpha \in Y}{\mathop{\oplus }}\,{{I}_{\alpha }}$ is such that IαIβ=Iαβ for that Iα is an ideal of ${{H}_{\alpha }}=k{{\Gamma }_{\alpha }}$ and Iβ is an ideal of ${{H}_{\beta }}=k{{\Gamma }_{\beta }}$ in Y,where ${{H}_{\alpha }}=k{{\Gamma }_{\alpha }}/{{I}_{\alpha }}$ for all α in Y,and $J=\underset{\alpha \in Y}{\mathop{\oplus }}\,{{J}_{\alpha }}$ is a semilattice graded sum such that JαJβ=Jαβ and Jnαα Iα J2α for nα≥2,with Jα as an ideal generated by the arrows in the subquiver Γα corresponding to the finite-dimensional basic sub-Hopf algebra ${{H}_{\alpha }}\cong k{{\Gamma }_{\alpha }}/{{I}_{\alpha }}$ for some α,β∈Y. Further,${{\Gamma }_{0}}=\underset{\alpha \in Y}{\mathop{\cup }}\,{{\left( {{\Gamma }_{\alpha }} \right)}_{0}}$,where Γ0 is the set of vertices of the quiver Γ and (Γα)0 is the set of vertices of the sub-quiver Γα for each α in Y. We describe the existence of finite quiver Γ with regards to a weak covering quiver as follows.

Let $S=\underset{\alpha \in Y}{\mathop{\cup }}\,{{G}_{\alpha }}$ be a finite Clifford monoid and Wα=(w1,α,w2,α,…,wnα) be a sequence of elements of the subgroup Gα,α∈Y. We call W=(Wα,Wβ,…,Wδ) a weight sequence for each gα in Gα,α∈Y,the weight sequence Wα and the sequence (gαw1,αg-1α,gαw2,αg-1α,…,gαwnαg-1α) are the same up to permutations for each αY,where Wα,Wβ,…, Wδ are the components of W. The relation between Wα and Wβ for α≥β in Y can be given in the following way: for any α≥β in Y,φα,β:GαGβ is a homomorphism form of Gα to Gβ given by φα,β(gα)=eβgαGαβ=Gβ for some gαGα,and the idempotent eβ is the identity element of Gβ,and φα,α is the identity automorphism on Gα for every α in Y. Let the restriction φα,β|WαWβ be a well defined map φWα,Wβ from Wα to Wβ for α≥β in Y. This implies that W and the sequence (gαWαg-1α,gβWβg-1β,…,gδWδg-1δ) are the same up to permutations. In particular,each of the sequences Wα,αY is separately closed under conjugation,and W does so componentwise.

Definition 2 A quiver ΓS(W) with the set of vertices represents by ΓS(W)0=∪α∈YΓGα(Wα)0,where ΓGα(Wα)0 is a set of vertices of the sub-quiver ΓGα(Wα) in the following ways:

(i) The vertices of ΓS(W) is the set of vertices v|v∈S=vgα|gαGαα∈Y,vgα|gαGα as the vertices of ΓGα(Wα) for each α∈Y;

(ii) The arrows are given as follows: if α ≥\ β,then there does not exist any arrow from vg-1α to ${{v}_{{{w}_{{{i}_{\beta ,\alpha }}}}\left( {{w}_{{{i}_{\beta }},\beta }}g_{\alpha }^{-1} \right)}}$. On the other hand,if α≥β,then the number of arrows (aβ,gα) from vg-1α to vwiβg-1α is the same as the number of arrows from vg-1α to vφβ(wiβg-1α),for all 1≤iβnβ and all α≥β; α,β∈ Y.

(iii) For an arrow aα:vgαvhα in ΓGα(Wα),let the left weight of aα be defined by l(aα)=hαg-1α. The group Gα is acting on ΓGα(Wα) on the left as follows:

For all gα in Gα,define gα·vhα=vhαg-1α and gα·(aα,hα)=(aα,gαhα). We call the quiver ΓGα(Wα) a sub-quiver of ΓS(W),and it is a covering quiver of Gα with respect to Wα,α∈Y. For an arrow aβ:vgαvgβ,α≥β,the left weight of aβ is defined by

$l\left( {{a}_{\beta }} \right){{g}_{\beta }}{{\varphi }_{\alpha \bullet \beta }}\left( {{g}_{\beta }}{{e}_{\beta }}g_{\alpha }^{-1} \right)={{g}_{\beta }}g_{\alpha }^{-1},\alpha \ge \beta ,\alpha ,\beta \in Y.$

Consequently,the Clifford monoid S is acting on ΓS(W) on the left as

${{g}_{\beta }}\bullet {{v}_{{{g}_{\alpha }}}}={{v}_{{{g}_{\alpha }}g_{\beta }^{-1}}},{{g}_{\beta }}\bullet \left( {{a}_{\beta }},{{h}_{\alpha }} \right)=\left( {{a}_{\beta }},{{g}_{\beta }}{{h}_{\alpha }} \right),$

where α≥β,α,βY.

Then,the quiver ΓS(W) is called a weak covering quiver with respect to weight sequence W,where each sub-quiver ΓGα(Wα) is a covering quiver which we call a sub-weak covering quiver of ΓS(W).

Remark 2 (i) If we compare the weak Hopf quiver Γ(S,r) of a Clifford monoid S with the ramification data r and the weak covering quiver ΓS(W),it can be easily verified that the two quivers are dual to each other and are isomorphic if they are finite.

(ii) A weak covering quiver ΓS(W) corresponding to a Clifford monoid S and a weight sequence W becomes a covering quiver if,and only if,the monoid S is a group. A relationship between the weak Hopf quiver and the weak covering quiver is established in the following result.

Proposition 1 A quiver is a weak covering quiver ΓS(W) if,and only if,it is a weak Hopf quiver Γ(S,r).

Proof Let ΓS(W) be a weak covering quiver with respect to W,where S is a Clifford monoid,and W=(Wα,Wβ,…,Wδ) be a sequence with all components Wα; α∈Y,with Wα=(w1,α,w2,α,…,wnα) as a sequence of distinct elements of the subgroup Gα which are closed under conjugation with elements in Gα. In fact,elements of Wα are disjoint union of elements of the conjugacy classes {Cα} of the group Gα. Define the ramification data ${{r}_{\alpha }}=\sum\limits_{{{C}_{\alpha }}}{{{r}_{\alpha ,{{C}_{\alpha }}}}{{C}_{\alpha }}}$ of Gα such that rα,Cα is the multiplicity of the conjugacy class Cα. Then there corresponds a Hopf quiver Γ(Gα,rα,Cα) for each sub-weak covering quiver ΓGα(Wα) of the weak covering quiver ΓS(W),and for each α in Y. It implies that we get a weak Hopf quiver Γ(S,r) for the weak covering quiver ΓS(W),where ${{r}_{\alpha }}=\sum\limits_{\alpha \in Y}{{{r}_{\alpha }}}$ is the ramification data corresponding to the sum of multiplicities for the components of W.

Conversely,suppose that Γ(S,r) is a weak Hopf quiver with ramification data $r=\sum\limits_{\alpha \in Y}{{{r}_{\alpha }}}$,where ${{r}_{\alpha }}=\sum\limits_{{{C}_{\alpha }}}{{{r}_{\alpha ,{{C}_{\alpha }}}}{{C}_{\alpha }}}$ of Gα,α∈Y. Since,Wα is disjoint union of rα,Cα copies of the conjugacy class Cα in Gα for every α in Y. Arrange the elements of Wα in some sequential order. Then,Wα becomes a weight sequence corresponding to Gα,which results in the existence of the covering quiver ΓGα(Wα) for each α in Y. Set W=(Wα)α∈Y,then W appears as the weight sequence for the Clifford monoid S,and we have the weak covering quiver ΓS(W) with respect to W,with the set of vertices $\Gamma s{{\left( W \right)}_{0}}=\underset{\alpha \in Y}{\mathop{\cup }}\,{{\Gamma }_{{{G}_{\alpha }}}}{{\left( {{W}_{\alpha }} \right)}_{0}}\cong S$.

Let ΓS(W) be a weak covering quiver with respect to a weight sequence W for a Clifford monoid S. Then,its path coalgebra kΓS(W) admits the structure of semilattice graded weak Hopf algebra by [6]. Thus,there is a strong relationship between the weak covering quiver ΓS(W) with respect to some weight sequence W corresponding to a Clifford monoid S,and the structures of semilattice graded weak Hopf algebras on its path algebra kΓS(W).

Theorem 2 If kΓ is a path algebra of a quiver Γ,then the following statements are equivalent.

(i) The path algebra kΓ admits a semilattice graded weak Hopf algebra structure such that each of its grading summands itself admits a graded Hopf algebra structure.

(ii) Γ is a weak covering quiver of some Clifford monoid S with some weight sequence.

Proof Let the path algebra $k\Gamma =\underset{\alpha \in L}{\mathop{\oplus }}\,{{\left( k\Gamma \right)}_{\alpha }}$ admit a semilattice graded algebra structure such that each grading summand (kΓ)α itself admits graded algebra structure. Then,(kΓ)α(kΓ)β (kΓ)αβ for all α,βαβ in a semilattice Y,and ${{\left( k\Gamma \right)}_{\alpha }}=\underset{n\ge 0}{\mathop{\oplus }}\,{{\left( k\Gamma \right)}_{n,\alpha }}$ for each α such that (kΓ)i,α(kΓ)j,β⊆(kΓ)i+j,αβ.

By [4] def.2.1,the path subalgebra (kΓ)α and the path algebra kΓ admit a coalgebra structure. Since (kΓ)α is a path coalgebra,there is a quiver Γα such that (kΓ)α=kΓα for every α in Y. Let Γ0 be the set of vertices of the quiver Γ. We set (Γ0)α=(Γα)0 as the set of vertices of the sub-quiver Γα. Let Γn denote the set of paths of length n in Γ. We set (Γn)α=(Γα)n as the set of paths of length n in the sub-quiver Γα.

If (kΓ)n,αn≥0 is the coradical filtration of coalgebra (kΓ)α,then (kΓ)0,α=(kΓ0)α=k0)α by the definitions of comultiplication Δ,counit ε for (kΓ)α[6, 12]. We can give a weak antipode similar to that given in proof of [6] th.4.5 and have a structure of semilattice graded weak Hopf algebra on the path algebra kΓ with each grading summand (kΓ)α admitting a graded Hopf algebra structure. In view of [6] th.4.5,Γ=Γ(S,r) is a weak Hopf quiver corresponding to a Clifford monoid with ramification data r. Then,by proposition 1,Γ is a weak covering quiver of the Clifford monoid S with some weight sequence W.

Conversely,let kΓ be a path algebra corresponding to a weak covering quiver Γ of some Clifford monoid S with a weight sequence W. By proposition 1,Γ is a weak Hopf quiver of the Clifford monoid S with some ramification data. By [6]th.4.5,kΓ admits a semilattice graded weak Hopf algebra structure such that each of its grading summands admits a Hopf algebra structure.

Using lemma 6 and proposition 1 ,it can be easily seen that a quiver Γ corresponding to a finite-dimensional basic semilattice graded weak Hopf algebra can be given by the weak covering quiver ΓS(W) for some Clifford monoid S and a weight sequence W of S.

We classify the path algebra corresponding to some weak covering quiver as a finite-dimensional basic semilattice graded weak Hopf algebra over an algebraically closed field k in the following theorem.

Theorem 3 Suppose $H=\underset{\alpha \in Y}{\mathop{\oplus }}\,{{H}_{\alpha }}$ is a finite-dimensional basic semilattice graded weak Hopf algebra over an algebraically closed field k. Further,if H is with a strongly graded radical,then there exist a finite Clifford monoid S and a weight W= (Wα,Wβ,…,Wδ) of S,where Wα=(w1,α,w2,α,…,wnα),αY,such that H$H\cong k\Gamma s\left( W \right)/I$ for an ideal I,where Jm I J2 and m≥2 for some ideal J generated by the arrows in ΓS(W).

Proof Let H be stated above and k be an algebraically closed field. Then,H becomes split,and there exists a Jacobson radical r=rad H which becomes a weak Hopf ideal by lemma 1(ii) such that $H/r\cong {{k}^{n}}$ for some integer n. Since H is semilattice graded,so H are H/r and r.

Assume H is with a strongly graded radical r. Then,$H/r=\underset{\alpha \in Y}{\mathop{\oplus }}\,{{H}_{\alpha }}/{{r}_{\alpha }}$ for $r=\underset{\alpha \in Y}{\mathop{\oplus }}\,{{r}_{\alpha }}$,where rα=rad Hα for each α in Y. Each Hα is a finite-dimensional basic sub-Hopf algebra over the same field k,so is split as well for each α in Y. Thus,${{H}_{\alpha }}/{{r}_{\alpha }}\cong {{k}^{{{n}_{\alpha }}}}$ for some integer nα for every α in Y,where $n=\sum\limits_{\alpha \in Y}{{{n}_{\alpha }}}$.

Further,by theorem 1 above,there exists a Clifford monoid S such that $H/r\cong {{\left( kS \right)}^{*}}$,where S=n.

By[13]lemma 2.1,corresponding to a basic and split finite-dimensional Hopf algebra Hα,there exists a finite group Gα,such that ${{H}_{\alpha }}/{{r}_{\alpha }}\cong {{\left( k{{G}_{\alpha }} \right)}^{*}}$,where Gα=nα for every α. Thus,$H/r\cong {{\left( kS \right)}^{*}}\cong \underset{\alpha \in Y}{\mathop{\oplus }}\,{{\left( k{{G}_{\alpha }} \right)}^{*}}$,where $S=\underset{\alpha \in Y}{\mathop{\cup }}\,{{G}_{\alpha }}$ is a Clifford monoid with GαGβ Gαβ for all α,β≥αβ in Y. Hence,H/r is semilattice graded with (kGα)*(kGβ)* (kGαβ)* for all α,β≥αβ in Y.

Since,Δ,ε and T are the comultiplication,the counit and the weak antipode for H,and φ:H→H/r is a surjection. The comultiplication Δ′ and the counit ε′ are the induced morphisms,and the weak antipode T′ is an induced automorphism in H/r.

The generating set {vs|s∈S} for the semilattice graded weak Hopf algebra H/r can be identified by the elements of the Clifford monoid S. For such a semilattice graded weak Hopf algebra,there exists a quiver Γ whose vertices can be identified by the generators of H/r,i.e.,by the set $\left\{ {{v}_{s}},s\in S \right\}=\underset{\alpha \in Y}{\mathop{\cup }}\,\left\{ {{v}_{{{s}_{\alpha }}}}{{s}_{\alpha }}\in {{G}_{\alpha }} \right\}$ and by the elements of S.

The actions of the Clifford monoid S on the quiver Γ are given by

$s.{{v}_{t}}={{v}_{ts}}',{{v}_{t}}\bullet s={{v}_{s't}}\left( \bmod ulor \right),$

where s,s′,t,t′∈S such that

${{v}_{s}}{{v}_{s'}}{{v}_{s}},{{v}_{s'}}{{v}_{s}}{{v}_{s'}}={{v}_{s'}},$

and

${{v}_{t}}{{v}_{t'}}{{v}_{t}},{{v}_{t'}}{{v}_{t}}{{v}_{t'}}={{v}_{t'}},$

or equivalently

$ss's=s,s'ss'=s'andtt'=t,t'tt'=t'.$

If W=(Wα,Wβ,…,Wδ) is a weight sequence,where Wα=(w1,α,w2,α,…,wnα) is a weight sequence of the elements of the subgroup Gα closed under conjugation for each αY and wiαGα;1≤iαnα,αY,then the arrows of Γ can be given by

$\left( {{a}_{\beta }},{{g}_{\alpha }} \right):v_{{{g}_{\alpha }}}^{-1}{{v}_{{{w}_{{{i}_{\beta }},{{\beta }^{g}}}}_{\alpha }^{-1}i\beta }}=1,2,\cdots ,{{n}_{\beta }},$

where gαGα,αβY with the action of S on the arrows of Γ as given in lemma 5 and lemma 6,thus the arrows of Γ corresponds to the generators of H/r2.

Consequently,we have (εid)°Δ(vs)=(idε)°Δ(vs) modulo r for the vertices,the formulas for the images of T modulo r2 and Δ modulo rad2(HH) for the arrows.

By the definition of weak covering quiver,one can identify the quiver Γ with the weak covering quiver ΓS(W). Consequently,by[1] cor.Ⅲ.1.10 and th.Ⅲ.1.9,there exists an ideal I where Jm I J2 and m≥2,for some ideal J generated by the arrows in ΓS(W) such that $H\cong k\Gamma s\left( W \right)/I$.

The following results give the presentation of the isomorphism classes of finite-dimensional basic and split semilattice graded weak hopf algebras corresponding to the isomorphism classes of the weak covering quivers and the isomorphism classes of the weak Hopf quivers.

Theorem 4 Let H and H′ be two finite-dimensional basic and split semilattice graded weak Hopf algebras with strongly graded radicals,and let S and S′ be the structured Clifford monoids of H and H′,respectively. Then,H and H′ are isomorphic if,and only if,the corresponding weak covering quivers ΓS(W) and ΓS(W′) are isomorphic,where W and W′ are their respective weight sequences.

Proof Suppose H and H′ stated above are isomorphic. Since $H\cong k\Gamma s\left( W \right)/I$ in view of theorem 3,where I and I′ are ideals of the path algebras kΓS(W) and kΓS(W′),respectively,which are generated by the paths of length 2 or greater. So that $I\cong I'$,which implies that kΓ$k\Gamma s\left( W \right)\cong k\Gamma s'\left( W' \right)$.

Conversely,suppose that the weak covering quivers ΓS(W) and ΓS(W′) are isomorphic,so that $k\Gamma s\left( W \right)\cong k\Gamma s'\left( W' \right)$ as algebras. If I and I′ are ideals of the path algebras kΓS(W) and kΓS(W′),such that Jm I J2,(J′)m I′ (J′)2 and m≥2 for some ideals J and J′ generated by the arrows in ΓS(W) and ΓS(W′),respectively. Then,$I\cong I'$,which implies that $k\Gamma s'\left( W' \right)/I'\cong k\Gamma s\left( W \right)/I$. Using theorem 3,we have $k\Gamma s\left( W \right)/I\cong k\Gamma s'\left( W' \right)/I'\cong H'$

Theorem 5 Let H and H′ be two finite-dimensional basic and split semilattice graded weak Hopf algebras with strongly graded radicals,and let S and S′ be the structured Clifford monoids of H and H′,respectively. Then,H and H′ are isomorphic if,and only if,the corresponding structure monoids S and S′ are isomorphic.

Proof Suppose that H and H′ are as stated above,and r and r′ are their respective strongly graded Jacobson radicals. Suppose that H and H′ are isomorphic,we have r≌r′,hence H/r≌H′/r′. In view of theorem 1,(kS)*≌H/r≌H′/r′≌(kS′)*. Since H and H′ are finite-dimensional,so S and S′ are finite Clifford monoids,hence they are isomorphic.

Conversely,if S and S′ are finite Clifford monoids that are isomorphic,then kS≌(kS)′ and so (kS)*≌(kS′)*. Using theorem 1,we have H/r≌(kS)*≌(kS′)*H′/r′,for the strongly graded Jacobson radicals r and r′ of H and H′,respectively. Since S and S′ are finite Clifford monoids and are isomorphic,we have r≌r′,hence H and H′ are isomorphic.

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