2. 浙江大学 数学科学学院, 浙江 杭州 310027
2. School of Mathematical Sciences, Zhejiang University, Hangzhou 310027, China
A finite-dimensional algebra A over a field k is defined to be basic if
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 Hom_{k}(H,H) such that id * T * id=id and T * id * T=T,where * is the convolution product in Hom_{k}(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
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 Γ_{G}(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,
We also introduce the notion of weak covering quiver Γ_{W}(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 Γ_{W}(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
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:H→k 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∈Hom_{k}(H,H) is such that T * id_{H}* T=T and id_{H}* T * id_{H}=id_{H},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 UⓧV 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 h∈H,u∈U,v∈V.
The field k=k_{ε} equips an H-module structure via ε and is a unit element in the class of H-modules. U^{*}=Hom_{k}(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(Hⓧ_{k}H) of Hⓧ_{k}H is equal to rⓧ_{k}H+Hⓧ_{k}r.
(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:H→H^{op} 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 H^{op} are weak Hopf algebras. Further,since T:H→H^{op} is an anti-morphism of k-algebars with T(xy)=T(y)T(x) for any x,y∈H. Thus,image of radical r under T is contained in the radical of H^{op},since r is also an ideal of the ring H. Then for the opposite ring H^{op},r=rad H=rad H^{op},and T(r)⊆rad H^{op}=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,H≌kΓ/I,where I is generated by the paths of length l≥2. The Jacobson radical r=rad H can be generated as 〈φ(a_{i})|a_{i}∈Γ_{1}〉 in kΓ/I. Since H/r≌k^{n} 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 End_{k}(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
Lemma 2 If
(i) r_{α} is the Jacobson radical of H_{α} for some α in Y;
(ii) r=⊕α∈Yr_{α} is semilattice graded.
Proof (i) The Jacobson radical of H is
(ii) Again,in view of lemma 1,we have that Jacobson radical r is a weak Hopf ideal. To prove that
Since r_{α}=r∩H_{α} =rad H_{α},x∈H_{α} H and y∈r_{β}=r∩H_{β},then y∈r. Therefore,xy∈r and xy∈H_{αβ}H_{β} H_{αβ},since r is an ideal of H. Consequently,xy∈r∩H_{αβ}=r_{αβ},which implies that H_{α}r_{β} r_{αβ}.
Further,if x∈r_{α}=r∩H_{α} and y∈r_{β} H_{β} H,we obtain xy∈r. As r is an ideal of H and xy∈H_{α}H_{β} H_{αβ},so xy∈r∩H_{αβ}=r_{αβ} and we have r_{α}H_{β} r_{αβ}.
Finally,if x∈r_{α}=r∩H_{α} and y∈r_{β}=r∩H_{β},then xy∈r,xy∈H_{α}H_{β} H_{αβ} H and xy∈r∩H_{αβ}=r_{αβ}. Thus,we have r_{α}r_{β} r_{αβ} for all α,β in Y,and the Jacobson radical
Obviously,
Lemma 3 If
Proof For the elements x+r_{α}∈H_{α}/r_{α} and y+r_{β}∈H_{β}/r_{β},where x is in H_{α} and y is in H_{β},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∈H_{α}H_{β} H_{αβ},xr_{β}∈H_{α}r_{β} r_{αβ} and r_{α}y∈r_{α}H_{β} r_{αβ},hence (x+r_{α})(y+r_{β})=xy+r_{αβ}∈H_{αβ}/r_{αβ}. Thus,we have (H_{α}/r_{α})(H_{β}/r_{β}) H_{αβ}/r_{αβ} for all α,β in Y.
Definition 1 If H is a finite-dimensional basic semilattice graded weak Hopf algebra with Jacobson radical
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
$\left( {{H}_{\alpha }}/{{r}_{\alpha }} \right)\left( {{H}_{\beta }}/{{r}_{\beta }} \right)\to {{H}_{\alpha \beta }}/{{r}_{\alpha \beta }}$ |
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=e_{i,α}|1≤i≤n_{α};α∈Y be a set of orthogonal idempotents in H,where ∑α∈Yn_{α}=n is the dimension of H/r. Then,E is also basis of H/r. Let B=g_{i,α}|1≤i≤n_{α};α∈Y be the corresponding dual basis of (H/r)^{*}. Since g_{i,α}∈Alg(H/r,k)=G((H/r)^{*}) (see^{[10]}th.2.3.1 ).
Thus,g_{i,α}|1≤i≤n_{α}=G((H_{α}/r_{α})^{*})=G_{α} that is a group for each α in Y and
$\Delta \left( gt,\gamma \right)=\sum\limits_{\alpha \beta =\gamma ,i+j=l}{{{g}_{i,a}}\otimes {{g}_{j,\beta }},}$ |
where αβ=γ and i+j=l if,and only if,g_{i,α}.g_{j,β}=g_{l,αβ}=g_{l,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/r→k such that g^{-1}∈Hom_{k}(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/r→k such that h·m=g(h)·m and T(h·m)=m·T(h) for all h∈H/r. If U_{g} 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 V≌U_{g} 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 U_{gi,α} is a 1-dimensional simple module generated by g_{i,α} for some 1≤i≤n_{α},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 U_{s},there exists a 1-dimensional H-module U_{t} such that U_{s}ⓧU_{t}ⓧU_{s}=U_{s} and U_{t}ⓧU_{s}ⓧU_{t}=U_{t}. The weak antipode T is in (H/r)^{*} such that its restriction T/_{(Hα/rα)*} is an antipode in the Hopf algebra (H_{α}/r_{α})^{*} for every α in Y.
Example 1 Let H=k^{S} 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=k^{S}=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
$\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 g_{α}∈G_{α},g_{β} ∈G_{β },g_{γ}∈G_{γ}; α,β,γ∈Y,and 1_{α} is the identity in G_{α} and an idempotent element of S.
The set of primitive orthogonal idempotents
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 {v_{s}}_{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:H→H/r is given by T(v_{s})=v_{t} for some t in S such that v_{t}v_{s}v_{t}=v_{t} and v_{s}v_{t}v_{s}=v_{s},for all s∈S.
(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(v_{s})=∑t∈Sv_{ss′}ⓧv_{t}
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 v_{s}. 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 v_{s-1}=v_{s′} such that v_{s′}v_{s}v_{s′}=v_{s′} and v_{s}v_{s′}v_{s}=v_{s} (or equivalently s′ss′=s′ and ss′s=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 J^{m} I J^{2},for some integer m and for an ideal J generated by all the arrows in Γ such that
We obtain some formulae to calculate the images of T modulo r^{2} (where r=rad H) and the images of Δ modulo rad^{2}(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 }V_{t } denotes the vector space v_{s}r/r^{2}v_{t} over k,then the following conditions hold:
(i) For x∈ _{s }V_{t }， we have Δ(x)=∑s′∈Ss′.xⓧv_{s′}+v_{s′}ⓧx.s′ modulo rad^{2}(HⓧH).
(ii) For all s′,t′ in S，we have dim_{k}(_{s}V_{t})=dim_{k}(_{t′-1s s′-1}V_{t′-1t s′-1}).
(iii) The set {st^{-1}|_{s}V_{t}≠{0}} is a conjugate closed subset of each maximal subgroup G of S such that s,t∈G.
(iv) If {v_{i}}^{m}_{i=1} is a basis for ⊕s∈S(_{s}V_{e}),where each v_{i} is in some vector space _{s }V_{e },then the set s.v_{i}^{m}_{i=1,s∈S} becomes a basis for r/r^{2}.
In the left and the right actions of S on _{s }V_{t } = v_{s}r/r^{2}v_{t},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 r^{2} for the elements of _{s }V_{t },which are arrows.
If T：r→r/r^{2} 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:H→H/r^{2} which satisfies T(x)=-(s·x)·t,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 }V_{t } and s′ in S.
2 Weak covering quiverGREEN et al^{[7]} introduced the covering quiver. We introduce the notion of weak covering quiver here.
Let
Let
Definition 2 A quiver Γ_{S}(W) with the set of vertices represents by Γ_{S}(W)_{0}=∪α∈YΓ_{G}_{α}(W_{α})_{0},where Γ_{Gα}(W_{α})_{0} is a set of vertices of the sub-quiver Γ_{G}_{α}(W_{α}) in the following ways:
(i) The vertices of Γ_{S}(W) is the set of vertices v|v∈S=v_{gα}|g_{α}∈G_{α}_{α∈Y},v_{gα}|g_{α}∈G_{α} as the vertices of Γ_{Gα}(W_{α}) for each α∈Y;
(ii) The arrows are given as follows: if α ≥＼ β,then there does not exist any arrow from v_{g-1α} to
(iii) For an arrow a_{α}:v_{gα}→v_{hα} in Γ_{G}_{α}(W_{α}),let the left weight of a_{α} be defined by l(a_{α})=h_{α}g^{-1}_{α}. The group G_{α} is acting on Γ_{G}_{α}(W_{α}) on the left as follows:
For all g_{α} in G_{α},define g_{α}·v_{hα}=v_{hαg-1α} and g_{α}·(a_{α},h_{α})=(a_{α},g_{α}h_{α}). We call the quiver Γ_{G}_{α}(W_{α}) a sub-quiver of Γ_{S}(W),and it is a covering quiver of G_{α} with respect to W_{α},α∈Y. For an arrow a_{β}:v_{gα}→v_{gβ},α≥β,the left weight of a_{β} 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 Γ_{S}(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,β∈Y.
Then,the quiver Γ_{S}(W) is called a weak covering quiver with respect to weight sequence W,where each sub-quiver Γ_{G}_{α}(W_{α}) is a covering quiver which we call a sub-weak covering quiver of Γ_{S}(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 Γ_{S}(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 Γ_{S}(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 Γ_{S}(W) if,and only if,it is a weak Hopf quiver Γ(S,r).
Proof Let Γ_{S}(W) be a weak covering quiver with respect to W,where S is a Clifford monoid,and W=(W_{α},W_{β},…,W_{δ}) be a sequence with all components W_{α}; α∈Y,with W_{α}=(w_{1,α},w_{2,α},…,w_{nα,α}) as a sequence of distinct elements of the subgroup G_{α} which are closed under conjugation with elements in G_{α}. In fact,elements of W_{α} are disjoint union of elements of the conjugacy classes {C_{α}} of the group G_{α}. Define the ramification data
Conversely,suppose that Γ(S,r) is a weak Hopf quiver with ramification data
Let Γ_{S}(W) be a weak covering quiver with respect to a weight sequence W for a Clifford monoid S. Then,its path coalgebra kΓ_{S}(W) admits the structure of semilattice graded weak Hopf algebra by ^{[6]}. Thus,there is a strong relationship between the weak covering quiver Γ_{S}(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 kΓ_{S}(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
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 Γ_{α} such that (kΓ)_{α}=kΓ_{α} for every α in Y. Let Γ_{0} be the set of vertices of the quiver Γ. We set (Γ_{0})_{α}=(Γ_{α})_{0} as the set of vertices of the sub-quiver Γ_{α}. Let Γ_{n} denote the set of paths of length n in Γ. We set (Γ_{n})_{α}=(Γ_{α})_{n} as the set of paths of length n in the sub-quiver Γ_{α}.
If (kΓ)_{n,α}_{n≥0} is the coradical filtration of coalgebra (kΓ)_{α},then (kΓ)_{0,α}=(kΓ_{0})_{α}=k(Γ_{0})_{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 Γ_{S}(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
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
Assume H is with a strongly graded radical r. Then,
Further,by theorem 1 above,there exists a Clifford monoid S such that
By^{[13]}lemma 2.1,corresponding to a basic and split finite-dimensional Hopf algebra H_{α},there exists a finite group G_{α},such that
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 {v_{s}|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
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=(W_{α},W_{β},…,W_{δ}) is a weight sequence,where W_{α}=(w_{1,α},w_{2,α},…,w_{nα,α}) is a weight sequence of the elements of the subgroup G_{α} closed under conjugation for each α∈Y and w_{iα,α}∈G_{α};1≤i_{α}≤n_{α},α∈Y,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 g_{α}∈G_{α},α≥β∈Y 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/r^{2}.
Consequently,we have (εⓧid)°Δ(v_{s})=(idⓧε)°Δ(v_{s}) modulo r for the vertices,the formulas for the images of T modulo r^{2} and Δ modulo rad^{2}(HⓧH) for the arrows.
By the definition of weak covering quiver,one can identify the quiver Γ with the weak covering quiver Γ_{S}(W). Consequently,by^{[1]} cor.Ⅲ.1.10 and th.Ⅲ.1.9,there exists an ideal I where J^{m} I J^{2} and m≥2,for some ideal J generated by the arrows in Γ_{S}(W) such that
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 Γ_{S}(W) and Γ_{S′}(W′) are isomorphic,where W and W′ are their respective weight sequences.
Proof Suppose H and H′ stated above are isomorphic. Since
Conversely,suppose that the weak covering quivers Γ_{S}(W) and Γ_{S′}(W′) are isomorphic,so that
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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