# Weak Hopf Algebra and Its Quiver Representation

Authors: Muhammad Naseer Khan; Ahmed Munir; Muhammad Arshad; Ahmed Alsanad; Suheer Al-Hadhrami
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

This study induced a weak Hopf algebra from the path coalgebra of a weak Hopf quiver. Moreover, it gave a quiver representation of the said algebra which gives rise to the various structures of the so-called weak Hopf algebra through the quiver. Furthermore, it also showed the canonical representation for each weak Hopf quiver. It was further observed that a Cayley digraph of a Clifford monoid can be embedded in its corresponding weak Hopf quiver of a Clifford monoid. This lead to the development of the foundation structures of weak Hopf algebra. Such quiver representation is useful for the classification of its path coalgebra. Additionally, some structures of module theory of algebra were also given. Such algebras can also be applied for obtaining the solutions of “quantum Yang–Baxter equation” that has many applications in the dynamical systems for finding interesting results.


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## Body

## 1. Introduction

A bialgebraH is equipped with the structures of algebra H,m and coalgebra. If H is a linear space over a field K, then H is called an algebra if H has a unit u:K⟶H and a multiplication m:H⊗H⟶H, such that mId⊗m=mm⊗Id (associativity) and Id=mu⊗Id=mId⊗u (unitary property), where Id is the identity map of H. H is called a coalgebra if H has a comultiplication Δ:H⟶H⊗H and a counit ε:H⟶K, such that Δ⊗IdΔ=Id⊗ΔΔ (coassociativity of Δ) and Id=ε⊗IdΔ=Id⊗εΔ (counitary property) [1]. Then, we have a unique element ρ∈HomkH,H, such that idH∗ρ=ρ∗idH=με, where “∗” is the convolution in HomkH,H. With this map ρ,H becomes a Hopf algebra. Montgomery [2] described the action of Hopf algebra on rings, Me [3] wrote a series of mathematics lecture notes, Redford [4] deliberated the structure of Hopf algebras with a projection, Daele and Wang [5] discussed the source and target algebras for weak multiplier Hopf algebras, Yang and Zhang [6] proposed the ore extensions for Sweedler’s Hopf algebra, Smith [7] formulated the quantum Yang–Baxter equation and quantum quasigroups, Nichita [8] introduced the Yang–Baxter equation with open problems, and Cibils and Rosso [9] introduced the Hopf quiver. According to them, a Hopf quiver is just a Cayley graph of a group. They discussed some matters regarding representations of Hopf algebra/quantum group and quiver. A quiver representation is a set Vi|i∈Q0 of k-vector spaces Vi having finite bases together with the set ∅a:Vta⟶Vha∈Q1 of k-linear maps. We denote a representation by R=Vi,∅a [10].A bialgebraH over a field k is called a weak Hopf algebra if there is an element T in the convolution algebra HomkH,H, such that id∗T∗id=ρ∗id and T∗id∗T=T, and T represents a weak antipode of H. Li obtained solutions for quantum Yang–Baxter equation using such weak Hopf algebra [1, 11, 12]. A weak Hopf algebra H with a weak antipode T is a semilattice graded weak Hopf algebra if H=⊕λεYHλ, where the graded sums Hλ; λεY are the subweak Hopf algebras (which are Hopf algebras) with antipodes restrictions T|Hλ for each λεY [13]. Then, there exist a homomorphism φλ,μ:Hλ⟶Hμ if λμ=μ, such that aεHλ and bεHμ, and the multiplication a⋅b in H is given by(1)a⋅b=φλ,λμaφμ,λμb.A Clifford monoidS is a regular semigroup S. Its center CS contains each of its idempotent. In other words, this is a semilattice of groups which is a collection of maximal subgroups Gλ:λεY of a regular monoid S, such that S=∪λεYGλ and GλGμ⊆Gλμ for all λ,μ=Y, where Y is a semilattice. For any λ,μ=Y with λμ=μ, there are group homomorphism φλ,μ:Gλ⟶Gμ with φλ,λ as an identity homomorphism on Gλ, and if λμ=μ and μν=ν, then φμ,νφλ,μ=φλ,ν. The multiplication in S for all a,b∈S is defined as above in H. The partial ordering “≤” in Y is given by “λ≤μ if and only if λμ=λ for all λ,μ∈Y.”Cibils introduced the Hopf quiver and discussed the structures of the Hopf algebra obtained corresponding to the Hopf quiver [9]. By [14], the categories of Hopf algebra are discussed for the representation has tensor structures induced from the graded Hopf structures of kΓ. By [15], the path coalgebra kΓ of a quiver Γ admits a coquasitriangular Majid algebra structure if and only if Γ is a Hopf quiver of the form ΓG,R with G abelian. Here, the authors gave a classification of the set of graded coquasitriangular Majid structures on connected Hopf quiver. Huang and Tao gave a thorough list of coquasitriangular structures of the graded Hopf algebra over a connected Hopf quiver [16]. Ahmed and Li introduced the concept of the so-called weak Hopf quiver and discussed some structures of its corresponding weak Hopf algebras and weak Hopf modules [17]. Some literature that help for better understanding of these algebra is listed. Auslander et al. [18] gave the theory of representation of artin algebra, China and Montgomery [19] defined the basic coalgebras, Cibils [20] found the tensor product of Hopf bimodules on a group, Nakajima [21] initiated the quiver varieties for ring and representation theorists, Simson [22] discussed the coalgebras, comodules, pseudocompact algebras, and tame comodule type, and Woodcock [23] put some remarks on the theory of representation of coalgebras.In this study, we introduce a notion of weak Hopf quiver representation that generalizes the Hopf quiver representation. We also prove that the Cayley digraph of a Clifford monoidS is embedded in the weak Hopf quiver of the algebra of the Clifford monoid which is also a weak Hopf algebra. Some calculations are made for obtaining the images of various mappings calculated by the tool of Mathematica.


## 2. Preliminaries

We include some necessary concepts of the related matter in this study to make the reader familiar with the matter of the work. First, we include the definition of weak Hopf quiver which is given as follows:Definition 1 (see [17]).
LetS=∪λ∈YGλ be a Clifford monoid, where Y is a semilattice of Gλ;λ∈Y, the subgroups of S.(1)
A ramification datar of S means a sum of rλ=∑cλ∈Cλ˜rCλCλ of subgroups Gλ;λ∈Y, i.e., r=∑λ∈Yrλ=∑λ∈Y∑cλ∈Cλ˜rCλCλ.(2)
Then,r could be viewed as a positive central element of the Clifford monoid ring of S, where Cλ˜ represents the collection of total conjugacy classes of subgroup Gλ for λ∈Y.
LetΓ be a quiver satisfying the following conditions:(a)
The set of vertices ofΓ just represents the set S(b)
Letx∈Gμ,y∈Gλ; x,y∈S and λ,μ∈Y; if μ≱λ, then there does not exists an arrow from x to y, and if μ≥λ, then the number of arrows from x to y is equal to that from φμ,λx to y which is equal to rCλ, if there exist cλ∈Cλ, such that y=cλφμ,λx.
Then,Γ is said to be the corresponding weak Hopf quiver of r. Γ0 is the set of vertices and Γ1 is the set of arrows of Γ.Definition 2 (see [17]).
Let for a quiverΓ and kΓ be the k-space with basis the set of all paths in Γ, where k is a field. Define kΓa by the algebra with multiplication and underlying k-space kΓ as(2)qp=bm,…,b1an,…,a1,iftan=sb1,0,otherwise,for the paths p=an,…,a1 and q=bn,…,b1. Then, kΓa becomes an associative algebra, known as path algebra of Γ [3,16].Definition 3 (see [4]).
LetΓ be a quiver (finite or infinite) and define kΓC to be a coalgebra with comultiplication Δ of kΓC defined by(3)Δp=p⊗sp+∑i=1n−1an,…,ai+1⊗ai,…,a1+tp⊗p,for any path p=an,…,a1:ai∈Γ1; i=1,…,n. For special case, a trivial path ei, the comultiplication is Δ and is described by Δei=ei⊗ei for each vertex i∈Γ0 and the counit ε is defined by(4)εp=0,ifn≥1,1,otherwise.
We usekΓ the path coalgebra of the quiver Γ.Lemma 1 (see [4]).
IfkΓ is the path coalgebra corresponding to the quiver Γ, then kΓ is pointed and GkΓ=Γ0. There is a necessary and sufficient condition between the semilattice-graded weak Hopf algebra and the existence of a weak Hopf quiver corresponding to a Clifford monoid with some ramification data.Theorem 1 (see [1]).
LetΓ represent a quiver; then, the following two statements are equivalent:(i)
The path coalgebrakΓ acknowledges a semilattice-graded weak Hopf algebra structure, such that all graded summands are themselves graded Hopf algebra(ii)
With respect to some ramification data,Γ is the weak Hopf quiver of some Clifford monoid S
The following proposition tells us that the collection of elements of group-like of path coalgebrakΓ of a weak Hopf quiver Γ is a Clifford monid.Proposition 1 (see [1]).
IfΓS,r is a weak Hopf quiver corresponding to a ramification data r of a Clifford monoid S, then Γ0 is the collection of elements of group-like of path coalgebra kΓ, and kΓ0≅kS, the Clifford monoid algebra of S is a subweak Hopf algebra of kΓ.Definition 4 (see [4]).
Supposeu and v represent the vertices in Γ, and k represents a field. The v,u-isotypic component of a kΓ0-bicomodule M is Mvu=m∈M|δLm=v⊗m,δRm=m⊗u. In particular, kΓnvu is the vector space of n-paths from vertex u to vertex v.


## 3. Structures of Weak Hopf Quivers

Here, we discuss the structures of weak Hopf quiver and its algebra. We start by the following example.


### 3.1. An Illustrative Example

LetY=α,β,γ,ρ,σ,δ be the semilattice with multiplication “·” as given in Table 1.Table 1
The semilatticeY=α,β,γ,ρ,σ,δ,⋅.
αβγρσδαααααααβαββαββγαβγαβγραααρρρσαββρσσδαβγρσδFor a ringR with identity R2×2 denotes the 2×2 full matrix ring over R, UR the group consisting of all units in R. Let Z be the integer numbers ring. For a prime p, Zp is a field, and UZp2×2 is just the 2×2 general linear group GL2Zp over Zp. Assume that Gα=eα and Gδ=eδ are the trivial groups, Gβ=GL2Z2,Gγ=UZ42×2,Gρ=GL2Z3,Gσ=UZ62×2. Then, Gu∩Gv=∅, for any u,v∈Y,u≠v, setting S=∪u∈YGu. The multiplication is defined as above on S makes S=∪u∈YGu a Clifford monoid with regards to the semilattice Y [11].The following mappings exist between the subgroups of the Clifford monoid.φδ,δ:Gδ⟶Gδ, defined by φδ,δeδ=eδφδ,σ:Gδ⟶Gσ, defined by φδ,σeδ=eσφδ,γ:Gδ⟶Gγ, defined by φδ,γeδ=eγφδ,β:Gδ⟶Gβ, defined by φδ,βeδ=eβφδ,ρ:Gδ⟶Gρ, defined by φδ,ρeδ=eρφδ,α:Gδ⟶Gα, defined by φδ,αeδ=eαWe denoteCλ as a conjugacy class of the group Gλ,λ∈Y. For each x∈Gδ and y∈Gδ, there exists cδ∈Cδ, such that y=cδφδ,δx. Since there is only one arrow (the loop) from Gδ to Gδ, therefore, rCα=1.φσ,σ:Gσ⟶Gσ,defined byφσ,σ:Z6⟶Z6,UZ6=1¯,5¯(5)φσ,σ1¯=1¯,φσ,σ2¯=2¯,φσ,σ3¯=3¯,φσ,σ4¯=4¯,φσ,σ5¯=5¯,φσ,σ6¯=0¯,φσ,γ:Gσ⟶Gγ,defined byφσ,γ:Z6⟶Z4,UZ4=1¯,3¯(6)φσ,γ1¯=1¯,φσ,γ2¯=2¯,φσ,γ3¯=3¯,φσ,γ4¯=0¯,φσ,γ5¯=1¯,φσ,γ6¯=2¯,φσ,β:Gσ⟶Gβ,defined byφσ,β:Z6⟶Z2,UZ2=1¯(7)φσ,β1¯=1¯,φσ,β2¯=0¯,φσ,σ3¯=1¯,φσ,σ4¯=0¯,φσ,σ5¯=1¯,φσ,σ6¯=0¯,φσ,ρ:Gσ⟶Gρ,defined byφσ,ρ:Z6⟶Z3,UZ3=1¯,2¯(8)φσ,ρ1¯=1¯,φσ,ρ2¯=2¯,φσ,ρ3¯=0¯,φσ,ρ4¯=1¯,φσ,ρ5¯=2¯,φσ,ρ6¯=0¯,φσ,α:Gσ⟶Gα,defined byφσ,αaσ=eα∀aσ∈Gσφγ,γ:Gγ⟶Gγ,defined byφγ,γ:Z4⟶Z4,UZ4=1¯,3¯(9)φγ,γ1¯=1¯,φγ,γ2¯=2¯,φγ,γ3¯=3¯,φγ,γ4¯=0¯,φγ,β:Gγ⟶Gβ,defined byφγ,β:Z4⟶Z2,UZ2=1¯(10)φγ,β1¯=1¯,φγ,β2¯=0¯,φγ,β3¯=1¯,φγ,β4¯=0¯,φγ,α:Gγ⟶Gα,defined byφγ,αaγ=eα∀aγ∈Gγφβ,β:Gβ⟶Gβ,defined byφβ,β:Z2⟶Z2,UZ2=1¯(11)φβ,β1¯=1¯,φβ,β2¯=0¯,φβ,ρ:Gβ⟶Gρ,defined byφβ,ρ:Z2⟶Z3,UZ3=1¯,2¯(12)φβ,ρ1¯=1¯,φβ,ρ2¯=2¯,φβ,ρ3¯=0¯,φβ,α:Gβ⟶Gα,defined byφβ,αaβ=eα∀aβ∈Gβφρ,ρ:Gρ⟶Gρ,defined byφρ,ρ:Z3⟶Z3,UZ3=1¯,2¯(13)φρ,ρ1¯=1¯,φρ,ρ2¯=2¯,φρ,ρ3¯=0¯,φρ,α:Gρ⟶Gα,defined byφρ,αaρ=eα,∀aρ∈Gρφα,α:Gα⟶Gα,defined byφα,αeα=eαFor each given mappingφλ,μ:Gλ⟶Gμ, if it exists, and for any x∈Gλ and y∈Gμ, there exists cμ∈Cμ, such that y=cμφλ,μx for all λ,μ∈Y,μ≥λ. The semilattice of the subgroups of the Clifford monoid along with the mappings among them is shown in Figure 1.Figure 1
Diagram of the semilattice of the subgroups of the Clifford monoid.In Figure1, the arrows show the mappings φλ,μ:Gλ⟶Gμ∀λ≥μ;λ,μ∈Y.(14)H=kS=kGλ.The weak Hopf quiver for the weak Hopf algebraH=kS=⊕λ∈YkGλ=⊕λ∈YHλ, where each Hλ=kGλ is a Hopf algebra. H is in fact a semilattice-graded weak Hopf algebra with HλHμ⊆Hμ, if and only if λ≥μ;λ,μ∈Y.The vertices and arrows of the weak Hopf quiverΓ=S,r corresponding to H is described in the following table instead of drawing its huge digraph, since there is a large number of vertices and arrows in this quiver. The mappings of the type φλ,μ:Gλ⟶Gμ,∀λ,μ∈Y which exist are shown by the symbol “⟶” in Table 2.Table 2
The vertices and arrows of the weak Hopf quiverΓ=S,r corresponding to the weak Hopf algebra H.
Range⟶Domain↓Gδ, Gδ=01Gσ, Gσ=288Gγ, Gγ=96Gβ, Gβ=06Gρ, Gρ=48Gα, Gα=01Number of arrows originating at each element of groupNumber of arrows originating from the whole of groupGδ, Gδ=01⟶⟶⟶⟶⟶⟶440440×1=440Gσ, Gσ=288⟶⟶⟶⟶⟶439439×288=12643Gγ, Gγ=96⟶⟶⟶103103×96=9888Gβ, Gβ=06⟶⟶⟶5555×6=330Gρ, Gρ=48⟶⟶4949×48=2352Gα, Gα=01⟶0101×01=01Number of arrows ending at each element of group01289385391343440Total number of originating arrows↓Number of arrows ending on the whole group01×01=01289×288=83232385×96=36960391×06=2346343×48=16464440×01=440Total number of ending⟶ arrows139443Particularly in the above quiver given in Section3.1, the number of arrows originating in ΓS,r is given by(15)N=440×1+439×288+103×96+55×6+49×48+1×1=139443.The number of arrows ending inΓS,r is given by(16)N=1×1+289×288+385×96+391×6+343×48+440×1=139443.We note that the originating number of arrows is equal to that ending in the quiver.LetN denotes the amount of arrows of quiver ΓS,r, Nλ denote the amount of arrows originating from the vertex represented by the element aλ of subgroup Gλ, and Nλ denotes the amount of arrows ending at the vertex corresponding to the element of subgroup Gλ. Then, we have the following lemma:Lemma 2.
(a)
The number of arrows originating inΓS,r is given by N=∑λ∈YNλGλ(b)
The number of arrows ending inΓS,r is given by N′=∑λ∈YNλGλ(c)
N=N′ = total numbers of arrows of the weak Hopf quiver ΓS,r.Proof.
The proofs of (a), (b), and (c) are obvious from Table2.
In view of Section3.1, the following results can immediately be identified and obtained in a weak Hopf quiver ΓS,r.


### 3.2. Results

Letx∈Gλ,y∈Gμ, and φλ,μ:Gλ⟶Gμ,λ≥μ;λ,μ∈Y. Then, there exists a unique arrow from x to y (or φλ,μ to y) and satisfies y=cμφλ,μx,cμ∈Cμ; therefore, rCμ=1∀μ∈Y.(i)
Ifrλ is the ramification data of group Gλ, then rλ=∑Cλ∈Cλ˜rCλCλ using (i)(ii)
The ramification data of the Clifford monoidS=∪λ∈YGλ is(17)r=∑λ∈Yrλ=∑λ∈y∑Cλ∈Cλ˜rCλCλ=∑λ∈ycλ∈C˜λCλ,WhereCλ represents the collection of total conjugacy classes of a group Gλ.(iii)
The number of arrows inΓ as obtained from Section 3.1 is 139443(iv)
The number of vertices of the weak Hopf quiverΓS,r from Section 3.1 is S=∑λεYGλ=440(v)
If there is an arrow from some elementxεGλ to some element y∈Gμ, then there are arrows from each xεGλ to yεGμ(vi)
The dimension of weak Hopf algebraH corresponding to Γ is the number of vertices of the weak Hopf quiver(vii)
The loops which exist are the arrows from each idempotent to itself. Thus, the number of loops is the order of the semilatticeY.(viii)
For a finite Clifford monoid,ΓS,r corresponding to H=kS has no loop if and only if r=0. Then, the quiver is a set S of number of isolated vertices. Otherwise, the weak Hopf quiver is a connected diagraph.(ix)
LetS and S′ be two finite Clifford monoids, and ΓS,r and Γ′S,´r′ be the weak Hopf quivers corresponding to two weak Hopf algebras H=kS and H′=kS, respectively. Then, the quivers Γ and Γ′ are isomorphic if and only if there is a bijective mapping φ:Γ⟶Γ′ between their sets of vertices, such that the number of arrows from v1 to v2, v1,v2∈Γ0, is equal to that from φv1 to φv2, where φv1,φv2∈Γ0´ [1].


## 3.1. An Illustrative Example

LetY=α,β,γ,ρ,σ,δ be the semilattice with multiplication “·” as given in Table 1.Table 1
The semilatticeY=α,β,γ,ρ,σ,δ,⋅.
αβγρσδαααααααβαββαββγαβγαβγραααρρρσαββρσσδαβγρσδFor a ringR with identity R2×2 denotes the 2×2 full matrix ring over R, UR the group consisting of all units in R. Let Z be the integer numbers ring. For a prime p, Zp is a field, and UZp2×2 is just the 2×2 general linear group GL2Zp over Zp. Assume that Gα=eα and Gδ=eδ are the trivial groups, Gβ=GL2Z2,Gγ=UZ42×2,Gρ=GL2Z3,Gσ=UZ62×2. Then, Gu∩Gv=∅, for any u,v∈Y,u≠v, setting S=∪u∈YGu. The multiplication is defined as above on S makes S=∪u∈YGu a Clifford monoid with regards to the semilattice Y [11].The following mappings exist between the subgroups of the Clifford monoid.φδ,δ:Gδ⟶Gδ, defined by φδ,δeδ=eδφδ,σ:Gδ⟶Gσ, defined by φδ,σeδ=eσφδ,γ:Gδ⟶Gγ, defined by φδ,γeδ=eγφδ,β:Gδ⟶Gβ, defined by φδ,βeδ=eβφδ,ρ:Gδ⟶Gρ, defined by φδ,ρeδ=eρφδ,α:Gδ⟶Gα, defined by φδ,αeδ=eαWe denoteCλ as a conjugacy class of the group Gλ,λ∈Y. For each x∈Gδ and y∈Gδ, there exists cδ∈Cδ, such that y=cδφδ,δx. Since there is only one arrow (the loop) from Gδ to Gδ, therefore, rCα=1.φσ,σ:Gσ⟶Gσ,defined byφσ,σ:Z6⟶Z6,UZ6=1¯,5¯(5)φσ,σ1¯=1¯,φσ,σ2¯=2¯,φσ,σ3¯=3¯,φσ,σ4¯=4¯,φσ,σ5¯=5¯,φσ,σ6¯=0¯,φσ,γ:Gσ⟶Gγ,defined byφσ,γ:Z6⟶Z4,UZ4=1¯,3¯(6)φσ,γ1¯=1¯,φσ,γ2¯=2¯,φσ,γ3¯=3¯,φσ,γ4¯=0¯,φσ,γ5¯=1¯,φσ,γ6¯=2¯,φσ,β:Gσ⟶Gβ,defined byφσ,β:Z6⟶Z2,UZ2=1¯(7)φσ,β1¯=1¯,φσ,β2¯=0¯,φσ,σ3¯=1¯,φσ,σ4¯=0¯,φσ,σ5¯=1¯,φσ,σ6¯=0¯,φσ,ρ:Gσ⟶Gρ,defined byφσ,ρ:Z6⟶Z3,UZ3=1¯,2¯(8)φσ,ρ1¯=1¯,φσ,ρ2¯=2¯,φσ,ρ3¯=0¯,φσ,ρ4¯=1¯,φσ,ρ5¯=2¯,φσ,ρ6¯=0¯,φσ,α:Gσ⟶Gα,defined byφσ,αaσ=eα∀aσ∈Gσφγ,γ:Gγ⟶Gγ,defined byφγ,γ:Z4⟶Z4,UZ4=1¯,3¯(9)φγ,γ1¯=1¯,φγ,γ2¯=2¯,φγ,γ3¯=3¯,φγ,γ4¯=0¯,φγ,β:Gγ⟶Gβ,defined byφγ,β:Z4⟶Z2,UZ2=1¯(10)φγ,β1¯=1¯,φγ,β2¯=0¯,φγ,β3¯=1¯,φγ,β4¯=0¯,φγ,α:Gγ⟶Gα,defined byφγ,αaγ=eα∀aγ∈Gγφβ,β:Gβ⟶Gβ,defined byφβ,β:Z2⟶Z2,UZ2=1¯(11)φβ,β1¯=1¯,φβ,β2¯=0¯,φβ,ρ:Gβ⟶Gρ,defined byφβ,ρ:Z2⟶Z3,UZ3=1¯,2¯(12)φβ,ρ1¯=1¯,φβ,ρ2¯=2¯,φβ,ρ3¯=0¯,φβ,α:Gβ⟶Gα,defined byφβ,αaβ=eα∀aβ∈Gβφρ,ρ:Gρ⟶Gρ,defined byφρ,ρ:Z3⟶Z3,UZ3=1¯,2¯(13)φρ,ρ1¯=1¯,φρ,ρ2¯=2¯,φρ,ρ3¯=0¯,φρ,α:Gρ⟶Gα,defined byφρ,αaρ=eα,∀aρ∈Gρφα,α:Gα⟶Gα,defined byφα,αeα=eαFor each given mappingφλ,μ:Gλ⟶Gμ, if it exists, and for any x∈Gλ and y∈Gμ, there exists cμ∈Cμ, such that y=cμφλ,μx for all λ,μ∈Y,μ≥λ. The semilattice of the subgroups of the Clifford monoid along with the mappings among them is shown in Figure 1.Figure 1
Diagram of the semilattice of the subgroups of the Clifford monoid.In Figure1, the arrows show the mappings φλ,μ:Gλ⟶Gμ∀λ≥μ;λ,μ∈Y.(14)H=kS=kGλ.The weak Hopf quiver for the weak Hopf algebraH=kS=⊕λ∈YkGλ=⊕λ∈YHλ, where each Hλ=kGλ is a Hopf algebra. H is in fact a semilattice-graded weak Hopf algebra with HλHμ⊆Hμ, if and only if λ≥μ;λ,μ∈Y.The vertices and arrows of the weak Hopf quiverΓ=S,r corresponding to H is described in the following table instead of drawing its huge digraph, since there is a large number of vertices and arrows in this quiver. The mappings of the type φλ,μ:Gλ⟶Gμ,∀λ,μ∈Y which exist are shown by the symbol “⟶” in Table 2.Table 2
The vertices and arrows of the weak Hopf quiverΓ=S,r corresponding to the weak Hopf algebra H.
Range⟶Domain↓Gδ, Gδ=01Gσ, Gσ=288Gγ, Gγ=96Gβ, Gβ=06Gρ, Gρ=48Gα, Gα=01Number of arrows originating at each element of groupNumber of arrows originating from the whole of groupGδ, Gδ=01⟶⟶⟶⟶⟶⟶440440×1=440Gσ, Gσ=288⟶⟶⟶⟶⟶439439×288=12643Gγ, Gγ=96⟶⟶⟶103103×96=9888Gβ, Gβ=06⟶⟶⟶5555×6=330Gρ, Gρ=48⟶⟶4949×48=2352Gα, Gα=01⟶0101×01=01Number of arrows ending at each element of group01289385391343440Total number of originating arrows↓Number of arrows ending on the whole group01×01=01289×288=83232385×96=36960391×06=2346343×48=16464440×01=440Total number of ending⟶ arrows139443Particularly in the above quiver given in Section3.1, the number of arrows originating in ΓS,r is given by(15)N=440×1+439×288+103×96+55×6+49×48+1×1=139443.The number of arrows ending inΓS,r is given by(16)N=1×1+289×288+385×96+391×6+343×48+440×1=139443.We note that the originating number of arrows is equal to that ending in the quiver.LetN denotes the amount of arrows of quiver ΓS,r, Nλ denote the amount of arrows originating from the vertex represented by the element aλ of subgroup Gλ, and Nλ denotes the amount of arrows ending at the vertex corresponding to the element of subgroup Gλ. Then, we have the following lemma:Lemma 2.
(a)
The number of arrows originating inΓS,r is given by N=∑λ∈YNλGλ(b)
The number of arrows ending inΓS,r is given by N′=∑λ∈YNλGλ(c)
N=N′ = total numbers of arrows of the weak Hopf quiver ΓS,r.Proof.
The proofs of (a), (b), and (c) are obvious from Table2.
In view of Section3.1, the following results can immediately be identified and obtained in a weak Hopf quiver ΓS,r.


## 3.2. Results

Letx∈Gλ,y∈Gμ, and φλ,μ:Gλ⟶Gμ,λ≥μ;λ,μ∈Y. Then, there exists a unique arrow from x to y (or φλ,μ to y) and satisfies y=cμφλ,μx,cμ∈Cμ; therefore, rCμ=1∀μ∈Y.(i)
Ifrλ is the ramification data of group Gλ, then rλ=∑Cλ∈Cλ˜rCλCλ using (i)(ii)
The ramification data of the Clifford monoidS=∪λ∈YGλ is(17)r=∑λ∈Yrλ=∑λ∈y∑Cλ∈Cλ˜rCλCλ=∑λ∈ycλ∈C˜λCλ,WhereCλ represents the collection of total conjugacy classes of a group Gλ.(iii)
The number of arrows inΓ as obtained from Section 3.1 is 139443(iv)
The number of vertices of the weak Hopf quiverΓS,r from Section 3.1 is S=∑λεYGλ=440(v)
If there is an arrow from some elementxεGλ to some element y∈Gμ, then there are arrows from each xεGλ to yεGμ(vi)
The dimension of weak Hopf algebraH corresponding to Γ is the number of vertices of the weak Hopf quiver(vii)
The loops which exist are the arrows from each idempotent to itself. Thus, the number of loops is the order of the semilatticeY.(viii)
For a finite Clifford monoid,ΓS,r corresponding to H=kS has no loop if and only if r=0. Then, the quiver is a set S of number of isolated vertices. Otherwise, the weak Hopf quiver is a connected diagraph.(ix)
LetS and S′ be two finite Clifford monoids, and ΓS,r and Γ′S,´r′ be the weak Hopf quivers corresponding to two weak Hopf algebras H=kS and H′=kS, respectively. Then, the quivers Γ and Γ′ are isomorphic if and only if there is a bijective mapping φ:Γ⟶Γ′ between their sets of vertices, such that the number of arrows from v1 to v2, v1,v2∈Γ0, is equal to that from φv1 to φv2, where φv1,φv2∈Γ0´ [1].


## 4. Representation of Weak Hopf Quiver

A Hopf quiver representation is defined in [15] and some structures are given in this regard. We generalize this notion as a weak Hopf quiver representation and discuss its structures. One can see also the quiver representation of a bialgebra [17].Definition 5.
(See [13]). A weak Hopf quiver representation is a class of vector spaces Vi,λ|i∈Γ0,λ∈Y of finite-dimensional k-vector spaces Vi,λ,i∈Γ0,λ∈Y together with a collection of mappings.(18)ϕλ,μm:Vtm,λ⟶Vsm,μ|m∈Γ1,tm,sm∈Γ0,μ≥λ,λ,μ∈Y.
We denoteVi,λ,Vj,μ;ϕλ,μm;i,j∈Γ0 by ℛλ,μ. A representation ℛ of the weak Hopf quiver is given by ℛ=ℛλ,μ;μ≥λ,λ,μ∈Y.
Letℛ=ℛλ,μ;μ≥λ,λ,μ∈Y and S=Sλ,μ;μ≥λ,λ,μ∈Y be two representations of weak Hopf quiver ΓS,r, where ℛλ,μ=Vi,λ,Vj,μ;ϕλ,μm and Sλ,μ=Wi,λ,Wj,μ;ψλ,μm. The representation Sλ,μ is a subrepresentation of ℛλ,μ if(a)
For alli,j∈Γ0,Wi,λ,Wj,μ are the subspaces of Vi,λ and Vj,μ, respectively, and(b)
For everym∈Γ1, the restriction of ϕλ,μm to Wtm,λ is the mapping φλ,μm|Wtm,λ and is given by φλ,μm|Wtm,λ:Wtm,λ⟶Wsm,μ.
Then,S=Sλ,μ:μ≥λ;λ,μ∈Y is called subrepresentsation of ℛ=ℛλ,μ:μ≥λ;λ,μ∈Y.
A nonzero representationV is called simple if the only subrepresentation of V is the zero representation and the V itself.
Given that a representationℛ=Vi,λ,φλ,λi of the quiver ΓS,r, we can obtain a representation φλ,λi:kΓ⟶End⊕i∈Γ0λ∈YVi,λ of kΓ, see also the framed representation in [13].
It suffices to define the representation onei’s and fj’s, and these generate the basis of a ring.(19)φλ,λiei≔Id|vi,φλ,λjfj:Vtjλ⟶Vhjλ,x↦φλ,λjx.
This gives an extension to a representation on all elements ofkΓ.
The direct sum of two weak Hopf quiver representations is given as follows:Definition 6.
(see [21]). If ℛ=ℛλ,μ;μ≥λ,λ,μ∈Y and S=Sλ,μ;μ≥λ,λ,μ∈Y be two representations of weak Hopf quiver ΓS,r, where ℛλ,μ=Vi,λ,Vj,μ;ϕλ,μm and Sλ,μ=Wi,λ,Wj,μ;ψλ,μm, then we define a direct-sum representation as follows:(20)ℛ⊕S=ℛλ,μ⊕Sλ,μ:μ≥λ;λ,μ∈Y,with χλ,μm=ϕλ,μm⊕ψλ,μm:μ≥λ;λ,μ∈Y by(a)
ui,λ=Vi,λ⊕Wi,λ for every i∈Γ0 and λ∈Y(b)
χλ,μm:Vtm,λ⊕Wtm,λ⟶Vsm,μ⊕Wsm,μ is defined by the matrix(21)Vλm00Wλm,for m∈Γ1 and λ∈Y.
Now, we define a morphism of a weak Hopf quiver representation to another weak Hopf quiver representation as follows.Definition 7.
(see [15]). If ℛ and S be two representations of the weak Hopf quiver ΓS,r, then Φ:ℛ⟶S as a representation morphism is a collection of k-linear maps φλ,μi:Vi,λ⟶Wi,μ|i∈Γ0&μ≥λ;λ,μ∈Y, where ℛ=ℛλ,μ:μ≥λ;λ,μ∈Y, S=Sλ,μ:μ≥λ;λ,μ∈Y, such that the following Figure 2 is commutative for all m∈Γ1.
Supposeφλ,μi:Vi,λ⟶Wi,μ is invertible for each i∈Γ0 and all μ≥λ;λ,μ∈Y, we have the morphism Φ:ℛ⟶S, which is called isomorphism from ℛ to S.
A representationℛ of a weak Hopf quiver Γ is decomposable if there exist two nonzero representations S and T, such that ℛ≅S⊕T, and a nonzero representation is indecomposable if it is not decomposable [15].
We introduce the notion of canonical representation ofΓ and observe that it is also a simple one.Figure 2
Commutative representation, whereψλ,μm:Wtm,λ⟶Wsm,μ.Definition 8.
(see [15]). A canonical representation ℛ=ℛλ,μ:μ≥λ;λ,μ∈Y for weak Hopf quiver ΓS,r is a collection of representations ℛλ,μ, such that(22)ℛλ,μ=Vi,λ=k,for onei∈Γ0,0,otherwise,φλ,μm=0,for allm∈Γ1,μ≥λ;λ,μ∈Y.
A canonical representationℛλ,μ must be a simple for all λ,μ,μ≥λ;λ,μ∈Y, since the only a subspace of each one is Vi,λ, the null space at every vertex.
LetΓ be a weak Hopf quiver having no oriented cycles. A representation ℛ of Γ is simple if and only if it is canonical.
IfΓS,r is a weak Hopf quiver without any oriented cycle, then there exists some vertex e1∈Γ0, which is not a tail of some arrows. This type of arrow is called a sink.
LetΓ be a weak Hopf quiver with no oriented cycle, and e1∈Γ0 be a vertex, such that tm≠e1, for all m∈Γ1.Proposition 2.
Letℛ be a canonical representation of a weak Hopf quiver ΓS,r. Then, the representation S=Sλ,μ:μ≥λ;λ,μ∈Y, where(23)Sλ,μ=Wi,λ=k,i=x,0,i≠x,φλ,μm=0,for allm∈Γ1,μ≥λ;λ,μ∈Y,for the weak Hopf quiver ΓS,r is a subrepresentation of ℛ.Proof.
Obviously, for eachi≠x,0=Wi,λ is a subspace of Vi,λ. Since Vx,λ is a nonzero k-vector space, k=Wx,λ⊆Vx,λ. Define p=pi,λ;λ∈Y:i∈Γ0 a representation morphism, such that pi,λ:Wi,λ⟶Vi,λ is the inclusion mapping. To verify that all mappings commute, m∈Γ1, such that tm≠x,Wtm,λ=0. So, ψλ,μm:Wtm,λ⟶Whm,μ has its domain as {0}, i.e., ψλ,μm=0. Similarly, ptm,λ:Wtm,λ⟶Vtm,λ is the inclusion of 0 that implies ptm,λ=0. Hence, for all m∈Γ1, such that tm≠x, we have phm,λ∘ψλ,μm=phm,λ∘0=0 and ψλ,μm∘Ptm,λ=φλ,μm∘0=0, so the diagram is commutative. For each m∈Γ1 with tm=x, we have that Vhm,λ=0. Hence, φλ,μm:Vtm,λ⟶Vhm,μ is φλ,μm:Vx,λ⟶Vx,μ=0, i.e., φλ,μm=0. Similarly, ψλ,μm=0, and phm,λ:0⟶0 is also the zero mapping. So, for all m∈Γ1, such that tm=x, we have phm,λ=ψλ,λm=0∘0=0. Hence, the diagram is commutative. Thus, S becomes a subrepresentation of ℛ.


## 5. Weak Hopf Quiver as Cayley Graph

LetS be a semigroup and C be a subset of S. Recall that the Cayley graph Cay S,C of S with the connection set C is defined as the digraph with a vertex set S and arc set ECayS,C=s,cs:s∈S,c∈C.In the following result, we give an embedding of a Cayley graph of a Clifford monoidS into the weak Hopf quiver of the corresponding weak Hopf algebra kS.Theorem 2.
Every Cayley graphCayS,C of a Clifford monoid S can be embedded into its corresponding weak Hopf quiver ΓS,r of the weak Hopf algebra H=kS=⊕λεYkGλ.Proof.
Define mappingφ:CayS,C⟶ΓS,r, such that φx=ex∈Γ0, for all x∈VS.
Letcyux represents the edge of the Cayley graph from vertex x to vertex y in EC.
Then,φcyux=cyvx∈Γ1,∀cyux∈EC, where y=cx for some c∈C,x,y∈S, and cyvx is the arrow in Γ1, such that y=cλφλ,μx for some cλ (if it exist) in Cλ, the conjugacy class of Gλ for all x∈Gλ,y∈Gμ;μ≥λ;λ,μ∈Y.
Clearly,φ is an injective mapping from CayS,C to the weak Hopf quiver ΓS,r.
Thus, the Cayley graph of a Clifford monoidS can be embedded into its corresponding weak Hopf quiver ΓS,r.


## 6. Conclusion

In this article, the formula that enumerates the arrows in the weak Hopf quiverΓS,r is devised. In addition, the verification of the fact is that the number of arrows originating and ending is equal in such quiver. It is further observed that a weak Hopf quiver representation appears as a generalization of the Hopf quiver representation. For each canonical representation, there exists a subrepresentation as given in Proposition 2.Furthermore, it is perceived that the Cayley digraph of a Clifford monoid is embedded in the corresponding weak Hopf quiver of its corresponding weak Hopf algebra.



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