# Double Weak Hopf Quiver and Its Path Coalgebra

Authors: Muhammad Naseer Khan; Munir Ahmed; Muhammad Arshad; Waleed Almutiry; Rashad A. R. Bantan; Mohammed Elgarhy
Journal: Journal of Function Spaces
 (2022)

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## Abstract

The main input of this research is the introduction of the concept of double weak Hopf quiver (DWHQ). In addition, the structures of weak Hopf algebra (WHA) are obtained through path coalgebra of the proposed quivers. Furthermore, the module and comodule structures on the said WHA are discussed. Moreover, the classification of the semilattice-graded WHA structures of the path coalgebra obtained from the so called DWHQ was presented. This contributes a step further in the development of the module and comodule structures of more general algebras. These structures are important in the physics for dynamic systems.


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## Body

## 1. Introduction

Hopf quiver (HQ) was introduced by Cibils and Rosso, who also explained the framework and structures of the Hopf algebra (HA) which was obtained corresponding to the HQ [1]. By [2], the classes of HA are deliberated for the representation has tensor structures that are induced by the graded Hopf structures of kΓ. Huang and Tao [3] give a thorough list of graded coquasitriangular HA over the HQ. Ahmed and Li introduced the concept of the so-called weak HQ (WHQ) and discussed some structures of its corresponding weak HA (WHA) and weak Hopf modules (WHM) [4]. Some literatures that aid in better understanding these algebras are listed in [1, 2, 5–10]. Nichita [5] presented the Yang-Baxter equation with open problems. Virgina et al. [6] defined the quivers, Yang [7] worked on the weak Hopf algebras corresponding to Cartan matrices, Cheng [8] described the classes of weak Hopf algebra, Cao [9] discusses the semilattice-graded WHA along with its quasibicrossed product, Huang and Yang [2] gave the green ring of minimal HQ, and Hai [10] introduced the coquasitriangluar Hopf algebras in braided categories. Furthermore, refer to [3, 4, 11–18] for representation theory, bimodules, and characteristics of Hopf Algebra; Huang and Tao [3] also worked on coquasitriangular structures on Hopf quivers. Ahmed and Li [4] proposed the weak Hopf quivers. Auslander et al. [11] came with the representation theory of artin algebra, Chin and Montgomery [12] defined the basic coalgebras. Cibils [13] found the tensor products of Hopf bimodules on some group. Nakajima [14] introduced the quiver varieties for representation theorists and ring. Simson [15] studied the comodules, coalgebras, tame comodule type, and pseudocompact algerbas. Woodcock [16] offered some remarks on the coalgebras’ representation theory. Alvarez et al. [17] proposed quasigroupoids and weak Hopf quasigroups, and Rodriguez and Raposo [18] formulated the weak crossed products over weak Hopf algebras. Naseer Khan et al. [19] worked on the WHA and its quiver representation. Li [20] solved the quantum Yang-Baxter equation, Montgomery [21] showed the action of HA on rings, Radford [22] described the projection of structures of HA, Daele and Wang [23] defined the multipliers of HA, Swedler [24], Yang and Zhang [25], and Smith [26] also extended the theory of Hopf Algebra.Dual weak HQ (DWHQ) is introduced in [27], where the structures of the path coalgebra of a dual weak Hopf quiver are discussed. This coalgebra equips the structure of a WHA.Dual weak Hopf quiver and its path algebra which is a semilattice-graded weak Hopf algebra (SLGWHA). Fora¯,b¯∈Ⱨ∗, where a¯∈G∗μ, and b¯∈Gλ∗, the multiplication a¯.b¯ in Ⱨ¯ is given by a¯.b¯=φ∗μ,λμa¯φ∗λ,λμb¯.We introduce the double weak Hopf quiver and obtain a WHA structure from the path coalgebra of this quiver. The path coalgebrakΓ¯ has the structure of cotensor coalgebra. We denoteΓ¯ by Γ¯S×S∗,R is called a double WHQ of a Clifford monoid. We have an algebra Ⱨ¯=⨁μ,λ∈YⱧμ⊗Ⱨλ∗=kS⊗kS∗≅kΓ¯S×S∗,R, where Ⱨμ=kGμ and Ⱨλ∗=kGλ∗.The weak Hopf algebraⱧ¯ is semilattice graded if Ⱨ¯=⨁μ,λϵYⱧμ⊗Ⱨ∗λ is a semilattice-graded sum, whereⱧμ,λ; μ,λϵY are subweak HAs which are HAs with antipodes the restriction T¯Ⱨμ,λ for every μ,λϵY [9]. There are weak Hopf algebra homomorphism φγ,δμ,λ:Gμ×Gλ∗⟶Gγ×Gδ∗, where φγ,δμ,λ=φμ,λ,φ∗γ,δ and Gα×Gβ;α,βϵY is a of S×S∗. For two homorphism φγ,δμ,λ, φσ,ργ,δ, their composition φσ,ργ,δφγ,δμ,λ=φσ,ρμ,λ from Ⱨλ⊗Ⱨμto Ⱨσ⊗Ⱨρ if λμ=μ.Letu,v∈S×S∗ the multiplication u.v in Ⱨ¯ is given by u.v=φγ,δμ,λuφσ,ργ,δv, where φγ,δμ,λ:Gμ×Gλ∗⟶Gγ×Gδ∗ is the mapping from the subgroup Gμ×Gλ∗ to Gγ×Gδ∗ for all μ≥γ or ;λ≤δ;λ,μ,γ,δ∈Y.(1)u.v=φγ,δμ,λuφσ,ργ,δv.A Clifford monoidS×S∗ is a regular semigroup S×S∗ with identity. Let u,v∈S×S∗ where u=gμ,g∗δ∈Gμ×Gδ∗ and v=gγ,g∗ρ∈Gγ×Gρ∗ then
(2)u.v=φγ,δμ,λuφσ,ργ,δv.The structure of comultiplication inⱧ¯=Ⱨ⊗Ⱨ∗ is defined by
(3)Δa=Δx⊗y¯=∑x′,x′′∈Ⱨy¯′,y¯′′∈Ⱨ∗x′⊗y¯′⊗x′′⊗y¯′′,for all a=x⊗y¯∈Ⱨ¯=Ⱨ⊗Ⱨ∗, where x∈Ⱨ & y¯∈Ⱨ∗. Since the weak antipode T in Ⱨ is an element of ⱧoɱκⱧ,Ⱨ and the weak antipode T∗exists in Ⱨ∗ which is an element of ⱧoɱκⱧ∗,Ⱨ∗ so that T¯ exists in Ⱨ¯=Ⱨ⊗Ⱨ∗ as an element of ⱧoɱκⱧ¯,Ⱨ¯. idⱧ¯∗T¯∗idⱧ¯=idⱧ¯ satisfying T¯∗idⱧ¯¯∗T¯=T¯. This makes Ⱨ¯ is a WHA. If Ⱨ is finite dimensional vectors space, so is Ⱨ∗ hence Ⱨ¯=Ⱨ⊗Ⱨ∗ is also a finite dimensional. Thus
(4)∴Ⱨ¯∗=Ⱨ¯.Here,kΓ¯0 is a linearization of Γ¯0 having the natural structure of coalgebra. The vector space kΓ¯ is said to be a kΓ¯0 bimodule and kΓ¯0 comodule with left and right comodule structure maps δLu=tu⊗u and δRu=u⊗su, respectively ∀u∈kΓ¯. The self-cotensor product of kΓ¯ is the kernel of
(5)δR⨂1−1⨂δL:kΓ¯⨂kΓ¯⟶kΓ¯⨂kΓ¯0⨂kΓ¯.The cotensor coalgebra overkΓ¯0 of the bicomodule kΓ¯ is a path coalgebra.For two verticesu and v of Γ¯ the (v,u)-isotypic component (IC) of a kΓ¯0-bicomodule, Ɱ¯ is
(6)vⱮ¯u=ɱ∈Ɱ¯δRɱ=ɱ⊗u,δLɱ=v⊗ɱ.Particularly,vkΓ¯nu corresponds to the vector space that has n-paths to v from u, see [1].A weak Hopf bimodule is a semigroup algebra provided that the classification of path coalgebra admits graded WHA structure. The representations of the WHA are called the weak Hopf bimodules (WHBM).By Cibils and Rosso [1], we obtain that there is a one-to-one correspondence between the elements of WHBM with a complete list-graded WHA structures on the path coalgebra kΓ¯ corresponding to the DWHQ Γ¯.Further, the group-like elements of the SLGWHA correspond to the vertices of the DWHQΓ¯. By [17, th. 4.5], the GⱧ¯=GkS⊗kS∗ of SLGWHA Ⱨ¯ is the Clifford monoid S×S∗. Thus, kS×S∗ becomes a SLGWHA.The remaining paper is arranged such that Section2 defines some basic concepts, Section 3 presents the main contributions of the paper, i.e., double weak Hopf quiver, its results, properties, and example. Section 4 concludes the paper.


## 2. Preliminaries

This section presents some basic concepts and definitions to the work such as algebra, coalgebra, bialgebra, module, comodule, and bimodule.Definition 1. (see [20]).
Ak algebra Ⱨ is a vector space over a field k. Ⱨ is called an algebra if Ⱨ has a multiplication ɱ:Ⱨ⊗Ⱨ⟶Ⱨ and a unit η:k⟶Ⱨ such that ɱId⊗ɱ=ɱɱ⊗Id (associativity) and Id=ɱu⊗Id=ɱId⊗u (unitary property), where Id is the identity map of Ⱨ.Definition 2. (see [20]).
Ⱨ is called a coalgebra if Ⱨ has a comultiplication Δ:Ⱨ⟶Ⱨ⊗Ⱨ and a counit ε:Ⱨ⟶k∋Id⊗ΔΔ=Δ⊗IdΔ (this property is known as coassociativity of Δ) and Id=ε⊗IdΔ=Id⊗εΔ (this property is known as counitary property).Definition 3. (see [20]).
A bialgebra is quintuple (Ⱨ,μ,Δ,η,ϵ) where Ⱨ is a vector space, μ is multiplication, Δ is comultiplication, η is unit, and ϵ is counit. An algebra is a k-vector space Ⱨ endowed with k-linear maps.
μ:Ⱨ⊗Ⱨ⟶Ⱨ, Δ:Ⱨ⟶Ⱨ⊗Ⱨ, ϵ:Ⱨ⟶ℂ, η:ℂ⟶Ⱨ.Definition 4. (see [21]).
If we have a unique elementρ∈ⱧoɱκⱧ,Ⱨ such that idⱧ∗ρ=ρ∗idⱧ=με, where "∗” is the convolution in ⱧoɱκⱧ,Ⱨ. Then, Ⱨ becomes a Hopf algebra with ρ as the antipode of Ⱨ.Definition 5. (see [20]).
A bialgebraⱧ over a field k is called a weak Hopf algebra if there is an element T in the convolution algebra ⱧoɱκⱧ,Ⱨ such that id∗T∗id=idand T∗id∗T=T, where T is a weak antipode of Ⱨ see.Definition 6. (see [9]).
A WHAⱧ with a weak antipode T is a semilattice graded if Ⱨ=⨁λϵYⱧλ, where the graded summandsⱧλ; λϵYare subweak HA. Obviously, these are HAs with antipodes the restrictions ΤⱧλ for eachλϵY. Then, there exist homomorphisms φλ,μ:Ⱨλ⟶Ⱨμ if λμ=μ such that aϵⱧλ andbϵⱧμ, the multiplication a.bin Ⱨ is given by a.b=φλ,λμaφμ,λμb.Definition 7. (see [13]).
LetⱧ,ɱ,v be a k-algebra. A left A-module is a pair Ɱ,γ, where Ɱ is a k-vector space, and γ:Ⱨ⊗Ɱ⟶Ɱ such that the diagrams in Figures 1(a) and 1(b).Figure 1
Commute.The Cat. of leftⱧ-module is denoted by ⱧM.Definition 8. (see [15]).
LetⱧ,Δ,ε be a k-algebra, a right Ⱨ-c-module is a is a k-vector space Ɱ with a k-linear map ρ:Ɱ⟶Ɱ⊗Ⱨ such that the diagrams in Figures 2(a) and 2(b).Figure 2
Commute.The Cat. of rightⱧ-comodule is denoted by MⱧDefinition 9. (see [15]).
AnⱧ-bimodule is a left and a right module Ɱ with structure maps μl:Ⱨ⊗Ɱ⟶Ɱ and μr:Ɱ⊗Ⱨ⟶Ɱ that satisfy the conditions in Figure 3.Figure 3
Left and right actions commute.


## 3. Double Weak Hopf Quiver

SupposeS=⋃μ∈YGμ is a Clifford monoid and Y represents semilattice. We take S as the basis for the underlying vector space of the weak Hopf algebra Ⱨ=kS which is of course a “SLGWHA”. Let S∗ be the dual basis of S and is a basis for the dual vector space Ⱨ∗=kS∗=⨁μ∈YkGμ∗=⨁μ∈YkGμ∗=⨁μ∈YⱧ∗μ=kS∗=kS∗, where G∗μ is dual basis of the basis Gμ of a vector space Ⱨ∗μ=kGμ, which equips the structure of a Hopf algebra for every μ∈Y. If x∈S and y¯∈S∗, then u=x,y¯ becomes an element of S×S∗ which is the basis of the vector space Ⱨ¯=kS⊗kS∗, where Ⱨ¯ is also a weak Hopf algebra.Definition 10.
For allu,v∈S×S∗ such that u=xμ,y¯λ∈Gμ×Gλ∗ and v=sγ,t¯δ∈Gγ×Gδ∗, where xμ∈Gμ, sγ∈Gγ; y¯λ∈Gλ∗, t¯δ∈Gδ∗. Then, the mapping φγ,δμ,λ:Gμ×Gλ∗⟶Gγ×Gδ∗. If μ≥γ or λ≤δ then the total arrows from u and φγ,δμ,λu to v are equal, which is equal to Rcμ,c¯λ, if there exists cμ,cλ∗∈Cμ×Cλ∗ such that v=cμ,cλ∗φγ,δμ,λu. R is ramification data (RD) of S×S∗, which is the sum of the RD Rcγ,cδ∗ of the group Gγ×Gδ∗;γ,δ∈Y.The ramification data of Clifford monoidS×S∗=⋃λ,μ∈Yxλ,y¯μ, where Y is a semilattice. The RD R of S×S∗ which is sum of the RD RCγ,Cδ∗ of the group Gγ×Gδ∗;γ,δ∈Y. Now, R=∑λ,μ∈YRμ,λ=∑λ,μ∈Y∑Cμ∈Cμ,Cλ∗∈C∗λRcμ,c¯λCμ×Cλ∗, where Rμ,λ=∑u=xμ,y¯λ∈Cμ×Cλ∗Rcμ,c¯λCμ×Cλ∗and Cμ×Cλ∗ is the conjugacy class of a subgroup Gμ×Gλ∗.(i)
Then,R is considered to be a positive central element of ring kS⊗kS∗. Moreover, Cμ and Cλ∗ denote the collections of all the conjugacy classes of Gμ and Gλ∗, respectively, for all μ,λ∈Y(ii)
IfR is ramification data of S×S∗ and Γ¯ is a quiver such that(a)
the set of vertices ofΓ¯S×S∗,R is the set of all elements of S×S∗(b)
for allu,v∈S×S∗ such that u=xμ,y¯λ∈Gμ×Gλ∗ and v=sγ,t¯δ∈Gγ×Gδ∗ where xμ∈Gμ, sγ∈Gγ; y¯λ∈Gλ∗, t¯δ∈Gδ∗and for μ≯γ and λ≮δ, ∄ any arrow from u to v. If μ≥γ or λ≤δ then the total arrows from u to v and φγ,δμ,λu to v are equal, which are equal to RCγ,Cδ∗; if there exists Cμ,Cλ∗∈Cμ×Cλ∗ such that v=cμ,cλ∗φγ,δμ,λu, where φγ,δμ,λ is a mapping given by φγ,δμ,λ:Gμ×Gλ∗⟶Gγ×Gδ∗. Then, Γ¯=Γ¯S×S∗,R is called a double weak Hopf quiver of R corresponding to the Clifford monoid S×S∗. The maps φμ,λμ,λ is an identity mapping, as a homomorphism of the group Gμ×Gλ∗. If φγ,δμ,λ and φσ,ργ,δ be any two homomorphisms as defined above then φσ,ργ,δφγ,δμ,λ=φσ,ρμ,λ and for any u,v∈S×S∗ we have u=xμ,y¯λ∈Gμ×Gλ∗ and v=sγ,t¯δ∈Gγ×Gδ∗, then u.v=φγ,δμ,λuφσ,ργ,δv; provided that γ≤μ iff μγ=γ and δ≤λ iff λδ=δ for the partial ordering “≤” in a semilattice Y, for all μ,λ,γ,ρ,σ,δ∈Y. We denoteΓ¯ by Γ¯S×S∗,R is called a double quiver of a Clifford monoid. We have an algebra Ⱨ¯=⨁μ,λ∈YⱧμ⊗Ⱨλ∗=kS⊗kS∗≅kΓ¯S×S∗,R, where Ⱨμ=kGμ and Ⱨλ∗=kGλ∗.Ⱨ¯ is a weak Hopf algebra if there exist an element T¯=T,T∗ in ⱧoɱκⱧ¯,Ⱨ¯ such that id∗T¯∗id=id and T¯∗id∗T¯=T¯, where ∗ is the convolution product in ⱧoɱκⱧ¯,Ⱨ¯ and a weak antipode of Ⱨ¯ is denoted by T¯, also see [7, 8, 20]. The WHA Ⱨ¯ with a weak antipode T¯ is a “SLGWHA” if Ⱨ¯=⨁μ,λϵYⱧμ⊗Ⱨ∗λ is a semilattice graded sum, whereⱧμ,λ;μ,λϵY are sub-WHA which are HA with antipodes the restriction T¯Ⱨμ,λ for each μ,λϵY [9]. There is WHA homomorphism φγ,δμ,λ:Gμ×Gλ∗⟶Gγ×Gδ∗, where φγ,δμ,λ=φμ,λ,φ∗γ,δ and Gα×Gβ;α,βϵY is subgroup of S×S∗. For two homorphism φγ,δμ,λ, φσ,ργ,δ their composition φσ,ργ,δφγ,δμ,λ=φσ,ρμ,λfrom Ⱨλ⊗Ⱨμto Ⱨσ⊗Ⱨρ if λμ=μ. Such that for all u=xμ,y¯λ∈Gμ×Gλ∗ and v=sγ,t¯δ∈Gγ×Gδ∗, where xμ∈Gμ, sγ∈Gγ; y¯λ∈Gλ∗, t¯δ∈Gδ∗, where the multiplication u.v in Ⱨ¯ is given by u.v=φγ,δμ,λuφσ,ργ,δv, where φγ,δμ,λ:Gμ×Gλ∗⟶Gγ×Gδ∗ is the mapping from the subgroup Gμ×Gλ∗ to Gγ×Gδ∗ for all μ≥γ or ;λ≤δ;λ,μ,γ,δ∈Y.(7)u.v=φγ,δμ,λxμ,y¯λφσ,ργ,δsγ,t¯δ=φμ,λ,φ∗γ,δxμ,y¯λφγ,δ,φ∗σ,ρsγ,t¯δ=φμ,λxμ,φ∗γ,δy¯λφγ,δsγ,φ∗σ,ρt¯δ.For two mappingsφσ,ργ,δφγ,δμ,λ=φσ,ρμ,λ.A Clifford monoidS×S∗ is a regular semigroup S×S∗ with identity. Let u,v∈S×S∗ where u=gμ,g∗δ∈Gμ×Gδ∗ and v=gγ,g∗ρ∈Gγ×Gρ∗ then
(8)u.v=φγ,δμ,λuφσ,ργ,δv=φμ,λ,φ∗γ,δgμ,g∗δφγ,δ,φ∗σ,ρgγ,g∗ρ=φμ,λgμ,φ∗γ,δg∗δφγ,δgγ,φ∗σ,ρg∗ρ=φμ,λgμφγ,δgγ,φ∗γ,δg∗δφ∗σ,ρg∗ρ=φγ,λgγ,φ∗γ,ρg∗ρ=gλ,g∗γ∈Gλ×Gγ∗whereμγ=γ&σρ=ρorγ≤μ&ρ≤σ.In the discussion below, we see that ifⱧ¯=kS⊗kS∗ is a WHA, then
(9)Ⱨ¯∗=kS⊗kS∗∗,Ⱨ¯∗=kS∗∗⨂kS∗=kS⨂kS∗=Ⱨ¯ (∴S is a finite semigroup); thus, Ⱨ¯ is self dual of Ⱨ¯, where S∗ is the dual basis of the WHA Ⱨ and is also a Clifford monoid. The multiplication in Ⱨ¯=Ⱨ⊗Ⱨ∗ is given by a.b=φγ,δμ,λaφσ,ργ,δb for a,b∈Ⱨ¯=Ⱨ⊗Ⱨ∗ of the form a=x⊗y¯&b=s⊗t¯. Then, their multiplication a.b=xs⊗t¯y¯ is in Ⱨ¯=Ⱨ⊗Ⱨ∗. The structure of comultiplication in Ⱨ¯=Ⱨ⊗Ⱨ∗ is defined by
(10)Δa=Δx⊗y¯=∑x′,x′′∈Ⱨy¯′,y¯′′∈Ⱨ∗x′⊗y¯′⊗x′′⊗y¯′′=∑x′,x′′∈Ⱨy¯′,y¯′′∈Ⱨ∗x′x′′⊗y¯′′y¯′,for all a=x⊗y¯∈Ⱨ¯=Ⱨ⊗Ⱨ∗, where x∈Ⱨ & y¯∈Ⱨ∗. Since the weak antipode T in Ⱨ is an element of ⱧoɱκⱧ,Ⱨ and the weak antipode T∗exists in Ⱨ∗ which is an element of ⱧoɱκⱧ∗,Ⱨ∗ so that T¯ exists in Ⱨ¯=Ⱨ⊗Ⱨ∗ as an element of ⱧoɱκⱧ¯,Ⱨ¯. idⱧ¯∗T¯∗idⱧ¯=idⱧ¯ satisfying T¯∗idⱧ¯∗T¯=T¯. This makes Ⱨ¯ is a weak Hopf algebra. If Ⱨ is finite dimensional vectors space, so is Ⱨ∗ hence Ⱨ¯=Ⱨ⊗Ⱨ∗ is also a finite dimensional. Thus,
(11)∴Ⱨ¯∗=Ⱨ⊗Ⱨ∗∗≅Ⱨ∗∗⊗Ⱨ∗≅Ⱨ⊗Ⱨ∗=Ⱨ.¯The idempotents ofS∗ lie in its center CS∗. S∗ is a regular monoid such that S∗=∪λϵYG∗λ and G∗λG∗μ⊆G∗λμ∀λ,μ∈Y, G∗λ for λ∈Y and is dual basis of the dual Ⱨ∗λ=kG∗λ of the Hopf algebra Ⱨλ=kGλ. Further, for any λ,μ∈Y with λμ=μ, there is a homomorphism φγ,δμ,λ:Gμ×Gλ∗⟶Gγ×Gδ∗ and if λμ=μ and μν=ν, then φ∗λ,μφ∗μ,ν=φ∗λ,ν with φμ,λμ,λ is an identity mapping of Gμ×Gλ∗. The partial ordering “≤” in Y is given by “μ≤λ⟺λμ=μ∀μ,λ∈Y.”The immediate observations regarding the WHQ and its algebra are as follows: The vertices ofΓ¯S×S∗,R are identified by the elements of S×S∗. To find the number of vertices of the double quiver Γ¯S×S∗,R, first, we consider the following example.Example 11.
LetY=α,β,γ,ρ,σ,δ Be a semilattice as given in Table 1Table 1
Semilattices.
.αβγρσδαααααααβαββαββγαβγαβγραααρρρσαββρσσδαβγρσδFor a ringR with identity, R2×2 denotes the 2×2 full matrix ring overR, UR is the group consisting of all units inR. Let Z be the ring of integers. For a primep, Zp is a field and UZp2×2 is just 2×2 general linear group GL2Zp over Zp.TakeS=∪uϵYGu where Y=α,β,γ,ρ,σ,δ as given in [20] where Gα=eα and Gδ=eδ are the trivial groups, Gβ=GL2Z2,Gγ=UZ42×2,Gρ=GL2Z3,Gσ=UZ62×2. Thus, kS=⨁u∈YkGu and kS∗=⨁u∈YkGu∗=⨁v∈YkGv∗. Thus, Ⱨ¯=kS⊗kS∗=⨁u∈YkGu⊗⨁v∈YkGv∗=⨁u∈YkGu⊗kGv∗ which bears the structures of algebra and coalgebra. Thus, Ⱨ¯ appears as the path coalgebra of the double quiver Γ¯S×S∗,R.In the above example, the number of vertices of DWHQ isɱ=440×440=193600.Total number of arrows in the double weak Hopf quiver is2043168. If Rcγ,cδ∗ denotes the number of arrows originating from u=xμ,y¯λ∈Gμ×Gλ∗ to v=sγ,t¯δ∈Gγ×Gδ∗ where μ,λ,γ,δ∈Y such that μ≥γ or λ≤δ.The number of vertices ofΓ¯S×S∗,R is ɱ=440×440=193600.The count of arrows in double weak Hopf quiverΓ¯S×S∗,R is obtained as 2043168. That can be observed from that Table 2.Table 2
The vertices and arrows of the DWHQΓ¯S×S∗,R corresponding to the WHA Ⱨ¯.
Range ➔|Gδ×Gδ∗|=|Gδ×Gσ∗|=|Gδ×Gγ∗|=|Gδ×Gβ∗|=|Gδ×Gρ∗|=|Gδ×Gα∗|=|Gσ×Gδ∗|=|Gσ×Gσ∗|=|Gσ×Gγ∗|=|Gσ×Gβ∗|=|Gσ×Gρ∗|=|Gσ×Gα∗|=|Gγ×Gδ∗|=|Gγ×Gσ∗|=|Gγ×Gγ∗|=|Gγ×Gβ∗|=|Gγ×Gρ∗|=|Gγ×Gα∗|=|Gβ×Gδ∗|=|Gβ×Gσ002A|=|Gβ×Gγ∗|=|Gβ×Gβ∗|=|Gβ×Gρ∗|=|Gβ×Gα∗|=|Gρ×Gδ∗|=|Gρ×Gσ∗|=|Gρ×Gγ∗|=|Gρ×Gβ∗|=|Gv×Gρ∗|=|Gρ×Gα∗|=|Gα×Gδ∗|=|Gα×Gσ∗|=|Gα×Gγ∗|=|Gα×Gβ∗|=|Gα×Gρ∗|=|Gα×Gα∗|=Sum ofarrowsDomain ↓128896648128882944276481728138242889627648921657646089661728576362886481382446082882304481288966481193600∣Gδ×Gδ∗| =11101112882882882882882889696969696966666664848484848481111112639|Gδ×Gσ∗| =2881288111128882944288288288288962764896969696617289696969648138244848484812881111129280∣Gδ×Gγ∗| =961288961112888294427648288288288962764892169696966172857666648138244608484848128896111170720|Gδ×Gβ∗| =612889661128882944276481728288288962764892165769696617285763666481382446082884848128896611172920|Gδ×Gρ∗| =48128896648128882944288172813824288962764896576460896617286362886481382448288230448128816481151895|Gδ×Gα∗| =11288966481288829442764817281382428896276489657646089661728576362886481382446082882304481288966481184480|Gσ×Gδ∗| =2881111112882882882882882889696969696966666664848484848481111112640|Gσ×Gσ∗| =82944128800002888294428828828828896276489696969661728666648138244848484812881111128916|Gσ×Gγ∗| =276481288960002888294427648288288288962764892169696966172857666648138244608484848128896111170717|Gσ×Gβ∗| =172812889660028882944276481728288288962764892165769696617285763666481382446082884848128896611172918|Gσ×Gρ∗| =1382419606480288829442881728138242889627648965769696617286362886481382448288230448128816481147094|Gσ×Gα∗| =288128896648128882944276481728138242889627648921657646089661728576362886481382446082882304481288966481193600∣Gγ×Gδ∗| =96100000100000969696969696666666100000111111621|Gγ×Gσ∗| =2764810000012880000962764896969696617286666128800001288111130758|Gγ×Gγ∗| =9216128896000128896000962764892169696966172857666612889600012889611141119|Gγ×Gβ∗| =57612889660012889660096276489216576969661728576360012889660012889661141640|Gγ×Gρ∗| =460812880648012880648096276489657646089661728636288012880648012881648136558|Gγ×Gα∗| =961288966481128896648196276489216576460896617285763628861288966481128896648146640|Gβ×Gδ∗| =6100000100000100000666666484848484848111111333|Gβ×Gσ∗| =17281288000012880000128800006172866664813824484848481288111116982|Gβ×Gγ∗| =576128896000128896000128896000617285766664813824460848484812889611122495|Gβ×Gβ∗| =3612889660012889660012889660061728576366648138244608288484812889661122788|Gβ×Gρ∗| =28812880648012880648012880648061728636288648138244828823044812881648120004|Gβ×Gα∗| =61288966481128896648112889664816172857636288648138244608288230448128896648125520|Gρ×Gδ∗| =48100000100000100000100000484848484848111111298|Gρ×Gσ∗| =13824128800001288000012880000128800004813824484848481288111115513|Gρ×Gγ∗| =46081288960001288960001288960001288960004813824460848484812889611120552|Gρ×Gβ∗| =2881288966001288966001288966001288966048138244608288484812889661120821|Gv×Gρ∗| =230412880648012880648012880648012880648048138244828823044812881648118277|Gρ×Gα∗| =48128896648112889664811288966481128896648148138244608288230448128896648123320|Gα×Gδ∗| =110000010000010000010000010000011111111|Gα×Gσ∗| =2881288000012880000128800001288000012880000128811111738|Gα×Gγ∗| =961288960001288960001288960001288960001288960001288961112313|Gα×Gβ∗| =61288966001288966001288966001288966001288966001288966112348|Gα×Gρ∗| =481288064801288064801288064801288064801288064801288164812060|Gα×Gα∗| =11288966481128896648112889664811288966481128896648112889664812640Sum of arrows1936003681621826112581123480835776168768121685798434601746419328755526102245761731156374647650630267622411642795525644840681939211543686461746126600362043168If there is an arrow from some elementu=xμ,y¯λ∈Gμ×Gλ∗ to some element v=sγ,t¯δ∈Gγ×Gδ∗ then there are arrows from each u=xμ,y¯λ to v=sγ,t¯δ. The dimension of the vector space corresponding to DWHQ Γ¯S×S∗,R is the number of vertices of the DWHQ. The arrows from each idempotent to itself are the existing loops. Therefore, the loops’ quantity is the order of the semilattice Y. A Clifford monoid S×S∗ which if finite implies that the DWHQ of the WHA Ⱨ¯=kS⊗kS∗ has no loop if and only if R=0. Then, the quiver is a digraph with S×S∗ number of isolated vertices. Otherwise, the DWHQ is a connected diagraph.LetN denote the number of arrows of the quiver Γ¯S×S∗,R. Let Nμ,λ denote the number of all the arrows originating from u=xμ,y¯λ which is the vertex of the subgroup Gμ×Gλ∗ and Nγ,δ denote the number of all the arrows ending at the vertex represented by the element v=sγ,t¯δ of the subgroup Gγ×G∗δ. Then, we have the following fact.Lemma 12.
IfΓ¯S×S∗,R Is a DWHQ with ɱ=S=S∗. Then,
(a)
The total number of vertices inΓ¯S×S∗,R is ɱ2(b)
Total number of arrows inΓ¯S×S∗,R is NProof.
(a) The total number of vertices of the DWHQ isΓ0¯=dimⱧ=dimkS⊗kS∗=ɱ.ɱ=ɱ2, where ɱ=dimkS=dimkS∗.☐Proof.
(b) Total no of arrows of DWHQΓ¯ originating from
(12)N=∑μ,λ∈YNμ,λGμ×G∗λ=R=∑Cγ∈Cγ,C∗δ∈C∗δγ,δ∈YRCγ,C∗δ,N′=∑γ,δ∈YNγ,δGγ×G∗δ=R=∑Cγ∈Cγ,C∗δ∈C∗δγ,δ∈YRCγ,C∗δ.☐Hence,N=Ń=TotalnumbersofarrowsoftheDWHQ.Lemma 13.
LetS=⋃μ∈YGμ be a Cllifford monoid and Ⱨ¯=kS⊗kS∗ be the WHA for the DWHQ Γ¯. Then, the following arguments are true:
(i)
IfⱮ is a left Ⱨ¯-comodule, then Ɱ=⨁u=xμ,y¯λ∈Gμ×Gλ∗uⱮ(ii)
IfⱮ is a right Ⱨ¯-comodule, then Ɱ=⨁v=sγ,t¯δ∈Gγ×Gδ∗Ɱv(iii)
IfⱮ is a Ⱨ¯-bicomodule, then Ɱ=⨁u=xμ,y¯λ∈Gμ×Gλ∗v=sγ,t¯δ∈Gγ×Gδ∗uⱮvProof.
For its proof, see Lemma 4.1 in [4].☐Lemma 14.
A weak Hopf (WH)Ⱨ¯-bimodule Ɱ can be decomposed into WH Ⱨ¯-subbimodules of Ɱ, i.e., Ɱ=⨁γ≤μλ≤δμ,λ,γ,δ∈YⱮμ,λγ,δ, where Ɱμ,λγ,δ=K− span vⱮu with v=cμ,cλ∗φγ,δμ,λu and vⱮu=vⱮw⊕xⱮy⨁⋯⨁zⱮu, where v=xγ,y¯δ,w=eδ,y¯η,x=eη,y¯σ,y=xσ,y¯ρ,⋯,z=xα,y¯β and u=xμ,y¯λ with λ≥β≥⋯≥ρ≥σ≥η≥δ and μ≥α≥⋯≥σ≥δ≥γ, where λ,β,⋯,ρ,σ,η,δ,μ,α,⋯,γ∈Y.Proof.
The proof clearly follows from [17, Lemma 4.1].☐Lemma 15.
IfⱮ Is a WH Ⱨ¯-bimodule, then
(13)dimvⱮu=Rμ,δ=∑Rcμ,c∗δCμ×Cδ∗,ifδ>μ,1,ifδ=μ,0,ifδ<μ.Proof.
It is obvious from [17, Lemma 4.3]☐From the following proposition, we obtain that the setΓ¯ of group-like elements of path coalgebra of a DWHQ Γ¯ is a Clifford monoid.Proposition 16.
IfΓ¯S×S∗,R is a double WHQ for the RD R of S×S∗, then Γ¯0=GkΓ¯ andΓ¯0=S×S∗. Moreover, kΓ¯0≃kS×S∗, the algebra of S×S∗ is a sub-WHA of kΓ¯.Proof.
Initially, we prove that the setGkΓ¯ is equal to Γ¯0. By the multiplication structure that is defined in kΓ¯, for every vertex i∈Γ¯0,ei∈KΓ¯0, we have Δei=ei⨂ei and ϵei=1. Thus, Γ¯0⊆GkΓ¯. Let u∈GkΓ¯ and take u=∑l≥0ul with ul∈Γ¯l. We prove that ul=0 for l>0. Suppose on contrary that n≠0 is the greatest l, with ul≠0. Thus,
(14)Δu=un⊗un+∑i,j>0i≠jui⊗uj,
whereun⊗u¯n∈kΓn⊗kΓ∗n and ui⊗u¯j∉kΓn⊗kΓ∗n for all i,j>0,i≠j.☐Notice thatΔu=∑l≥0Δul=ul⊗sul+∑l≥1vl⋯vl+1⊗vi⋯vl+tul⊗ul has no term that belongs tokΓn⊗kΓ∗n, a contradiction. Thus, u∈Γ¯0 implies that GkΓ¯⊆Γ¯0, and we get GkΓ¯=Γ¯0. By the definition, the collection of vertices Γ¯0 signifies the collections of components of a Clifford monoid S×S∗. Thus, Γ¯0=S×S∗=GkΓ¯ andkΓ¯0≃kS×S∗≃kS⊗kS∗ is a WHA by [8]. The path coalgebra kΓ¯ is pointed; therefore, kΓ¯0 becomes the sum of its simple subcoalgebras. Thus, kΓ¯0 is a subcoalgebra of kΓ¯ which is also a WHA. Thus, kΓ¯0 is sub-WHA of kΓ¯. Here, the path coalgebra kΓ∗ is dual of the coalgebra kΓ of the quiver Γ of the Clifford monoid S. Each summand kGλ∗ of kΓ∗=⨁λ∈YkGλ∗ is the dual of the coalgebra kGλ.There is a necessary and sufficient condition between the SLGWHAⱧ¯=kΓ¯ and the existence of a DWHQ Γ¯S×S∗,R corresponding to S×S∗ with some ramification data R. We have the following theorem for this argument.Theorem 17.
IfΓ¯S×S∗,R is a DWHQ, then the following arguments are equivalent:
(i)
kΓ¯ equips a SLGWHA structure such that each of its graded summands is itself a graded weak Hopf algebra(ii)
Γ¯ is DWHQ Γ¯S×S∗,R of S×S∗ with ramification data RProof.
(i) LetΓ¯S×S∗,R be an orbitrary double Weak Hopf quiver of kΓ¯ admitting the structure of SLGWHA.
Ⱨ¯=⨁μ,λ∈YⱧμ⊗Ⱨ∗λ, with ΔⱧ¯μγ,δλ⊆Ⱨ¯μ,λ⊗Ⱨ¯γ,δ where Ⱨ¯α,β=Ⱨα⊗Ⱨ∗β∀α,β∈Y,
withⱧμ⊗Ⱨ∗λⱧγ⊗Ⱨ∗δ⊆Ⱨμγ⊗Ⱨ∗δλ,μ,λ,δ,γ∈Y such that μ≥γ and λ≤δ.
=Ⱨγ⊗Ⱨ∗λ, which is graded.
Thus,εⱧ¯\GⱧ¯=0, and εGⱧ¯=1 and for each a=x⊗y¯∈Ⱨ¯, where x∈Ⱨ=kΓ and y¯∈Ⱨ∗=kΓ∗
, is also graded, see [4]☐Moreover,GⱧ¯=∪λ,uϵYGμ×Gλ∗. Hence, Γ¯0=GⱧ¯ denotes the vertices of the quiver Γ¯ and the IC vkΓ¯u=p∈kΓ¯δLp=v⊗p,δRp=p⊗u gives the comodule structure in Ⱨ¯=kΓ¯, and in particular vkΓ1u remains constant iff u,v∈Ⱨα⊗Ⱨ∗α∀α∈Y. Thus, S×S∗=GⱧ¯=GⱧ×GⱧ∗=Γ×Γ∗=Γ¯. Thus, Γ¯ is a double weak Hopf quiver.Conversely, ifΓ¯ is a double weak Hopf quiver of some Clifford monoid S¯=S×S∗ with some ramification data R. i.e., S¯=∪λ,uϵYGμ×Gλ∗ is a Clifford monoid with respect to ramification data R as given in Definition 1 above. The vector space kΓ¯ equips the structures of algebra and coalgebra, and there exists Γ¯∈ⱧoɱkΓ¯,kΓ¯∗ as a weak antipode satisfying idKΓ¯¯∗kΓ¯∗idKΓ¯¯=idKΓ¯¯ & T¯∗idKΓ¯¯∗T¯=T¯. Thus, kΓ¯ is a weak Hopf such that kΓ¯=kΓ⊗kΓ∗=∑μ,λ∈YkGμ⊗kG∗λ which is semilattice graded kGμ⊗kG∗λkGγ⊗kG∗δ⊆kGμ,γ⊗kG∗δ,λ, where each grading summand kGμ⊗kG∗λ is a Hopf algebra ∀μ,λ∈Y. Thus, kΓ¯ appears as the kΓ0=kS×S∗ bimodule and bicomodule.MoreoverΔkΓ¯μ,λ,δ,γ⊆kΓ¯μ,λ⊗kΓ¯γ,δ and GkΓ¯=∪λ,uϵYGμ×Gλ∗=Γ0=S×S∗=S¯.Thus,kΓ¯=Ⱨ¯ is a “semilattice-graded weak Hopf algebra”.If we use the product of arrowsa and b of a WHQ as
(15)a.b=ta,ba,sb+a,tbsa,b,and Def. 3.4 in [1], we have the following result for weak Hopf quiver.Theorem 18.
LetS be a Clifford monoid and k be a field, B be a kS-weak Hopf module and let C be an associated cotensor coalgebra corresponding to the WHA structure by [17, Th. 4.5]. Identify the coalgebra C with the path coalgebra of the WHQ of Ɱ by taking a basis of the IC of the bicomodule Ɱ. The lengths of the paths α and β are n and ɱ, respectively.
Then,α.β=∑d∈Dnn+ɱα.βd, where Dnn+ɱ=n+ɱn.Proof.
Considerα and β as the paths of length n and ɱ, respectively. Let d∈Dnn+ɱ and d¯∈Dɱn+ɱ the complement sequence of d obtained by interchanging 0 and 1 mutually. Now, consider
(16)α.βd=dαɱ+n.d¯βɱ+n⋯dα1.d¯β1which and belong to the IC of type tαtβ,sαsβ and lies in the ɱ+n−cotensor power of Ɱ.☐In casedi=1 then d¯βi is a vector that acts on the right on dαi (the arrow) and the action results as dαi.d¯βi. If on the other hand di=0, then dαi.d¯βi is the result of left action of dαi (the vertex) on the arrow d¯αi.The source of the first term of the elementα.βd is the
(17)sdα1.d¯β1=sdα1sd¯β1=sαsβ,and sequence of the terms of α.βd is a concatenated.Now, using Th. 3.8 and Lemma 3.10 in [1], we have the required result.A similar result can be stated for the co-algebraC∗ and bicomodule Ɱ∗ for the dual WHA Ⱨ∗=kS∗ of a dual WHQ Γ∗, which is given as follows, by using Th. 3.9 and [17, Th. 4.5].Theorem 19.
LetS Be the Clifford monoid and S∗ be its module as the dual basis of the vector space kS. Let B∗ be kS∗ weak Hopf module and let C∗ be the associated sual cotensor coalgebra corresponding to the DWHQ Γ∗. Identify the dual coalgebra C∗ with the path coalgebra of the DWHQ Γ∗ of the bicomodule Ɱ∗ by selecting a basis of IC of Ɱ∗. The lengths of the paths α¯ and β¯ are n and ɱ, respectively. Then,
(18)β¯.α¯=∑d¯∈D¯ɱn+ɱβ¯.α¯d¯where d¯∈D¯ɱn+ɱandd¯¯∈D¯¯nn+ɱ=Dnn+ɱProof.
Proof is obvious using above Th. 2.9 and [9, Th. 3.8].☐By using the coalgebras and bicomodules ofΓ and Γ∗, we can obtain the coalgebra C¯, bimodule, and bicomodule Ɱ̅ of the double weak Hopf quiver Γ¯=Γ×Γ∗ and using the above two theorem, Th. 2.9 and Th. 2.10, we have the result as given bellow.Theorem 20.
LetS be the Clifford monoid, S∗ be its dual and k be a field. Let Ɱ¯ Be a kS⨂kS∗ weak Hopf bimodule and bicomodule and C¯ be the associated cotensor coalgebra corresponding to the WHA structure of the coalgebra kΓ¯. Identify coalgebra C¯ with the path coalgebra of the DWHQ Γ¯ of Ɱ by selecting a basis of the IC of Ɱ¯. The lengths of the paths α~ and β~ are n and ɱ, respectively. Then,
(19)α~.β~=∑d~∈D~n,ɱn+ɱα~.β~d~,where α~=α,α¯, β~=β,β¯, D~n,ɱn+ɱ=Dnn+ɱ,D¯ɱn+ɱ and d~=d,d¯.Proof.
Proof is obvious using the proofs of Theorems18 and 19.☐


## 4. Conclusion

In this article, the generalized form of a quiver, called the double weak Hopf algebra (DWHQ), was introduced. Further, proposed quivers were used to obtain their path coalgebras that appeared as the weak Hopf algebra (WHA). Additionally, the study also discussed the module structures of path coalgebra of the double weak quivers and some of their interesting properties. The proposed work contributes in the module theory of a weak Hopf algebra that corresponds to the double weak Hopf quiver.



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