Weak Hopf Algebra and Its Quiver Representation

-is 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. -is 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.


Introduction
A bialgebra H 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 m(Id ⊗ m) � m(m ⊗ Id) (associativity) and Id � m(u ⊗ Id) � m(Id ⊗ 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 ⊗ Id)Δ � (Id ⊗ Δ)Δ (coassociativity of Δ ) and Id � (ε ⊗ Id)Δ � (Id ⊗ ε)Δ (counitary property) [1]. en, we have a unique element ρ ∈ Hom k (H, H), such that id H * ρ � ρ * id H � με, where " * " is the convolution in Hom k (H, 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. ey discussed some matters regarding representations of Hopf algebra/quantum group and quiver. A quiver representation is a set V i |i ∈ Q 0 of k-vector spaces V i having finite bases together with the set ∅ a : V t(a) ⟶ V h(a) ∈ Q 1 of k-linear maps. We denote a representation by R � (V i , ∅ a ) [10].
A bialgebra H over a field k is called a weak Hopf algebra if there is an element T in the convolution algebra Hom k (H, 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 � ⊕ λεY H λ , where the graded sums H λ ; λεY are the subweak Hopf algebras (which are Hopf algebras) with antipodes restrictions T| H λ for each λεY [13]. en, there exist a homomorphism φ λ,μ : H λ ⟶ H μ if λμ � μ, such that aεH λ and bεH μ , and the multiplication a · b in H is given by (1) A Clifford monoid S is a regular semigroup S. Its center C(S) contains each of its idempotent. In other words, this is a semilattice of groups which is a collection of maximal subgroups G λ : λεY of a regular monoid S, such that S � ∪ λεY G λ and G λ G μ ⊆G λμ for all λ, μ � Y, where Y is a semilattice. For any λ, μ � Y with λμ � μ, there are group homomorphism φ λ,μ : G λ ⟶ G μ with φ λ,λ as an identity homomorphism on G λ , and if λμ � μ and μ] � ], then e multiplication in S for all a, b ∈ S is defined as above in H. e 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 Γ(G, 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 monoid S 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.

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]). Let S � ∪ λ∈Y G λ be a Clifford monoid, where Y is a semilattice of G λ ; λ ∈ Y, the subgroups of S.
(1) A ramification data r of S means a sum of r λ � (2) en, r could be viewed as a positive central element of the Clifford monoid ring of S, where C λ represents the collection of total conjugacy classes of subgroup G λ for λ ∈ Y.
Let Γ be a quiver satisfying the following conditions: (a) e set of vertices of Γ just represents the set S (b) Let x ∈ G μ , y ∈ G λ ; 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 φ μ,λ (x) to y which is equal to en, Γ is said to be the corresponding weak Hopf quiver of r. Γ 0 is the set of vertices and Γ 1 is the set of arrows of Γ.
Definition 2 (see [17]). Let for a quiver Γ and kΓ be the k-space with basis the set of all paths in Γ, where k is a field. Define kΓ a by the algebra with multiplication and underlying k-space kΓ as for the paths p � a n , . . . , a 1 and q � b n , . . . , b 1 . en, kΓ a becomes an associative algebra, known as path algebra of Γ [3,16].
Definition 3 (see [4]). Let Γ be a quiver (finite or infinite) and define kΓ C to be a coalgebra with comultiplication Δ of kΓ C defined by for any path p � a n , . . . , a 1 : a i ∈ Γ 1 ; i � 1, . . . , n. For special case, a trivial path e i , the comultiplication is Δ and is described by Δ(e i ) � e i ⊗ e i for each vertex i ∈ Γ 0 and the counit ε is defined by We use kΓ the path coalgebra of the quiver Γ.
Lemma 1 (see [4]). If kΓ is the path coalgebra corresponding to the quiver Γ, then kΓ is pointed and G(kΓ) � Γ 0 . ere is a necessary and sufficient condition between the semilatticegraded weak Hopf algebra and the existence of a weak Hopf quiver corresponding to a Clifford monoid with some ramification data.
Theorem 1 (see [1]). Let Γ represent a quiver; then, the following two statements are equivalent: (i) e path coalgebra kΓ acknowledges a semilatticegraded 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 e following proposition tells us that the collection of elements of group-like of path coalgebra kΓ of a weak Hopf quiver Γ is a Clifford monid.
Proposition 1 (see [1]). If Γ(S, r) is a weak Hopf quiver corresponding to a ramification data r of a Clifford monoid S, then Γ 0 is the collection of elements of group-like of path coalgebra kΓ, and kΓ 0 � kS, the Clifford monoid algebra of S is a subweak Hopf algebra of kΓ.
Definition 4 (see [4]). Suppose u and v represent the vertices in Γ, and k represents a field. e

Structures of Weak Hopf Quivers
Here, we discuss the structures of weak Hopf quiver and its algebra. We start by the following example.
For a ring R with identity R 2×2 denotes the 2 × 2 full matrix ring over R, U(R) the group consisting of all units in R. Let Z be the integer numbers ring. For a prime p, Z p is a field, and U(Z 2×2 p ) is just the 2 × 2 general linear group GL 2 (Z p ) over Z p . Assume that G α � e α and G δ � e δ are the trivial groups, e multiplication is defined as above on S makes S � ∪ u∈Y G u a Clifford monoid with regards to the semilattice Y [11]. e following mappings exist between the subgroups of the Clifford monoid.
We denote C λ as a conjugacy class of the group G λ , λ ∈ Y.
For each given mapping φ λ,μ : G λ ⟶ G μ , if it exists, and for any x ∈ G λ and y ∈ G μ , there exists c μ ∈ C μ , such that y � c μ φ λ,μ (x) for all λ, μ ∈ Y, μ ≥ λ. e semilattice of the subgroups of the Clifford monoid along with the mappings among them is shown in Figure 1.
In Figure 1, the arrows show the mappings φ λ,μ : e weak Hopf quiver for the weak Hopf algebra e vertices and arrows of the weak Hopf quiver Γ � (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. e mappings of the type φ λ,μ : G λ ⟶ G μ , ∀λ, μ ∈ Y which exist are shown by the symbol " ⟶ " in Table 2.
Particularly in the above quiver given in Section 3.1, the number of arrows originating in Γ(S, r) is given by e number of arrows ending in Γ(S, r) is given by We note that the originating number of arrows is equal to that ending in the quiver.
Let N denotes the amount of arrows of quiver Γ(S, r), N λ denote the amount of arrows originating from the vertex represented by the element a λ of subgroup G λ , and N λ denotes the amount of arrows ending at the vertex corresponding to the element of subgroup G λ . en, we have the following lemma:

numbers of arrows of the weak Hopf quiver Γ(S, r).
Proof. e proofs of (a), (b), and (c) are obvious from Table 2.
In view of Section 3.1, the following results can immediately be identified and obtained in a weak Hopf quiver Γ(S, r).
(i) If r λ is the ramification data of group G λ , then r λ � C λ ∈C λ r C λ C λ using (i) (ii) e ramification data of the Clifford monoid S � ∪ λ∈Y G λ is Where C λ represents the collection of total conjugacy classes of a group G λ . (iii) e number of arrows in Γ as obtained from Section 3.1 is 139443 (iv) e number of vertices of the weak Hopf quiver Γ(S, r) from Section 3.1 is |S| � λεY |G λ | � 440 (v) If there is an arrow from some element xεG λ to some element y ∈ G μ , then there are arrows from each xεG λ to yεG μ (vi) e dimension of weak Hopf algebra H corresponding to Γ is the number of vertices of the weak Hopf quiver (vii) e loops which exist are the arrows from each idempotent to itself. us, the number of loops is the order of the semilattice Y.

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].
We denote (V i,λ , V j,μ ); ϕ m λ,μ ; i, j ∈ Γ 0 by R λ,μ . A representation R of the weak Hopf quiver is given by and V j,μ , respectively, and (b) For every m ∈ Γ 1 , the restriction of ϕ m λ,μ to W t(m),λ is the mapping φ m λ,μ | W t(m),λ and is given by A nonzero representation V is called simple if the only subrepresentation of V is the zero representation and the V itself.
Given that a representation R � (V i,λ , φ i λ,λ ) of the quiver Γ(S, r), we can obtain a representation see also the framed representation in [13]. It suffices to define the representation on e i 's and f j 's, and these generate the basis of a ring.
is gives an extension to a representation on all elements of kΓ.
Definition 7 (see [15]). If R and S be two representations of the weak Hopf quiver Γ(S, r), then Φ: R ⟶ S as a representation morphism is a collection of k-linear maps such that the following Figure 2 is commutative for all m ∈ Γ 1 . Suppose φ i λ,μ : V i,λ ⟶ W i,μ is invertible for each i ∈ Γ 0 and all μ ≥ λ; λ, μ ∈ Y, we have the morphism Φ: R ⟶ S, which is called isomorphism from R to S.
A representation R 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 of Γ and observe that it is also a simple one.
Let Γ be a weak Hopf quiver having no oriented cycles. A representation R of Γ is simple if and only if it is canonical.
If Γ(S, r) is a weak Hopf quiver without any oriented cycle, then there exists some vertex e 1 ∈ Γ 0 , which is not a tail of some arrows. is type of arrow is called a sink.
Let Γ be a weak Hopf quiver with no oriented cycle, and e 1 ∈ Γ 0 be a vertex, such that t(m) ≠ e 1 , for all m ∈ Γ 1 .

Proposition 2. Let R be a canonical representation of a weak Hopf quiver Γ(S, r).
en, the representation for the weak Hopf quiver Γ(S, r) is a subrepresentation of R.
Define p � p i,λ ; λ ∈ Y: i ∈ Γ 0 a representation morphism, such that p i,λ : W i,λ ⟶ V i,λ is the inclusion mapping. To verify that all mappings commute, m ∈ Γ 1 , such that

Weak Hopf Quiver as Cayley Graph
Let S 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 E(Cay(S, C)) � (s, cs): s ∈ S, c ∈ C { }. In the following result, we give an embedding of a Cayley graph of a Clifford monoid S into the weak Hopf quiver of the corresponding weak Hopf algebra kS. Theorem 2. Every Cayley graph Cay(S, C) of a Clifford monoid S can be embedded into its corresponding weak Hopf quiver Γ(S, r) of the weak Hopf algebra H � kS � ⊕ λεY kG λ .
Let c y u x represents the edge of the Cayley graph from vertex x to vertex y in E(C).
en, φ(c y u x ) � c y v x ∈ Γ 1 , ∀c y u x ∈ E(C), where y � cx for some c ∈ C, x, y ∈ S, and c y v x is the arrow in Γ 1 , such that y � c λ φ λ,μ (x) for some c λ (if it exist) in C λ , the conjugacy class of G λ for all x ∈ G λ , y ∈ G μ ; μ ≥ λ; λ, μ ∈ Y. Clearly, φ is an injective mapping from Cay(S, C) to the weak Hopf quiver Γ(S, r).
us, the Cayley graph of a Clifford monoid S can be embedded into its corresponding weak Hopf quiver Γ(S, r).

Conclusion
In this article, the formula that enumerates the arrows in the weak Hopf quiver Γ(S, 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.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that there are no conflicts of interest.