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.
1. Introduction
A bialgebra is equipped with the structures of algebra and coalgebra. If is a linear space over a field , then is called an algebra if has a unit and a multiplication , such that (associativity) and (unitary property), where is the identity map of . is called a coalgebra if has a comultiplication and a counit , such that (coassociativity of ) and (counitary property) [1]. Then, we have a unique element , such that , where “” is the convolution in . With this map 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 of -vector spaces having finite bases together with the set of -linear maps. We denote a representation by [10].
A bialgebra over a field is called a weak Hopf algebra if there is an element in the convolution algebra , such that and , and represents a weak antipode of . Li obtained solutions for quantum Yang–Baxter equation using such weak Hopf algebra [1, 11, 12]. A weak Hopf algebra with a weak antipode is a semilattice graded weak Hopf algebra if , where the graded sums ; are the subweak Hopf algebras (which are Hopf algebras) with antipodes restrictions for each [13]. Then, there exist a homomorphism if , such that and , and the multiplication in is given by
A Clifford monoid is a regular semigroup . Its center contains each of its idempotent. In other words, this is a semilattice of groups which is a collection of maximal subgroups of a regular monoid , such that and for all , where is a semilattice. For any with , there are group homomorphism with as an identity homomorphism on , and if and , then . The multiplication in for all is defined as above in . The partial ordering “” in is given by “ if and only if for all .”
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 . By [15], the path coalgebra of a quiver admits a coquasitriangular Majid algebra structure if and only if is a Hopf quiver of the form with 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 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]). Let be a Clifford monoid, where is a semilattice of , the subgroups of .(1)A ramification data of means a sum of of subgroups , i.e., .(2)Then, could be viewed as a positive central element of the Clifford monoid ring of , where represents the collection of total conjugacy classes of subgroup for .Let be a quiver satisfying the following conditions:(a)The set of vertices of just represents the set (b)Let ; and ; if , then there does not exists an arrow from to , and if , then the number of arrows from to is equal to that from to which is equal to , if there exist , such that .Then, is said to be the corresponding weak Hopf quiver of . is the set of vertices and is the set of arrows of .
Definition 2 (see [17]). Let for a quiver and be the -space with basis the set of all paths in , where is a field. Define by the algebra with multiplication and underlying -space asfor the paths and . Then, becomes an associative algebra, known as path algebra of [3,16].
Definition 3 (see [4]). Let be a quiver (finite or infinite) and define to be a coalgebra with comultiplication of defined byfor any path ; . For special case, a trivial path , the comultiplication is and is described by for each vertex and the counit is defined byWe use the path coalgebra of the quiver .
Lemma 1 (see [4]). If is the path coalgebra corresponding to the quiver , then is pointed and . 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]). Let represent a quiver; then, the following two statements are equivalent:(i)The path coalgebra 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, is the weak Hopf quiver of some Clifford monoid The following proposition tells us that the collection of elements of group-like of path coalgebra of a weak Hopf quiver is a Clifford monid.
Proposition 1 (see [1]). If is a weak Hopf quiver corresponding to a ramification data of a Clifford monoid , then is the collection of elements of group-like of path coalgebra , and , the Clifford monoid algebra of is a subweak Hopf algebra of .
Definition 4 (see [4]). Suppose and represent the vertices in , and represents a field. The -isotypic component of a -bicomodule is . In particular, is the vector space of -paths from vertex to vertex .
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
Let be the semilattice with multiplication “·” as given in Table 1.
For a ring with identity denotes the full matrix ring over , the group consisting of all units in . Let be the integer numbers ring. For a prime , is a field, and is just the general linear group over . Assume that and are the trivial groups, . Then, , for any , setting . The multiplication is defined as above on makes a Clifford monoid with regards to the semilattice [11].
The following mappings exist between the subgroups of the Clifford monoid. , defined by , defined by , defined by , defined by , defined by , defined by
We denote as a conjugacy class of the group . For each and , there exists , such that . Since there is only one arrow (the loop) from to , therefore, .
For each given mapping , if it exists, and for any and , there exists , such that for all . The 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 .
The weak Hopf quiver for the weak Hopf algebra , where each is a Hopf algebra. is in fact a semilattice-graded weak Hopf algebra with , if and only if .
The vertices and arrows of the weak Hopf quiver corresponding to 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 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 is given by
The number of arrows ending in is given by
We note that the originating number of arrows is equal to that ending in the quiver.
Let denotes the amount of arrows of quiver , denote the amount of arrows originating from the vertex represented by the element of subgroup , and denotes the amount of arrows ending at the vertex corresponding to the element of subgroup . Then, we have the following lemma:
Lemma 2. (a)The number of arrows originating in is given by (b)The number of arrows ending in is given by (c) = total numbers of arrows of the weak Hopf quiver .
Proof. The 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 .
3.2. Results
Let , and . Then, there exists a unique arrow from to (or to ) and satisfies ; therefore, .(i)If is the ramification data of group , then using (i)(ii)The ramification data of the Clifford monoid is Where represents the collection of total conjugacy classes of a group .(iii)The number of arrows in as obtained from Section 3.1 is (iv)The number of vertices of the weak Hopf quiver from Section 3.1 is (v)If there is an arrow from some element to some element , then there are arrows from each to (vi)The dimension of weak Hopf algebra corresponding to 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 semilattice .(viii)For a finite Clifford monoid, corresponding to has no loop if and only if . Then, the quiver is a set of number of isolated vertices. Otherwise, the weak Hopf quiver is a connected diagraph.(ix)Let and be two finite Clifford monoids, and and be the weak Hopf quivers corresponding to two weak Hopf algebras and , respectively. Then, the quivers and are isomorphic if and only if there is a bijective mapping between their sets of vertices, such that the number of arrows from to , , is equal to that from to , where [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 of finite-dimensional -vector spaces together with a collection of mappings.We denote by . A representation of the weak Hopf quiver is given by .
Let and be two representations of weak Hopf quiver , where and . The representation is a subrepresentation of if(a)For all are the subspaces of and , respectively, and(b)For every , the restriction of to is the mapping and is given by .Then, is called subrepresentsation of .
A nonzero representation is called simple if the only subrepresentation of is the zero representation and the itself.
Given that a representation of the quiver , we can obtain a representation of , see also the framed representation in [13].
It suffices to define the representation on ’s and ’s, and these generate the basis of a ring.This gives an extension to a representation on all elements of .
The direct sum of two weak Hopf quiver representations is given as follows:
Definition 6. (see [21]). If and be two representations of weak Hopf quiver , where and , then we define a direct-sum representation as follows:with by(a) for every and (b) is defined by the matrixfor and .
Now, we define a morphism of a weak Hopf quiver representation to another weak Hopf quiver representation as follows.
Definition 7. (see [15]). If and be two representations of the weak Hopf quiver , then as a representation morphism is a collection of -linear maps , where , , such that the following Figure 2 is commutative for all .
Suppose is invertible for each and all , we have the morphism , which is called isomorphism from to .
A representation of a weak Hopf quiver is decomposable if there exist two nonzero representations and , such that , and a nonzero representation is indecomposable if it is not decomposable [15].
We introduce the notion of canonical representation of and observe that it is also a simple one.
Definition 8. (see [15]). A canonical representation for weak Hopf quiver is a collection of representations , such thatA canonical representation must be a simple for all , since the only a subspace of each one is , the null space at every vertex.
Let be a weak Hopf quiver having no oriented cycles. A representation of is simple if and only if it is canonical.
If is a weak Hopf quiver without any oriented cycle, then there exists some vertex , which is not a tail of some arrows. This type of arrow is called a sink.
Let be a weak Hopf quiver with no oriented cycle, and be a vertex, such that , for all .
Proposition 2. Let be a canonical representation of a weak Hopf quiver . Then, the representation , wherefor the weak Hopf quiver is a subrepresentation of .
Proof. Obviously, for each is a subspace of . Since is a nonzero -vector space, . Define a representation morphism, such that is the inclusion mapping. To verify that all mappings commute, , such that . So, has its domain as {0}, i.e., . Similarly, is the inclusion of that implies . Hence, for all , such that , we have and , so the diagram is commutative. For each with , we have that . Hence, is , i.e., . Similarly, , and is also the zero mapping. So, for all , such that , we have . Hence, the diagram is commutative. Thus, becomes a subrepresentation of .
5. Weak Hopf Quiver as Cayley Graph
Let be a semigroup and be a subset of . Recall that the Cayley graph Cay of with the connection set is defined as the digraph with a vertex set and arc set .
In the following result, we give an embedding of a Cayley graph of a Clifford monoid into the weak Hopf quiver of the corresponding weak Hopf algebra .
Theorem 2. Every Cayley graph of a Clifford monoid can be embedded into its corresponding weak Hopf quiver of the weak Hopf algebra .
Proof. Define mapping , such that , for all .
Let represents the edge of the Cayley graph from vertex to vertex in .
Then, , where for some , and is the arrow in , such that for some (if it exist) in , the conjugacy class of for all .
Clearly, is an injective mapping from to the weak Hopf quiver .
Thus, the Cayley graph of a Clifford monoid can be embedded into its corresponding weak Hopf quiver .
6. Conclusion
In this article, the formula that enumerates the arrows in the weak Hopf quiver 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
The authors declare that there are no conflicts of interest.
Acknowledgments
The authors are grateful to the Deanship of Scientific Research, King Saud University, for funding through Vice Deanship of Scientific Research Chairs.