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.
Highlights
A bialgebra H is equipped with the structures of algebra (H, m) and coalgebra
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
If Γ(S, r) is a weak Hopf quiver corresponding to a ramification data r of a Clifford monoid S, 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
Summary
A bialgebra H is equipped with the structures of algebra (H, m) and coalgebra. If H is a linear space over a field K, 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.
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