Double Weak Hopf Quiver and Its Path Coalgebra

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.