Quivers And Hopf Algebras

In this paper, starting from a quiver and its representation theory, we study the structures and representations of Artinian algebras, Hopf algebras and quantum doubles .For an arbitrary finite quiver Q, we introduce its set-representation categories Set-RepQ and study the dual-relation between the Hopf algebra structures on path algebra kQ^a and Hopf algebra structures on path coalgebra kQ^c for a covering quiver Q.Moreover, since there is a Hopf algebra structure on path coalgebra kQ^c if and only if Q is a Hopf quiver, we consider what kinds of generalized path coalgebras hold Hopf algebra structures. Through defining the natural quiverâ–³_A of an Artinian algebra A, a special class of Artinian algebras can be described via the corresponding generalized path algebras.To generalized the classical quantum double D(H) = (H^op)* (?) H from a Hopf algebra H, we destroy the balance between the left and right H's by defining a non-balanced quantum double Dc(H) = (C^op)^* (?) H from two Hopf algebras C and H. Some properties such as quasi-triangularity, semisimplicity, module category and representation type are considered. In particular, when C and H are both group algebras, Dc(H) is isomorphic to a Hopf algebra defined on a quotient kQ^a/