关键词: Hopf 箭图, 覆盖箭图, 对偶Hopf代数, Schurian 箭图, 分次自同构群

Abstract:

In [3] and [6], the Hopf algebra structures of path algebra and path coalgebra on a Hopf quiver and a covering quiver respect to a weight sequence respectively were introduced independently. The main aim of this paper is to show the dually one-to-one correspondent relations between their structures (see Theorems 2.1 and 2.4).
As applications, firstly, the authors obtain some important results about the Hopf algebra structure on the quotient of path algebra on a cycle; then, they prove that the Sweedler's fourdimensional Hopf algebra H₄ is not only quasi-triangular but also co-quasi-triangular. Lastly, they characterize the graded automorphism group of the Hopf algebras on the path algebra of a Schurian covering quiver, according to that on the path coalgebra of a Schurian Hopf quiver.

Key words: Hopf quiver, Covering quiver, Dual Hopf algebra, Schurian, Graded automorphism group