Acta mathematica scientia,Series A ›› 2009, Vol. 29 ›› Issue (2): 505-516.

• Articles • Previous Articles     Next Articles

Dual Hopf Algebras from a Quiver and Dual Quiver Quantum Groups

  

  1. (Department of Mathematics, Zhejiang University,  Hangzhou  310027)
  • Received:2007-11-01 Revised:2009-02-07 Online:2009-04-25 Published:2009-04-25
  • Supported by:

    国家自然科学基金(10571153, 10871170)和浙江省自然科学基金重大项目(D7080064)资助

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

CLC Number: 

  • 16D20
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