由箭图构造的对偶Hopf代数和量子群

Dual Hopf Algebras from a Quiver and Dual Quiver Quantum Groups

在文献[3]和[6]中,Hopf箭图的路代数上的Hopf代数结构和覆盖箭图的路余代数上的Hopf代数结构分别被给出.该文通过一个箭图是Hopf箭图当且仅当它是箭图覆盖这一结论,来讨论同一箭图上给出的这两种Hopf代数结构之间的对偶关系(见定理3和定理4).作为应用,作者先得到关于定向圈的路代数的商上的Hopf代数结构的一些性质,然后证明了Sweedler的4维-Hopf代数不仅是拟三角的而且是余拟三角的.最后,作者刻画了Schurian覆盖箭图的路代数上的Hopf代数的分次自同构群.

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.