Bilinear Forms on the Green Rings of Finite Dimensional Hopf Algebras

Abstract

In this paper, we study the Green ring and the stable Green ring of a finite dimensional Hopf algebra by means of bilinear forms. We show that the Green ring of a Hopf algebra of finite representation type is a Frobenius algebra over $\mathbb {Z}$ with a dual basis associated to almost split sequences. On the stable Green ring we define a new bilinear form which is more accurate to determine the bi-Frobenius algebra structure on the stable Green ring. We show that the complexified stable Green algebra is a group-like algebra, and hence a bi-Frobenius algebra, if the bilinear form on the stable Green ring is non-degenerate.