Bilinear forms on Green rings of finite dimensional Hopf algebras

TL;DR

This paper investigates the algebraic structures of Green rings associated with finite-dimensional Hopf algebras, revealing Frobenius and bi-Frobenius properties through bilinear forms and their non-degeneracy.

Contribution

It introduces a new bilinear form on the stable Green ring to determine its bi-Frobenius property and characterizes the complexified stable Green ring as a group-like algebra.

Findings

Green ring of finite type Hopf algebra is Frobenius over Z

New bilinear form on stable Green ring helps identify bi-Frobenius property

Complexified stable Green ring is a group-like algebra if form is non-degenerate

Abstract

In this paper, we study the Green ring and the stable Green ring of a Hopf algebra $H$ 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 property of the stable Green ring. We show that the complexified stable Green ring (or algebra) is a group-like algebra, and hence a bi-Frobenius algebra, if the bilinear form on the stable Green ring is non-degenerate.