Green Rings of Finite Dimensional Pointed Rank One Hopf algebras of Nilpotent Type

TL;DR

This paper classifies modules over a specific class of finite-dimensional Hopf algebras, describes their tensor product decompositions, and characterizes the structure of their Green rings, revealing properties like commutativity, Frobenius, and symmetry.

Contribution

It explicitly determines the Green ring structure of finite dimensional pointed rank one Hopf algebras of nilpotent type, including generators, relations, and algebraic properties.

Findings

Green ring is commutative and generated by one element over the Grothendieck ring.

Green ring is Frobenius and symmetric with dual bases from almost split sequences.

Stable Green ring is isomorphic to a quotient of the Green ring, forming a bi-Frobenius algebra.

Abstract

Let $H$ be a finite dimensional pointed rank one Hopf algebra of nilpotent type. We first determine all finite dimensional indecomposable $H$-modules up to isomorphism, and then establish the Clebsch-Gordan formulas for the decompositions of the tensor products of indecomposable $H$-modules by virtue of almost split sequences. The Green ring $r(H)$ of $H$ will be presented in terms of generators and relations. It turns out that the Green ring $r(H)$ is commutative and is generated by one variable over the Grothendieck ring $G_0(H)$ of $H$ modulo one relation. Moreover, $r(H)$ is Frobenius and symmetric with dual bases associated to almost split sequences, and its Jacobson radical is a principal ideal. Finally, we show that the stable Green ring, the Green ring of the stable module category, is isomorphic to the quotient ring of $r(H)$ modulo the principal ideal generated by the projective cover of the trivial module. It turns out that the complexified stable Green algebra is a group-like algebra and hence a bi-Frobenius algebra.