Families of Green rings for abelian restricted Lie algebras

TL;DR

This paper characterizes the representation type of abelian restricted Lie algebras by examining how their Green rings of restricted representations change with different cocommutative Hopf algebra structures, revealing a correspondence for tame cases.

Contribution

It establishes a link between the representation type of abelian restricted Lie algebras and the variation of their Green rings under different Hopf algebra structures, identifying a unique correspondence for tame cases.

Findings

Tame algebras have a correspondence between Hopf subalgebras and minimal thick tensor-ideals.

Wild algebras do not exhibit such a correspondence.

The Green ring structure is determined by the Lie algebra and Hopf algebra structures.

Abstract

We find equivalent conditions determining the representation type of abelian restricted Lie algebras in terms of how their Green ring of restricted representations varies with respect to different cocommutative Hopf algebra structures on its restricted universal enveloping algebra. Each compatible cocommutative Hopf algebra structure on a tame algebra is shown to have a correspondence between a certain set of Hopf subalgebras, and the set of minimal thick tensor-ideals having identical ring structure when determined by either the Hopf algebra structure or the base Lie algebra structure (up to a choice of character group). Those of wild representation type are shown never to have such a correspondence.