Examples of support varieties for Hopf algebras with noncommutative tensor products

TL;DR

This paper explores unusual behaviors in the representation theory of certain Hopf algebras, focusing on support varieties and ideal classifications, revealing complexities in tensor product properties.

Contribution

It provides explicit examples of Hopf algebras with noncommutative tensor products and classifies their stable module category ideals.

Findings

Nonprojective modules can have projective tensor powers

Tensor product varieties may not be contained in the intersection of individual varieties

Classification of ideals in stable module categories

Abstract

The representations of some Hopf algebras have curious behavior: Nonprojective modules may have projective tensor powers, and the variety of a tensor product of modules may not be contained in the intersection of their varieties. We explain a family of examples of such Hopf algebras and their modules, and classify left, right, and two-sided ideals in their stable module categories.