Tensor ideals for quantum groups via minimal tilting complexes

TL;DR

This paper establishes a precise correspondence between certain tensor ideals in quantum group modules and tilting modules using minimal tilting complexes, highlighting differences with algebraic groups in positive characteristic.

Contribution

It introduces an explicit bijection between thick tensor ideals in quantum group modules and tilting modules via minimal tilting complexes, clarifying structural differences with algebraic groups.

Findings

Bijection between tensor ideals and tilting modules in quantum groups.

Explanation of why similar constructions fail for algebraic groups.

Insight into the structure of tensor ideals in quantum group representations.

Abstract

We use minimal tilting complexes to construct an explicit bijection between the set of thick tensor ideals with the two-out-of-three property in the category of finite-dimensional modules over a quantum group at a root of unity and the set of thick tensor ideals in the subcategory of tilting modules. We also explain why the analogous construction for rational representations of a reductive algebraic group over a field of positive characteristic does not give rise to a bijection.