Auslander-Reiten theory of Frobenius-Lusztig kernels

TL;DR

This paper investigates the structure of the Auslander-Reiten quiver components of Frobenius-Lusztig kernels, revealing they are classified by infinite Dynkin diagrams, with specific results for small quantum groups.

Contribution

It establishes the classification of stable Auslander-Reiten quiver components for Frobenius-Lusztig kernels, including the shape of periodic and non-periodic components for small quantum groups.

Findings

Components are classified by three infinite Dynkin diagrams.

Periodic components are homogeneous tubes in small quantum groups.

Non-periodic components have shape Z[A_infinity] in certain cases.

Abstract

In this paper we show that the tree class of a component of the stable Auslander-Reiten quiver of a Frobenius-Lusztig kernel is one of the three infinite Dynkin diagrams. For the special case of the small quantum group we show that the periodic components are homogeneous tubes and that the non-periodic components have shape Z[A_\infty] if the component contains a module for the infinite-dimensional quantum group.