A category for the adjoint representation

TL;DR

This paper constructs an abelian category with functors that realize the adjoint representation of a simply-laced quantum group, linking algebraic, geometric, and representation-theoretic structures.

Contribution

It introduces a new abelian category and functors that realize quantum group actions, connecting modular representation theory and the McKay correspondence.

Findings

Realization of quantum group actions in a new abelian category

Connection between algebra A and root systems

Application to modular representation theory and McKay correspondence

Abstract

We construct an abelian category C and exact functors in C which on the Grothendieck group descend to the action of a simply-laced quantum group in its adjoint representation. The braid group action in the adjoint representation lifts to an action in the derived category of C. The category C is the direct sum of a semisimple category and the category of modules over a certain algebra A, associated to a Dynkin diagram. In the second half of the paper we show how these algebras appear in the modular representation theory and in the McKay correspondence and explore their relationship with root systems.