Homomorphisms between standard modules over finite type KLR algebras

TL;DR

This paper proves that in finite type KLR algebras, standard modules are homologically rigid, with no non-zero homomorphisms between distinct modules and only injective endomorphisms, impacting their extension and representation theory.

Contribution

It establishes the absence of non-zero homomorphisms between distinct standard modules and characterizes endomorphisms, advancing understanding of module homomorphisms in finite type KLR algebras.

Findings

No non-zero homomorphisms between distinct standard modules

All non-zero endomorphisms are injective

Applications to extensions and modular representation theory

Abstract

Khovanov-Lauda-Rouquier algebras of finite Lie type come with families of standard modules, which under the Khovanov-Lauda-Rouquier categorification correspond to PBW-bases of the positive part of the corresponding quantized enveloping algebra. We show that there are no non-zero homomorphisms between distinct standard modules and all non-zero endomorphisms of a standard module are injective. We obtain applications to extensions between standard modules and modular representation theory of KLR algebras.