Prime Representations from a Homological Perspective

TL;DR

This paper investigates the homological properties of simple finite-dimensional prime representations of quantum affine algebras, revealing a link between self extensions and primeness, especially in the sl(2) case.

Contribution

It introduces a homological criterion for prime representations based on self extensions and provides evidence supporting its general applicability.

Findings

Every nontrivial simple module has a nontrivial self extension.

Prime representations in sl(2) are characterized by a unique nontrivial self extension.

A large class of prime representations satisfy the proposed homological property.

Abstract

We begin the study of simple finite-dimensional prime representations of quantum affine algebras from a homological perspective. Namely, we explore the relation between self extensions of simple representations and the property of being prime. We show that every nontrivial simple module has a nontrivial self extension. Conversely, if a simple representation has a unique nontrivial self extension up to isomorphism, then its Drinfeld polynomial is a power of the Drinfeld polynomial of a prime representation. It turns out that, in the sl(2) case, a simple module is prime if and only if it has a unique nontrivial self extension up to isomorphism. It is tempting to conjecture that this is true in general and we present a large class of prime representations satisfying this homological property.