Koszul modules of Kac-Moody Lie algebras

TL;DR

This paper introduces Koszul modules for graded Kac-Moody Lie algebras, establishing criteria for their finite length, bounding dimensions, and describing nilpotent modules, thus advancing understanding of their structure.

Contribution

It provides a precise criterion for finite length of Koszul modules and characterizes all nilpotent Kac-Moody Koszul modules, offering new structural insights.

Findings

Criterion for finite length of Koszul modules

Bound on the second graded component dimension

Complete description of nilpotent Kac-Moody Koszul modules

Abstract

We introduce Koszul modules associated with (graded) Kac-Moody Lie algebras. We provide a precise criterion for when these modules are of finite length. As an exemplary application we deduce a bound on the dimension of the second graded component for a certain class of graded Kac-Moody Lie algebras. We also provide an exact description of all nilpotent Kac-Moody Koszul modules.