Weight modules for current algebras

TL;DR

This paper classifies all simple weight modules with bounded multiplicities for current algebras formed from finite-dimensional simple Lie algebras and finite type commutative algebras.

Contribution

It provides a complete classification of simple weight modules with bounded multiplicities for current algebras, extending understanding of their module structure.

Findings

Complete classification of simple weight modules with bounded multiplicities.

Identification of conditions for module simplicity.

Framework applicable to various Lie algebra and algebra combinations.

Abstract

For any finite-dimensional simple Lie algebra $\mathfrak{g}$ and commutative associative algebra $S$ of finite type, we give a complete classification of the simple weight modules of $\mathfrak{g}\otimes S$ with bounded weight multiplicities.