Simple modules over truncated current Lie algebras of the Witt algebra

TL;DR

This paper classifies simple modules over truncated current Lie algebras derived from the Witt algebra, focusing on modules with specific p-character heights, advancing understanding of their structure.

Contribution

It provides a complete classification of simple modules with p-character of height at most one and explores modules with higher p-character heights.

Findings

Classified simple modules with p-character height ≤ 1.

Analyzed modules with p-character height > 1.

Enhanced understanding of module structures over truncated Witt algebra.

Abstract

Let $\Bbbk$ be an algebraically closed field of characteristic $p>3$, and let $W$ denote the $p$-dimensional Witt algebra, the first example of a non-classical simple Lie algebra. For a non-negative integer $\ell$, consider the associated truncated current Lie algebra $W_\ell=W \otimes \Bbbk[t]/(t^{\ell+1})$. In this paper, we first study simple $W_\ell$-modules having $p$-character $\chi$ of height at most one, and provide a complete classification of such modules up to isomorphism. We then investigate a family of simple $W_\ell$-modules whose $p$-characters have height greater than one.