Extensions of simple modules for the Witt algebra

TL;DR

This paper classifies extensions of simple modules over the Witt algebra for specific p-characters, advancing understanding of their module structure in modular representation theory.

Contribution

It provides a new classification of module extensions for the Witt algebra with certain p-characters, extending previous classifications of irreducible modules.

Findings

Classification of extensions for p-characters of height -1, 0, 1, and p-1

Enhanced understanding of module structures over the Witt algebra

Builds on prior classifications of irreducible modules

Abstract

The irreducible representations of the Witt algebra $W$ are completely known. A classification of the irreducible $U_\chi(W)$--modules was first established by Chang and later simplified by Strade. The aim of this article is to give a classification of the extensions of the simple $U_\chi(W)$--modules having $p$--character $\chi$ of height $-1,0,1$ and $p-1$ where $p$ denotes the characteristic of the ground field.