Cohomology of $\mathfrak{sl}_3$ and $\mathfrak{gl}_3$ with Coefficients in Simple Modules and Weyl Modules in Positive Characteristics

TL;DR

This paper computes the cohomology of the Lie algebras _3 and _3 with coefficients in simple and Weyl modules over fields of characteristic greater than 3, providing new insights into their structure.

Contribution

It provides explicit calculations of cohomology for _3 and _3 with coefficients in key modules in positive characteristic.

Findings

Cohomology of _3 computed for simple modules.

Cohomology of _3 computed for Weyl modules.

Descriptions of _3 cohomology over algebraically closed fields.

Abstract

We calculate the cohomology of $\mathfrak{sl}_3(k)$ over an algebraically closed field $k$ of characteristic $p>3$ with coefficients in simple modules and Weyl modules. We also give descriptions of the corresponding cohomology of $\mathfrak{gl}_3(k)$.