Cohomology of simple modules for ${\mathfrak{sl}}_3(k)$ in characteristic $3$

TL;DR

This paper computes the cohomology of simple modules for the Lie algebra of type A2 over a field of characteristic 3, revealing unique structures and their implications for the algebra's cohomology.

Contribution

It provides explicit calculations of cohomology for simple modules of _2 in characteristic 3, including peculiar modules and the adjoint module, advancing understanding of modular Lie algebra cohomology.

Findings

Identified two unique simple modules in characteristic 3.

Calculated cohomology of these modules and the adjoint module.

Analyzed the cohomology of the quotient algebra Lie of _2.

Abstract

In this paper, we calculate cohomology of a classical Lie algebra of type $A_2$ over an algebraically field $k$ of characteristic $p=3$ with coefficients in simple modules. To describe their structure, we will consider them as modules over an algebraic group $SL_3(k).$ In the case of characteristic $p=3,$ there are only two peculiar simple modules: a simple module isomorphic to the quotient module of the adjoint module by the center, and a one-dimensional trivial module. The results on the cohomology of simple nontrivial module are used for calculate the coomology of the adjoint module. We also calculate cohomology of the simple quotient algebra Lie of $A_2$ by the center.