The first cohomology of sl(2,1) with coefficients in $\chi$-reduced Kac modules and simple modules

TL;DR

This paper computes the first cohomology groups of the Lie superalgebra sl(2,1) over fields with characteristic p > 2, focusing on hi-reduced Kac modules and simple modules, using weight space decompositions.

Contribution

It provides the first explicit determination of the first cohomology for these modules over sl(2,1) in characteristic p > 2, expanding understanding of their structure.

Findings

First cohomology groups are explicitly computed for hi-reduced Kac modules.

First cohomology groups are explicitly computed for simple modules.

Results depend on weight space decompositions relative to the Cartan subalgebra.

Abstract

Over a field of characteristic p > 2, the first cohomology of the special linear Lie superalgebra sl(2,1) with coefficients in all \c{hi}-reduced Kac modules and simple modules is determined by use of the weight space decompositions of these modules relative to the standard Cartan subalgebra of sl(2,1).