The second cohomology of simple SL_2-modules

TL;DR

This paper computes the second cohomology groups of all irreducible modules for the simple algebraic group SL_2 over an algebraically closed field of positive characteristic, advancing understanding of their cohomological properties.

Contribution

It provides a complete determination of the second cohomology for all irreducible SL_2-modules, a result not previously established.

Findings

Explicit second cohomology groups for all irreducible modules of SL_2

New insights into the structure of these modules' cohomology

Enhanced understanding of the representation theory of SL_2 in positive characteristic

Abstract

Let $G$ be the simple algebraic group $SL_2$ defined over an algebraically closed field $K$ of characteristic $p>0$. In this paper, we find the second cohomology of all irreducible representations of $G$