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

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.