Cohomology of simple modules for sl3(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 that 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 calculating the cohomology of the adjoint module. We also calculate cohomology of the simple quotient algebra Lie of A ₂ by the center.