On Restricted Cohomology of Modular Classical Lie Algebras and Their Applications

In this paper, we study the restricted cohomology of Lie algebras of semisimple and simply connected algebraic groups in positive characteristics with coefficients in simple restricted modules and their applications in studying the connections between these cohomology with the corresponding ordinary cohomology and cohomology of algebraic groups. Let G be a semisimple and simply connected algebraic group G over an algebraically closed field of characteristic p > h , where h is a Coxeter number. Denote the first Frobenius kernel and Lie algebra of G by G 1 and g , respectively. First, we calculate the restricted cohomology of g with coefficients in simple modules for two families of restricted simple modules. Since in the restricted region the restricted cohomology of g is equivalent to the corresponding cohomology of G 1 , we describe them as the cohomology of G 1 in terms of the cohomology for G 1 with coefficients in dual Weyl modules. Then, we give a necessary and sufficient condition for the isomorphisms H n ( G 1 , V ) ≅ H n ( G , V ) and H n ( g , V ) ≅ H n ( G , V ) , and a necessary condition for the isomorphism H n ( g , V ) ≅ H n ( G 1 , V ) , where V is a simple module with highest restricted weight. Using these results, we obtain all non-trivial isomorphisms between the cohomology of G , G 1 , and g with coefficients in the considered simple modules.