Extension functors of modular Lie algebras (Q807732)

One often observes fundamental differences between the cohomological aspects of modular and nonmodular representation theory of Lie algebras. For instance, the fact that for any nonzero modular Lie algebra L, \(H^ 1(L,M)\neq 0\) for a suitably chosen finite L-module M contrasts markedly with Weyl's classical theorem. The author first introduces the notion of a generalized reduced Verma module which unifies various induction functors hitherto considered. The functors \(Ext^ n(M\),-) and \(Ext^ n(\)-,M) with values in generalized reduced Verma modules are investigated. Using the product theory of extension functors, several reduction theorems are established which elicit the conceptual source of the structural peculiarities of modular cohomology theory. Nonvanishing results are obtained which ensure the existence of finite dimensional modular representations \(\rho\) : \(L\to gl(N)\) such that \(Ext^ i(M,N)\neq 0\neq Ext^ i(N,M)\), \(0\leq i\leq \dim L\). Finally, a description of the ordinary and relative cohomology of a \({\mathbb{Z}}\)-graded Lie algebra L with coefficients in generalized reduced Verma modules is given.