Central extensions and [formula omitted] of the graded lie algebras of cartan type

Abstract

Let F be an algebraically closed field of characteristic p > 0, L= ⊕ i = − r s L ( i) a graded Lie algebra of Cartan type over F. We give a description of the dual adjoint module L ∗ by means of the mixed product and the coinduced module such that the computation of H 1(L, L ∗) can be reduced to that of the first cohomology of L (0). Then we determine the central extensions and H 2( L, F) by computing H 1(L, L ∗) .