On cohomology of a class of nonclassical restricted simple Lie algebras

In this note, we study the cohomology of the nonclassical restricted Lie algebras W(n), n ϵ+ over a field of characteristic p > 0, which are by definition the Lie algebras of derivations on the truncated polynomial algebras [x1,...,xn]/(x1p,...,x np) (called the Jacobson-Witt algebras), and simple unless p = 2 and n = 1. We show a vanishing theorem for the cohomology of W(n) with coefficients in the trivial module , which says that Hq(W(n)) = 0 for all q = 1,...,min{n,p}- 1. Moreover, we compute the cohomology of W(n) with coefficients in its defining truncated polynomial algebra under a certain assumption on the characteristic of the ground field.