LAUSR

In this note, we study the cohomology of the nonclassical restricted Lie algebras W(n), n ∈ ℕ+ 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.