On cohomology of a class of nonclassical restricted simple Lie algebras

摘要

In this note, we study the cohomology of the nonclassical restricted Lie algebras W( n), n is an element of N+ over a field F of characteristic p > 0, which are by definition the Lie algebras of derivations on the truncated polynomial algebras Fx(1),..., xn/(x(1)(p),..., x(n)(p)) (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 F, which says that H-q(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.