The groups of automorphisms of the Witt W n and Virasoro Lie algebras

Abstract

Let L n = K[x 1 ±1,..., x n ±1] be a Laurent polynomial algebra over a field K of characteristic zero, W n:= DerK(L n) the Lie algebra of K-derivations of the algebra L n, the so-called Witt Lie algebra, and let Vir be the Virasoro Lie algebra which is a 1-dimensional central extension of the Witt Lie algebra. The Lie algebras W n and Vir are infinite dimen- sional Lie algebras. We prove that the following isomorphisms of the groups of Lie algebra automorphisms hold: AutLie(Vir) AutLie(W 1) {±1} K*, and give a short proof that AutLie(W n) AutK-alg(L n) GLn(Z) K *n.