The groups of automorphisms of the Witt Wn and Virasoro Lie algebras

Let Ln = K[x1±1,..., xn±1] be a Laurent polynomial algebra over a field K of characteristic zero, Wn:= DerK(Ln) the Lie algebra of K-derivations of the algebra Ln, 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 Wn and Vir are infinite dimen- sional Lie algebras. We prove that the following isomorphisms of the groups of Lie algebra automorphisms hold: AutLie(Vir) [graphic not available: see fulltext] AutLie(W1) [graphic not available: see fulltext] {±1}[graphic not available: see fulltext]K*, and give a short proof that AutLie(Wn) [graphic not available: see fulltext] AutK-alg(Ln) [graphic not available: see fulltext] GLn(Z) [graphic not available: see fulltext]K*n.