Automorphisms and twisted forms of generalized Witt Lie algebras

Abstract:

We prove that the automorphisms of the generalized Witt Lie algebras $W(m,{\mathbf {n}})$ over arbitrary commutative rings of characteristic $p \geq 3$ all come from automorphisms of the algebras on which they are defined as derivations. By descent theory, this result then implies that if a Lie algebra over a field becomes isomorphic to $W(m,{\mathbf {n}})$ over the algebraic closure, it is a derivation algebra of the type studied long ago by Ree. Furthermore, all isomorphisms of those derivation algebras are induced by isomorphisms of their underlying associative algebras.