Classification of generalized Witt algebras over algebraically closed fields

Abstract

Let Φ \Phi be a field of characteristic p > 0 p > 0 and m , n 1 , … , n m m,{n_1}, \ldots ,{n_m} be integers ≧ 1 \geqq 1 . A Lie algebra W ( m : n 1 , … , n m ) W(m:{n_1}, \ldots ,{n_m}) over Φ \Phi is defined. It is shown that if Φ \Phi is algebraically closed then W ( m : n 1 , … , n m ) W(m:{n_1}, \ldots ,{n_m}) is isomorphic to a generalized Witt algebra, that every finite-dimensional generalized Witt algebra over Φ \Phi is isomorphic to some W ( m : n 1 , … , n m ) W(m:{n_1}, \ldots ,{n_m}) , and that W ( m : n 1 , … , n m ) W(m:{n_1}, \ldots ,{n_m}) is isomorphic to W ( s : r 1 , … , r s ) W(s:{r_1}, \ldots ,{r_s}) if and only if m = s m = s and r i = n σ ( i ) {r_i} = {n_{\sigma (i)}} for 1 ≦ i ≦ m 1 \leqq i \leqq m where σ \sigma is a permutation of { 1 , … , m } \{ 1, \ldots ,m\} . This gives a complete classification of the finite-dimensional generalized Witt algebras over algebraically closed fields. The automorphism group of W ( m : n 1 , … , n m ) W(m:{n_1}, \ldots ,{n_m}) is determined for p > 3 p > 3 .