Classification of generalized Witt algebras over algebraically closed fields

Abstract:

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