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Transactions of the American Mathematical Society

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Classification of generalized Witt algebras over algebraically closed fields


Author: Robert Lee Wilson
Journal: Trans. Amer. Math. Soc. 153 (1971), 191-210
MSC: Primary 17B20; Secondary 16A72
DOI: https://doi.org/10.1090/S0002-9947-1971-0316523-6
MathSciNet review: 0316523
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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$.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1971-0316523-6
Keywords: Generalized Witt algebras
Article copyright: © Copyright 1971 American Mathematical Society

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