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Derivations, isomorphisms, and second cohomology of generalized Witt algebras

Authors: Dragomir Ž. Đ Oković and Kaming Zhao
Journal: Trans. Amer. Math. Soc. 350 (1998), 643-664
MSC (1991): Primary 17B40, 17B65; Secondary 17B56, 17B68
MathSciNet review: 1390977
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Abstract: Generalized Witt algebras, over a field $F$ of characteristic $0$, were defined by Kawamoto about 12 years ago. Using different notations from Kawamoto’s, we give an essentially equivalent definition of generalized Witt algebras $W=W(A,T,\varphi )$ over $F$, where the ingredients are an abelian group $A$, a vector space $T$ over $F$, and a map $\varphi :T\times A\to K$ which is linear in the first variable and additive in the second one. In this paper, the derivations of any generalized Witt algebra $W=$ $W(A,T,\varphi )$, with the right kernel of $\varphi$ being $0$, are explicitly described; the isomorphisms between any two simple generalized Witt algebras are completely determined; and the second cohomology group $H^2(W,F)$ for any simple generalized Witt algebra is computed. The derivations, the automorphisms and the second cohomology groups of some special generalized Witt algebras have been studied by several other authors as indicated in the references.

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Additional Information

Dragomir Ž. Đ Oković
Affiliation: Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada

Kaming Zhao
Affiliation: Institute of Systems Science, Academia Sinica, Beijing, 100080, China
Address at time of publication: Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706-1388

Keywords: Simple Lie algebras, derivations, 2-cocycles, automorphism group
Received by editor(s): January 2, 1996
Received by editor(s) in revised form: April 8, 1996
Additional Notes: The first author was supported in part by the NSERC Grant A-5285. The second author was supported by Academia Sinica of P.R. China.
Article copyright: © Copyright 1998 American Mathematical Society