Derivations, isomorphisms, and second cohomology of generalized Witt algebras

Authors:
Dragomir Z. DJ Okovic and Kaming Zhao

Journal:
Trans. Amer. Math. Soc. **350** (1998), 643-664

MSC (1991):
Primary 17B40, 17B65; Secondary 17B56, 17B68

DOI:
https://doi.org/10.1090/S0002-9947-98-01786-3

MathSciNet review:
1390977

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Abstract | References | Similar Articles | Additional Information

Abstract: Generalized Witt algebras, over a field of characteristic , were defined by Kawamoto about 12 years ago. Using different notations from Kawamoto's, we give an essentially equivalent definition of generalized Witt algebras over , where the ingredients are an abelian group , a vector space over , and a map which is linear in the first variable and additive in the second one. In this paper, the derivations of any generalized Witt algebra

, with the right kernel of being , are explicitly described; the isomorphisms between any two simple generalized Witt algebras are completely determined; and the second cohomology group 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.

**1.**R. K. Amayo and I. Stewart,*Infinite-dimensional Lie Algebras*, Noordhoff, Leyden, 1974.MR**53:570****2.**R. Farnsteiner,*Derivations and central extensions of finitely generated Lie algebras*, J. Algebra**118**(1988), 33-45.MR**89h:17018****3.**T. Ikeda,*Derivations and central extensions of a generalized Witt algebra*, in Nonassociative Algebras and Related Topics, World Scientific, Singapore, 1991, 47-57.MR**93c:17038****4.**T. Ikeda and N. Kawamoto,*On the derivations of generalized Witt algebras over a field of characteristic zero*, Hiroshima Math. J.**20**(1990), 47-55.MR**91d:17021****5.**N. Jacobson,*Lie Algebras*, Dover, New York, 1979.MR**80k:17001****6.**I. Kaplansky,*Seminar on simple Lie algebras*, Bull. Amer. Math. Soc.**60**(1954), 470-471.**7.**N. Kawamoto,*Generalizations of Witt algebras over a field of characteristic zero*, Hiroshima Math. J.**16**(1986), 417-426.MR**88d:17017****8.**N. Kawamoto,*On G-graded automorphisms of generalized Witt algebras*, in Contemporary Mathematics, Second International Conference on Algebra (L.A. Bokut, A.I. Kostrikin and S.S. Kutateladze, editors),**184**(1995), 225-230.MR**97b:17017****9.**R.V. Moody and A. Pianzola,*Lie Algebras with Triangular Decompositions*, J. Wiley, New York, 1995.MR**96d:17025****10.**J.M. Osborn,*New simple infinite dimensional Lie algebras of characteristic*, J. Algebra**185**(1996), 820-835. CMP**1997#4****11.**J.M. Osborn,*Derivations and isomorphisms of Lie algebras of characteristic*(preprint).**12.**J.M. Osborn,*Automorphisms of the Lie algebras in characteristic*(preprint).

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Additional Information

**Dragomir Z. DJ Okovic**

Affiliation:
Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada

Email:
dragomir@herod.uwaterloo.ca

**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

Email:
zhao@math.wisc.edu

DOI:
https://doi.org/10.1090/S0002-9947-98-01786-3

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.\endgraf The second author was supported by Academia Sinica of P.R. China.

Article copyright:
© Copyright 1998
American Mathematical Society