✦ LIBER ✦

On the Simplicity of Lie Algebras of Derivations of Commutative Algebras

✍ Scribed by David A. Jordan

Publisher: Elsevier Science
Year: 2000
Tongue: English
Weight: 72 KB
Volume: 228
Category: Article
ISSN: 0021-8693
DOI: 10.1006/jabr.2000.8286

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✦ Synopsis

Let R be a commutative algebra over a field k. We prove two related results on the simplicity of Lie algebras acting as derivations of R. If D is both a Lie subalgebra and R-submodule of Der k R such that R is D-simple and either char k = 2 or D is not cyclic as an R-module or D R = R, then we show that D is simple. This extends a previous result from the author (1986, J. London Math. Soc. ( 2) 33, 33-39) so as to include characteristic 2. If is a Lie subalgebra of Der k R then we show that R ⊗ k is simple if and only if R is -simple, the action of R ⊗ k on R is faithful, and, if char k = 2 and dim k = 1, R = R. This generalizes the weaker of two forms of a result by D. S. Passman (1998, J. Algebra 34, 682-692), where is abelian. However, a stronger form, in which the action of R ⊗ k is replaced by that of R ⊗ k , does not generalize.

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