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Simple Lie Algebras of Special Type

✍ Scribed by Jeffrey Bergen; D.S Passman

Publisher: Elsevier Science
Year: 2000
Tongue: English
Weight: 190 KB
Volume: 227
Category: Article
ISSN: 0021-8693
DOI: 10.1006/jabr.1999.8225

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✦ Synopsis

Let K be a field, let A be an associative, commutative K-algebra, and let be a nonzero K-vector space of commuting K-derivations of A. Then, with a rather natural definition, A = A ⊗ K = A becomes a Lie algebra, a Witt type algebra. In addition, there is a map div:

A → A called the divergence and its kernel S = A is a Lie subalgebra, a special type algebra. In this paper, we study S from a ring theoretic point of view, obtaining sufficient conditions for the Lie simplicity of S S . While the main result here is somewhat cumbersome to state, it does handle a number of examples in a fairly efficient manner. Furthermore, some of the preliminary lemmas are of interest in their own right and may, in time, lead to a more satisfactory answer.

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