Simple Lie algebras of special type
type: Item
instance of: scholarly article
title: Simple Lie algebras of special type
zbMATH Open document ID: 1002.17006
author: Jeffrey Bergen
author: D. S. Passman
published in: Journal of Algebra
publication date: 2001-10-29T00:00:00Z
review text: Let \(K\) be an arbitrary field, \(A\) a commutative \(K\)-algebra, \(\Delta\) a nonzero \(K\)-vector space of commuting \(K\)-derivations of \(A\), and \(W\) the Lie subalgebra \(A\otimes_K\Delta= A\Delta\). The divergence map \(\operatorname {div}: W\rightarrow A\) is defined to be the \(K\)-linear extension of div\((a\delta)=\delta(a)\) for all \(a\in A\), \(\delta\in \Delta\). The kernel \(S\) of div is a Lie subalgebra of \(W\), called a special type Lie algebra. In the article under review, the authors study \(S\) from a ring-theoretic point of view and obtain sufficient conditions for the simplicity of \([S,S]\). These conditions are somewhat technical and cumbersome. But the results are effective in proving the simplicity of a number of interesting examples.
Mathematics Subject Classification ID: 17B20
Mathematics Subject Classification ID: 17B05
zbMATH DE Number: 1448637
zbMATH Keywords: Lie algebra of special type
zbMATH Keywords: divergence
zbMATH Keywords: derivation
zbMATH Keywords: Witt type algebra
reviewed by: Guangyu Shen
MaRDI profile type: Publication
full work available at URL: jabr.1999.8225
OpenAlex ID: W2040835006
Wikidata QID: Q115395777
cites work: Generalized Cartan type \(W\) Lie algebras in characteristic zero
cites work: Simple Lie Rings of Derivations of Commutative Rings
cites work: On the Ideals of a Lie Algebra of Derivations
cites work: Generalizations of Witt algebras over a field of characteristic zero
cites work: Q3786586
cites work: New simple infinite-dimensional Lie algebras of characteristic 0
cites work: Simple Lie algebras of Witt type
cites work: Q4039778
DOI: 10.1006/JABR.1999.8225
description: scientific article; zbMATH DE number 1448637