Some new simple Lie algebras in characteristic 2
type: Item
instance of: scholarly article
title: Some new simple Lie algebras in characteristic 2
zbMATH Open document ID: 1229.17024
author: B. Eick
published in: Journal of Symbolic Computation
publication date: 2010-08-13T00:00:00Z
review text: In a previous paper [Linear Algebra Appl.\ 382, 195--209 (2004; Zbl 1045.17001)] the author describes effective algorithms for computing the automorphism groups of finite-dimensional solvable Lie algebras over finite fields and for testing isomorphisms between these algebras. The first part of the paper under review contains deterministic and probabilistic algorithms for the same tasks but now for not necessarily solvable Lie algebras based on certain nice presentations of the algebras under consideration. The algorithms are particularly effective for simple or almost simple Lie algebras and therefore complement the ones in the previous paper. Recently, the classification of finite-dimensional simple Lie algebras has been completed for algebraically closed fields of characteristic \(p>3\). It is known from a wealth of examples that a classification in characteristic \(2\) or \(3\) will look very different and so any ideas on how such classifications might look like would be helpful. \textit{M. Vaughan-Lee} [LMS J.\ Comput.\ Math.\ 9, 174--192 electronic only (2006; Zbl 1162.17009)] classified the simple Lie algebras of dimension at most \(9\) over the field \(\mathbb{F}_2\) with two elements. In the second part of her paper the author shows how the isomorphism test from the first part can be used in a computer-based randomized search for new simple Lie algebras of dimension up to \(20\) over \(\mathbb{F}_2\). It is pointed out that the author's approach does not yield a complete classification but that it enabled the author to find two simple Lie algebras in dimension \(15\) and four simple Lie algebras in dimension \(16\) that were previously unknown. The new simple Lie algebras in dimension \(15\) arose as constituents of subalgebras of a simple Lie algebra found by \textit{I.\ Kaplansky} thirty years ago [Some simple Lie algebras of characteristic 2, in: Lie algebras and related topics, Proc.\ Conf., New Brunswick/USA 1981, Lect.\ Notes Math.\ 933, 127--129 (1982…
reviewed by: Jörg Feldvoss
Mathematics Subject Classification ID: 17B50
Mathematics Subject Classification ID: 17-04
Mathematics Subject Classification ID: 17-08
Mathematics Subject Classification ID: 17B40
Mathematics Subject Classification ID: 17B20
zbMATH DE Number: 5770524
zbMATH Keywords: Modular Lie algebra
zbMATH Keywords: simple Lie algebra
zbMATH Keywords: automorphism group
zbMATH Keywords: isomorphism testing
zbMATH Keywords: classification in small characteristics
describes a project that uses: Sophus
MaRDI profile type: Publication
full work available at URL: j.jsc.2010.05.003
OpenAlex ID: W2009890000
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DOI: 10.1016/J.JSC.2010.05.003
description: scientific article; zbMATH DE number 5770524