Narrow algebras and groups
type: Item
instance of: scholarly article
title: Narrow algebras and groups
zbMATH Open document ID: 0933.17011
DOI: 10.1007/BF02366350
published in: Journal of Mathematical Sciences (New York)
publication date: 1999-08-16T00:00:00Z
review text: This is a survey on algebras of Gelfand-Kirillov dimension 1 and related pro-\(p\)-groups, also some new results are proved. First, the authors study associative algebras and Lie algebras in characteristic zero. Pro-\(p\)-groups and \(\mathbb N\)-graded Lie algebras cannot be simple. Instead appears the notion of being just-infinite, namely, not having non-trivial normal subgroups (ideals) of infinite index (codimension). Understanding just-infinite filtered or \(\mathbb N\)-graded Lie algebras of Gelfand-Kirillov dimension 1 and pro-\(p\)-groups of finite width is still a formidable challenge. The authors propose a conjecture on a structure of just-infinite filtered Lie algebras \(L=L_1\supset L_2\dots\) over an algebraically closed field of zero characteristic such that \(\dim L_i/L_{i+1}\) are uniformly bounded. As a partial result the authors classify just-infinite \(\mathbb N\)-graded Lie algebras \(L=\bigoplus_{i\geq 1}L_i\) in characteristic zero under assumption \(\dim L_i=d\), \(i\geq n_0\), with ``polynomial multiplication''. Namely, they prove that such algebras are commensurable with the Witt algebra \(W_1\) or with the positive part of some loop algebra. (Two algebras are said commensurable if they contain isomorphic subalgebras of finite codimension). Second, results on pro-\(p\)-groups, finite \(p\)-groups, and graded Lie algebras in characteristic \(p\) are surveyed. Main objects here are groups and algebras of finite coclass.
Mathematics Subject Classification ID: 17B70
Mathematics Subject Classification ID: 20E18
Mathematics Subject Classification ID: 20D15
Mathematics Subject Classification ID: 20F40
Mathematics Subject Classification ID: 16P90
zbMATH DE Number: 1315229
zbMATH Keywords: Gelfand-Kirillov dimension
zbMATH Keywords: graded Lie algebras
zbMATH Keywords: pro-\(p\)-groups
zbMATH Keywords: coclass
author: Aner Shalev
author: E. Zelmanov
reviewed by: Victor Petrogradsky
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02366350
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