Verbal width in the Nottingham group and related Lie algebras
type: Item
instance of: scholarly article
title: Verbal width in the Nottingham group and related Lie algebras
zbMATH Open document ID: 1472.20063
author: J. Martínez Carracedo
author: Consuelo Martínez
published in: Journal of Algebra
publication date: 2021-03-09T00:00:00Z
full work available at URL: Journal of Algebra Nottingham Group.pdf
review text: \textit{M. A. Virasoro} introduced his celebrated algebraic structure (known as Virasoro algebra) in order to describe some physical phenomena, which are related to the harmonic oscillator in [``Subsidiary conditions and ghosts in dual-resonance models'', Phys. Rev. D. 1, No. 10, 2933--2936 (1970; \url{doi:10.1103/PhysRevD.1.2933})]. Virasoro algebras appeared in different ways in many dynamical systems, because their structure is very interesting and deserves a separate interest, not only in mathematical physics, but also in group theory and Lie theory. It should be mentioned (from the mathematical perspective) that a well-known result of \textit{M. Lazard} [Ann. Sci. Éc. Norm. Supér. (3) 71, 101--190 (1954; Zbl 0055.25103)] explores in fact a correspondence between nilpotent groups and nilpotent Lie rings. The present contribution shows some peculiarities of Virasoro algebras in connection with a profinite group, known as Nottingham group, and introduced by \textit{D. L. Johnson} [J. Aust. Math. Soc., Ser. A 45, No. 3, 296--302 (1988; Zbl 0666.20016)]. Virasoro algebras and some of their subalgebras are in fact related to the Nottingham group. Theorem 3.7 of the paper under review shows that the Nottingham group in zero characteristic is verbally elliptic, that is, all the words of such a group have finite width. Theorems 2.2 and 2.3 show that arbitrary multilinear polynomials have corresponding properties of width in the context of Virasoro algebras. Theorem 2.4 describes additional properties of structural nature for Virasoro algebras.
Mathematics Subject Classification ID: 20E18
Mathematics Subject Classification ID: 20F14
Mathematics Subject Classification ID: 17B15
Mathematics Subject Classification ID: 20F10
zbMATH DE Number: 7318820
zbMATH Keywords: lower central series
zbMATH Keywords: graded Lie algebra
zbMATH Keywords: Nottingham group
zbMATH Keywords: Virasoro algebra
zbMATH Keywords: Witt algebra
zbMATH Keywords: verbal width
reviewed by: Francesco Russo
MaRDI profile type: Publication
OpenAlex ID: W3127994878
Wikidata QID: Q115350394
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cites work: Subgroups of finite index in profinite groups
cites work: On the verbal width of finitely generated pro-\(p\) groups.
cites work: Ellipticity of words
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cites work: Q3818488
DOI: 10.1016/J.JALGEBRA.2021.02.003
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description: scientific article; zbMATH DE number 7318820