The residual nilpotence of the multiplicative group of a skew field generated by universal enveloping algebras
type: Item
instance of: scholarly article
title: The residual nilpotence of the multiplicative group of a skew field generated by universal enveloping algebras
zbMATH Open document ID: 0636.16007
DOI: 10.1016/0021-8693(88)90144-5
author: A. I. Lichtman
published in: Journal of Algebra
publication date: 1988-01-01T00:00:00Z
review text: Let L be a Lie algebra over a field k and U(L) its universal associative envelope. This algebra U(L) can be embedded in a skew field \(D=D(L)\) [cf. the reviewer, Proc. Lond. Math. Soc., III. Ser. 11, 511-530 (1961; Zbl 0104.032)]. The author's aim is a study of the multiplicative group \(D^*\); he proves here \(D^*\cong k^*\times D_ 1\) where the group \(D_ 1\) is either residually torsion-free nilpotent, when char k\(=0\), or a residually nilpotent p-group of bounded exponent when char k\(=p>0\). In the special case when L is soluble-by-finite dimensional, U(L) is an Ore domain and the assertion holds for its field of fractions.
Mathematics Subject Classification ID: 16Kxx
Mathematics Subject Classification ID: 17B35
Mathematics Subject Classification ID: 16U60
Mathematics Subject Classification ID: 20E26
zbMATH DE Number: 4036011
zbMATH Keywords: Lie algebra
zbMATH Keywords: universal associative envelope
zbMATH Keywords: skew field
zbMATH Keywords: multiplicative group
zbMATH Keywords: residually torsion-free nilpotent
zbMATH Keywords: residually nilpotent p-group
zbMATH Keywords: soluble-by-finite dimensional
zbMATH Keywords: Ore domain
reviewed by: Paul M. Cohn
MaRDI profile type: Publication
full work available at URL: 0021-8693(88)90144-5
OpenAlex ID: W2051012006
cites work: On the Embedding of Rings in Skew Fields
cites work: Q5682962
cites work: Matrix rings and linear groups over a field of fractions of enveloping algebras and group rings. I
cites work: Growth in enveloping algebras
cites work: Generalized nilpotent algebras and their adjoint groups
description: scientific article; zbMATH DE number 4036011