Free group algebras in the field of fractions of differential polynomial rings and enveloping algebras
type: Item
instance of: scholarly article
title: Free group algebras in the field of fractions of differential polynomial rings and enveloping algebras
zbMATH Open document ID: 0932.16029
published in: Journal of Algebra
publication date: 1999-12-14T00:00:00Z
review text: Let \(K\) be a characteristic 0 division ring, let \(\delta\) be a nonzero derivation of \(K\), and let \(R=K[X,X^{-1};\delta]\) be the subring of the division ring of fractions \(K(X;\delta)\) generated by \(K\), \(X\) and \(X^{-1}\). If \(A\) is a nonempty subset of \(K\), then the main result here is a collection of rather technical, but verifiable, conditions which guarantee that the elements \(aX^{-1}\) with \(a\in A\) freely generate a \(\mathbb{Q}\)-subalgebra of \(R\). With this in hand, the paper goes on to obtain a number of especially nice consequences. For example, Theorem C asserts that if \(D\) is a characteristic 0 division ring with an uncountable center and if \(D\) has an element \(x\) with \(D\neq C_D(x)\oplus[x,D]\), then \(D\) contains a noncommutative free group algebra. Furthermore, suppose \(L\) is a characteristic 0 Lie algebra and let \(D(L)\) denote the division ring generated by the enveloping algebra \(U(L)\) under the embedding given by \textit{P.~M.~Cohn} [Proc. Lond. Math. Soc., III. Ser. 11, 511-530 (1961; Zbl 0104.03203)]. Using this notation, Theorem D asserts that if \(L\neq L'+C_L(x)\) for some \(x\in L\setminus L'\), then \(D(L)\) contains a noncommutative free group algebra.
Mathematics Subject Classification ID: 16S36
Mathematics Subject Classification ID: 16K40
Mathematics Subject Classification ID: 16S30
Mathematics Subject Classification ID: 16S34
Mathematics Subject Classification ID: 16S10
zbMATH DE Number: 1218664
zbMATH Keywords: free rings
zbMATH Keywords: free group algebras
zbMATH Keywords: division rings
zbMATH Keywords: differential polynomial rings
zbMATH Keywords: enveloping algebras
author: M. Shirvani
author: J. Z. Gonçalves
reviewed by: D. S. Passman
MaRDI profile type: Publication
full work available at URL: jabr.1997.7283
OpenAlex ID: W2002415768
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DOI: 10.1006/JABR.1997.7283
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description: scientific article; zbMATH DE number 1218664