Growth of Lie algebras and the existence of the classical of fractions for the universal enveloping algebras
type: Item
instance of: scholarly article
title: Growth of Lie algebras and the existence of the classical of fractions for the universal enveloping algebras
zbMATH Open document ID: 0883.17009
author: I. A. Yanson
published in: Moscow University Mathematics Bulletin
publication date: 1998-03-05T00:00:00Z
review text: The purpose of the paper under review is to relate the growth of a Lie algebra \(L\) with the existence of a classical ring of fractions of its universal enveloping algebra \(U(L)\). The main result is the following. Let \({\mathcal V}\) be a variety of Lie algebras such that all finitely generated relatively free algebras in \({\mathcal V}\) are of subexponential growth. If \(L\) is a Lie algebra with a solvable ideal \(I\) such that \(L/I\in{\mathcal V}\), then the universal enveloping algebra \(U(L)\) has a classical ring of fractions. In particular, this result holds for the variety \({\mathcal V}\) generated by a finite dimensional algebra. In his approach the author uses the Ore condition and techniques typical for the quantitative study of Lie algebras with polynomial identities and their universal enveloping algebras. The proof of the main result is also based on the theorem from [\textit{A. A. Kirillov, M. L. Kontsevich} and \textit{A. I. Molev}, Algebras of intermediate growth (Russian), Preprint of the Keldysh Institute of Applied Mathematics, No. 39, Moscow (1983)], stating that a finitely generated associative algebra without zero divisors and of subexponential growth has a classical ring of fractions.
Mathematics Subject Classification ID: 17B35
Mathematics Subject Classification ID: 16S30
Mathematics Subject Classification ID: 16U20
Mathematics Subject Classification ID: 17B01
Mathematics Subject Classification ID: 16P90
zbMATH DE Number: 1028476
zbMATH Keywords: growth of Lie algebras
zbMATH Keywords: relatively free Lie algebras
zbMATH Keywords: Lie algebras with polynomial identity
zbMATH Keywords: classical ring of fractions
zbMATH Keywords: universal enveloping algebra
reviewed by: Vesselin Drensky
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1028476