Sur les corps liés aux algèbres enveloppantes des algèbres de Lie
type: Item
instance of: scholarly article
title: Sur les corps liés aux algèbres enveloppantes des algèbres de Lie
zbMATH Open document ID: 0144.02104
DOI: 10.1007/BF02684800
published in: Publications Mathématiques
publication date: 1966-01-01T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/103872
review text: Let \(G\) be a Lie algebra over a field \(L\) of characteristic 0. \(\mathfrak U(G)\) be the enveloping algebra of \(G\). \(\mathfrak U(G)\) is, in general, non-commutative and is an Ore ring without zero divisors, i.e., for any two non-zero elements, there exist common left and right multiples. Therefore, \(\mathfrak U(G)\) has a quotient field \(\mathfrak D(G)\) which is called Lie field of \(G\). In this note, the authors construct for any pair of non-negative integers \((n, k)\), a field \(\mathfrak D_{n,k}(L)\) containing \(L\) such that the dimension of it over \(L\) is \(2n+k\) and the dimension of its center over \(L\) is \(k\). Then they state a conjecture that if \(G\) is an algebraic Lie algebra over \(L\), then \(\mathfrak D(G)\) is isomorphic to \(\mathfrak D_{n,k}(L)\) for some pair \((n, k)\). Here, the conjecture is solved affirmatively for the Lie algebras \(\mathfrak{gl}(n, L)\), \(\mathfrak{sl}(n, L)\) and the nilpotent Lie algebras. Further, they give examples of two Lie algebras over the field \(R\) of real numbers which are both not algebraic such that the Lie field of the one is not isomorphic to any \(\mathfrak D_{n,k}(R)\) and that of the other is isomorphic to \(\mathfrak D_{2,0}(L)\).
reviewed by: Eiichi Abe
Mathematics Subject Classification ID: 17B35
Mathematics Subject Classification ID: 17B20
Mathematics Subject Classification ID: 17B30
zbMATH DE Number: 3231960
zbMATH Keywords: enveloping algebra
zbMATH Keywords: algebraic Lie algebra
zbMATH Keywords: nilpotent Lie algebras
Wikidata QID: Q105488794
author: I. M. Gel'fand
author: A. A. Kirillov
MaRDI profile type: Publication
cites work: Sur les représentations unitaires des groupes de Lie nilpotents. II
cites work: Sur l'algèbre enveloppante d'une algèbre de Lie nilpotente
cites work: Q5340167
OpenAlex ID: W185924885
description: scientific article