Wronskian envelope of a Lie algebra
type: Item
instance of: scholarly article
title: Wronskian envelope of a Lie algebra
zbMATH Open document ID: 1334.17005
DOI: 10.1155/2013/341631
author: Laurent Poinsot
published in: Algebra
publication date: 2015-12-11T00:00:00Z
review text: Summary: The famous Poincaré-Birkhoff-Witt theorem states that a Lie algebra, free as a module, embeds into its associative envelope -- its universal enveloping algebra -- as a sub-Lie algebra for the usual commutator Lie bracket. However, there is another functorial way -- less known -- to associate a Lie algebra to an associative algebra and inversely. Any commutative algebra equipped with a derivation \(a \mapsto a'\), that is, a commutative differential algebra, admits a Wronskian bracket \(W(a,b)=ab'-a'b\) under which it becomes a Lie algebra. Conversely, to any Lie algebra a commutative differential algebra is universally associated, its Wronskian envelope, in a way similar to the associative envelope. This contribution is the beginning of an investigation of these relations between Lie algebras and differential algebras which is parallel to the classical theory. In particular, we give a sufficient condition under which a Lie algebra may be embedded into its Wronskian envelope, and we present the construction of the free Lie algebra with this property.
Mathematics Subject Classification ID: 17B05
Mathematics Subject Classification ID: 17B01
zbMATH DE Number: 6519780
zbMATH Keywords: Lie algebras
zbMATH Keywords: differential algebras
Wikidata QID: Q58919579
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1155/2013/341631
OpenAlex ID: W2167890345
description: scientific article; zbMATH DE number 6519780