Algebra

Volume 2013, Article ID 341631, 8 pages

http://dx.doi.org/10.1155/2013/341631

## Wronskian Envelope of a Lie Algebra

Université Paris 13, Sorbonne Paris Cité, LIPN, CNRS, UMR 7030, 93430 Villetaneuse, France

Received 6 March 2013; Accepted 23 April 2013

Academic Editor: Rutwig Campoamor-Stursberg

Copyright © 2013 Laurent Poinsot. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

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 , that is, a commutative differential algebra, admits a Wronskian bracket 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.