Wronskian Envelope of a Lie Algebra

Abstract EN

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↦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.