Wronskian Envelope of a Lie Algebra

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. 1. Introduction Any (associative) algebra (and more generally a Lie-admissible algebra), say , admits a derived structure of Lie algebra under the commutator bracket , . This actually describes a forgetful functor, more precisely an algebraic functor, from associative to Lie algebras. This functor admits a left adjoint that enables to associate to any Lie algebra its universal associative envelope. In this way the theory of Lie algebras may be explored through (but not reduced to) that of associative algebras. A Lie algebra which embeds into its universal enveloping algebra is referred to as special. The famous Poincaré-Birkhoff-Witt theorem states that any Lie algebra which is free as a module (and therefore, any Lie algebra over a field) is special. When the Lie algebra is Abelian, then its universal enveloping algebra reduces to the symmetric algebra of its underlying module structure, and thus any commutative Lie algebra is trivially special. However, there is another way to associate an associative algebra to a Lie algebra, and reciprocally, in a functorial way. The idea does not consist anymore to consider noncommutative algebras under commutators but differential commutative algebras together with the so-called Wronskian determinant. A derivation of an algebra is a linear map that satisfies the usual Leibniz rule. An algebra with a distinguished derivation is said to be a differential algebra. For any such pair may be defined a bilinear map by . When the algebra is commutative, then is alternating and satisfies the Jacobi identity so that it