Wronskian Envelope of a Lie Algebra

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.

Prime Lie algebras satisfying the standard Lie identity of degree 5

“…Using result of Yu. Razmyslov, L. Poinsot proved (see [3,Th. 16]) that subalgebras of the direct product, where each component is either simple and satisfies St 5 or is abelian, are Wronskian special.…”

“…More precisely, the Lie algebra of special derivations of a commutative algebra always satisfy the standard Lie identity of degree 5. The problem of existence of such embedding is a long-standing problem (see [2,4,3]), which is closely related to the Lie algebra of vector fields on the affine line (see [2]). It was solved by Razmyslov in [2] for simple Lie algebras satisfying this identity (see also [3, Th.…”

“…The problem of existence of such embedding is a long-standing problem (see [2,4,3]), which is closely related to the Lie algebra of vector fields on the affine line (see [2]). It was solved by Razmyslov in [2] for simple Lie algebras satisfying this identity (see also [3, Th. 16]).…”

“…In his papers [3,4], L. Poinsot used a language of category theory to describe the correspondence between differential algebra and its Lie algebra of special derivations. It turns out that the correspondence is functorial.…”

“…This functor has left adjoint, so for every Lie algebra L we assign a differential algebra. Poinsot calls the Lie algebra of special derivations of this differential algebra the Wronskian envelope of L, and Theorem 9 in [3] (see also Proposition 49 in [4]) describes the Wronskian envelope of L in terms of generators and relations. Proposition 1 shows that the canonical homomorphism from L to its Wronskian envelope need not be injective (see also Example 50 in [4]).…”