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.
Highlights
Any algebra, say A, admits a derived structure of Lie algebra under the commutator bracket [a, b] = ab − ba, a, b ∈ A
When the Lie algebra is Abelian, its universal enveloping algebra reduces to the symmetric algebra of its underlying module structure, and any commutative Lie algebra is trivially special
As in the classical case, this functor admits a left adjoint that allows us to define a Wronskian envelope for a Lie algebra, that is, a universal commutative and differential algebra
Summary
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. When the Lie algebra is Abelian, its universal enveloping algebra reduces to the symmetric algebra of its underlying module structure, and any commutative Lie algebra is trivially special. There is another way to associate an associative algebra to a Lie algebra, and reciprocally, in a functorial way. As in the classical case, this functor admits a left adjoint that allows us to define a Wronskian envelope for a Lie algebra, that is, a universal commutative and differential algebra. Special Lie algebras, in this new setting, satisfy a nontrivial relation similar to a relation that holds in Lie algebras of vector fields
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