Differential (Lie) algebras from a functorial point of view

It is well-known that any associative algebra becomes a Lie algebra under the commutator bracket. This relation is actually functorial, and this functor, as any algebraic functor, is known to admit a left adjoint, namely the universal enveloping algebra of a Lie algebra. This correspondence may be lifted to the setting of differential (Lie) algebras. In this contribution it is shown that, also in the differential context, there is another, similar, but somewhat different, correspondence. Indeed any commutative differential algebra becomes a Lie algebra under the Wronskian bracket W(a,b)=ab′−a′bW(a,b)=ab′−a′b. It is proved that this correspondence again is functorial, and that it admits a left adjoint, namely the differential enveloping (commutative) algebra of a Lie algebra. Other standard functorial constructions, such as the tensor and symmetric algebras, are studied for algebras with a given derivation.