symmetric monoidal (∞,1)-category of spectra
Given a Lie algebra internal to some symmetric monoidal -linear category , an enveloping monoid (or enveloping algebra) of in is any morphism of Lie algebras in where is a monoid (= algebra) in , and is the underlying object of equipped with the Lie bracket . In further we will just write for . A morphism of enveloping algebras is a morphism of monoids completing a commutative triangle of morphisms in , i.e. . With an obvious composition of morphisms, the enveloping algebras of form a category. A universal enveloping algebra of in is any universal initial object in the category of enveloping algebras of ; it is of course unique up to an isomorphism if it exists. If it exists for all Lie algebras in , then the rule can be extended to a functor which is the left adjoint to the functor defined above and the morphism is the unit of the adjunction.
In the more general context of higher algebra there is a notion of universal enveloping E-n algebra of an L-infinity algebra for all which generalizes the notion of universal associative algebra envelope of a Lie algebra. See at universal enveloping E-n algebra.
The existence of the universal enveloping algebra is easy in many concrete symmetric monoidal categories, e.g. the symmetric monoidal category of bounded chain complexes (giving the universal enveloping dg-algebra of a dg-Lie algebra), but not true in general.
First of all if admits countable coproducts, form the tensor algebra on the object ; this is a monoid in . In most standard cases, one can also form the smallest 2-sided ideal (i.e. -subbimodule) in monoid among those ideals whose inclusion into is factorizing the map ; if the coequalizers exist in then we can form the quotient object and there is an induced monoid structure in it. Under mild conditions on , the natural morphism is an universal enveloping monoid of in . If is an abelian tensor category and some flatness conditions on the tensor product are satisfied, then the enveloping monoid is a monic morphism in and .
The isomorphism problem for enveloping algebras is about the fact that the universal enveloping monoids of two Lie algebras of are isomorphic as associative monoids in , but this does not imply that the Lie algebras are isomorphic. This is even not true in general for the Lie -algebras (in classical sense), even if is a field of characteristics zero. It is known however in that case that the dimension of the finite-dimensional Lie -algebra can be read off from its universal enveloping -algebra as its Gel’fand-Kirillov dimension .
Suppose the universal enveloping algebras of Lie algebras exist in a -linear symmetric monoidal category and the functorial choice realizing the above construction with tensor products is fixed. For example, this is true in the category of -modules where is a commutative ring. Then the projection where is the trivial Lie algebra induces the counit . The coproduct is induced by the diagonal map whereas the antipode . One checks that these morphisms make into a Hopf algebra in .
If the category is simply the vector spaces over a field , then for , after we identify with its image in , , i.e. the elements in are the primitive elements in .
The Poincaré–Birkhoff–Witt theorem states that the associated graded algebra of an enveloping algebra in characteristics zero is canonically isomorphic to a symmetric algebra , and is isomorphic to as a coalgebra, via the projection map .
The universal enveloping algebra of the tangent Lie algebra of a finite-dimensional Lie group over real or complex numbers is canonically isomorphic to the algebra of the left invariant differential operators on .