Universal enveloping algebra

The universal enveloping algebra (also universal envelope ) is a term from the mathematical branch of the theory of Lie algebras. It is an associative algebra, which shows that we may look upon as the commutator Lieklammer always, even in Lie algebras that do not come from an associative algebra.

Definition

It is a Lie algebra ( over a field ). A universal enveloping algebra consists of a unitary associative algebra and Liealgebrenhomomorphismus ( The Liealgebrastruktur was added to associative algebra by the commutator ) so that:

Properties

• The most important statement of universal enveloping algebras is the set of Poincaré - Birkhoff -Witt ( after Henri Poincaré, Garrett Birkhoff and Ernst Witt, as PBW abbreviated ): Is a base of and the canonical map, so are the monomials

• In particular, is injective, and every Lie algebra is an associative algebra subalgebra.
• Moduli under a Lie algebra are the same as modules under its universal enveloping algebra.

Construction

You can specify the universal envelope explicitly as a quotient of the tensor algebra by the two-sided ideal, the elements of the form

Is generated for. Note: In contrast to the corresponding structures of the external algebra or symmetric algebra, this ideal is not homogeneous, ie bears no induced graduation.

Examples

• Is abelian, then the universal enveloping algebra is isomorphic to the symmetric algebra over.

• Theory of Lie algebras