Universal Enveloping Algebra

Poincaré–Birkhoff–Witt Theorem

Universal Enveloping Algebra

A universal enveloping algebra of a Lie algebra is a pair where is a Lie algebra that is also a unital associative algebra with the standard Lie bracket (i.e. commutator of multiplication), and is a Lie algebra homomorphism. It satisfies the following universal property: for any associative algebra with the standard Lie bracket and any Lie algebra homomorphism , there exists a unique associative algebra homomorphism such that the following diagram commutes: In other words, is the “most general” associative algebra that can map to via Lie algebra homomorphisms.

The existence and description of universal enveloping algebras is given by the Poincaré–Birkhoff–Witt theorem:

Poincaré–Birkhoff–Witt Theorem

Suppose is a Lie algebra over with a totally ordered basis . Then its universal enveloping algebra exists and has a basis consisting of the ordered monomials. That is, the set of all elements of the form for and integers is a basis of .

Corollary

The canonical map of a Lie algebra to its universal enveloping algebra is an injection.

Modules over a Lie Algebra

Now, every Lie algebra has a universal enveloping algebra, which is in particular a ring. So we can talk about modules over it:

Module over a Lie Algebra

Suppose is a Lie algebra over a field . A module over is a module over its universal enveloping algebra . Specifically, it is an -vector space together with a scalar multiplication , satisfying:

• ;
• ;
• ,

for all , and .

Modules over a Lie algebra are Exactly Representations

A representation of a Lie algebra is a homomorphism for some vector space . This is equivalent to giving the structure of a module over by defining for and . Therefore, we shall often use the terms “module over a Lie algebra” and “representation of a Lie algebra” interchangeably.