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 of universal enveloping algebras can be shown using the tensor algebra construction and taking an appropriate quotient: Let be the two-sided ideal of the tensor algebra generated by elements of the form for all . Then the universal enveloping algebra of is given by the quotient algebra , together with the canonical map that sends each element of to its equivalence class in the quotient. Furthermore, a concrete description of universal enveloping algebras can be 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 some is a basis of . In other words, the associated graded algebra is isomorphic to the symmetric algebra .

Proof Sketch It is clear that the ordered monomials span , because any monomial that is not well ordered can be rewritten using the relations in the ideal to express it as a linear combination of well-ordered monomials. The non-trivial part is to show that these ordered monomials are linearly independent. We utilize the filtration of the tensor algebra by degree:Let be the inclusion map. Then annihilates the ideal , so it induces a well-defined map . Let us denote for each non-negative integer . For the induction hypothesis, assume we have constructed maps annihilating on for all such that their images are spanned by ordered monomials of degree at most . Then we define recursively by and where is the first index with . The goal is to show that annihilates . Instead of doing this in full rigor, we illustrate the idea when and annihilates . Suppose are basis vectors. Then we consider the worst case where the monomial is in completely reversed order. We have where the last equality follows from the Jacobi identity. The general case can be handled similarly by repeated application of the Jacobi identity. Consequently, induces a well-defined map whose image is spanned by ordered monomials of degree at most . By taking the direct limit as , we obtain a map annihilating , so induces . Since acts as the identity on ordered monomials, the subalgebra generated by ordered monomials intersects trivially. Thus, the ordered monomials are linearly independent in .

Remark

As stated in the theorem, equivalently, the PBW theorem can be formulated abstractly as the isomorphism . Let us verify this from the concrete version. Recall that the symmetric algebra is defined as the quotient of the tensor algebra by the ideal generated by elements of the form for all . That is, is the smallest commutative algebra containing , so the order of multiplication does not matter in , and is isomorphic to the subalgebra generated by ordered monomials. On the other hand, , so has a natural filtration induced from the degree filtration of : where is the image of under the quotient map . Then the associated graded algebra has components given by . Note that as has been quotiented out, the relation holds for all , so in , we have However, the last term lies in because the bracket can be expressed as a linear combination of basis elements of , so the entire product has degree at most . Therefore, in the quotient , we have which shows that multiplication in is commutative. Therefore, by the universal property of the symmetric algebra, there exists a unique graded algebra morphism extending the identity on . PBW in ordered-monomial form says the elements with form a basis of , and any word can be rewritten into an ordered word plus terms of strictly lower filtration degree. Passing to , the lower-degree terms vanish, so equals the symbol of the ordered word, showing spans . Moreover, if a nontrivial linear combination of symbols of ordered monomials of length vanished in , the corresponding linear combination in would lie in contradicting that ordered monomials of fixed length are part of a basis of . Hence those symbols are linearly independent, and is an isomorphism on each graded piece, so is an isomorphism of graded algebras.

Corollary

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

Proposition

Suppose and are Lie algebras over a field . Then we have the following canonical isomorphism:

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 . Almost equivalently, it is an -vector space together with a scalar multiplication , satisfying:

• ;
• ;
• ,

for all , and .

To make the “almost equivalence” precise, given a module over , we define the scalar multiplication of on by restricting the action of to . Conversely, given a vector space with a scalar multiplication of satisfying the above properties, we can uniquely extend this action to all of using the universal property, thereby making into a module over .

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.

The Hopf Algebra Structure

Proposition

The universal enveloping algebra of a Lie algebra carries a natural structure of a cocommutative Hopf algebra. Specifically, we define

• comultiplication by for all and ,
• counit by for all and ,
• antipode by for all and .

In particular, is precisely the Lie algebra generated by Lie-like elements, i.e., .

Proof We verify the axioms of a Hopf algebra one by one. First, the counitality laws hold because for any , Next, the coassociativity laws hold because for any , Finally, the antipode axiom holds because for any , Furthermore, the cocommutativity holds because for any , where is the twist map defined by .