The same machine, a different ring
A Lie algebra has a bracket rather than an associative product, so it is not a ring and its modules are not modules in the usual sense. The universal enveloping algebra U(g) fixes this: it is the associative algebra generated by g with the bracket realised as a commutator, and g-representations are exactly U(g)-modules. Lie algebra cohomology is then Ext over U(g), and everything proved for group cohomology transfers with the group ring replaced by the enveloping algebra.
Learning objectives
- Define a Lie algebra and a representation.
- State the universal property of the enveloping algebra.
- Quote the Poincaré–Birkhoff–Witt theorem and its consequences.
- Identify the augmentation and the trivial module.
Section 01Lie algebras and modules
A Lie algebra over a field K is a vector space with a bilinear bracket that is alternating and satisfies the Jacobi identity:
A g-module is a vector space A with a bilinear action satisfying x·(y·a) − y·(x·a) = [x, y]·a.
| Groups | Lie algebras |
|---|---|
| Group G | Lie algebra g |
| Group ring ℤ[G] | Universal enveloping algebra U(g) |
| Trivial module ℤ | Trivial module K, the ground field |
| Augmentation ideal IG | Augmentation ideal Ug |
| Hn(G, A) = Extnℤ[G](ℤ, A) | Hn(g, A) = ExtnU(g)(K, A) |
| Bar resolution | Chevalley–Eilenberg resolution |
| Semidirect product | Semidirect sum of Lie algebras |
Section 02The enveloping algebra
U(g) is the quotient of the tensor algebra on g by the relations xy − yx = [x, y]. Its universal property:
for any associative algebra A with its commutator bracket. So U is left adjoint to the functor turning an associative algebra into a Lie algebra — another adjunction doing structural work.
If x1, …, xn is an ordered basis of g, the ordered monomials in the xi form a basis of U(g). Two consequences matter here: g embeds in U(g), and U(g) is free as a module over the enveloping algebra of any subalgebra — which is what makes change-of-rings arguments work.
Section 03Augmentation and the trivial module
The augmentation ε: U(g) → K sends g to zero; its kernel Ug is the augmentation ideal, generated by g. As with groups, the fundamental sequence
drives dimension shifting, and Ug/(Ug)² ≅ g/[g, g], the abelianisation — giving H1(g, K) immediately.
U(g) is an algebra over a field, and for finite-dimensional g it has finite global dimension equal to dim g. Group rings over ℤ behave less well — a finite group ring in modular characteristic has infinite global dimension. So Lie algebra cohomology terminates where group cohomology often does not.
ReferenceFrequently asked questions
Is U(g) finite-dimensional?
Only when g = 0. By PBW it is a polynomial-sized algebra: for g of dimension n it has a basis of ordered monomials, so it is infinite-dimensional but Noetherian and of finite global dimension when g is finite-dimensional.
Why not work directly with the Lie algebra?
Because homological algebra requires an associative ring to define modules, resolutions and Ext. The enveloping algebra is the minimal associative ring whose modules are exactly the representations, so it is the natural home for the theory.
Does PBW hold in all characteristics?
Yes for Lie algebras over a field, and more generally when g is free as a module over the base ring. In characteristic p the representation theory diverges substantially from characteristic zero, and restricted Lie algebras become the relevant objects.
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