Universal Enveloping AlgebraEdit

Universal enveloping algebras sit at the crossroads of Lie theory and representation theory, providing a bridge between Lie algebras and associative algebras. Given a Lie algebra Lie algebra g over a field k, its universal enveloping algebra U(g) is the most general associative algebra that contains a copy of g and turns Lie brackets into commutators. Formally, U(g) is constructed as the quotient of the tensor algebra T(g) by the ideal generated by elements x⊗y − y⊗x − [x,y], where x,y ∈ g and [x,y] denotes the Lie bracket.

The universal property is central: for any representation ρ of g on a vector space V, there is a unique algebra homomorphism U(g) → End(V) that extends ρ. In this sense, studying modules over U(g) is a way to study representations of the Lie algebra g. This perspective makes U(g) a fundamental object in representation theory and its connections to geometry and physics. See also the category O for a special class of U(g)-modules that captures many important representation-theoretic features.

U(g) comes equipped with a natural filtration by degree, with the associated graded algebra isomorphic to the symmetric algebra S(g). This is encapsulated in the Poincaré–Birkhoff–Witt theorem, a cornerstone result that ensures a concrete, combinatorial handle on U(g). In particular, if {x1, x2, ..., xn} is a basis of g, then monomials in these generators, taken in a fixed order, give a basis of U(g) as a vector space. See Poincaré–Birkhoff–Witt theorem for the precise statement and the standard PBW basis construction.

Construction

Start with the tensor algebra T(g) = ⊕_{m≥0} g^{⊗m}, which is the free associative algebra generated by g.
Impose the Lie bracket relations by quotienting out the two-sided ideal I generated by elements x⊗y − y⊗x − [x,y].
Define U(g) = T(g)/I. The inclusion x ↦ x (mod I) identifies g with a Lie subalgebra of the underlying associative algebra, and the Lie bracket is recovered as the commutator [x,y] = xy − yx in U(g).
The universal property states that for any associative algebra A and any Lie algebra homomorphism ρ: g → A, there exists a unique algebra homomorphism φ: U(g) → A with φ∘i = ρ, where i: g → U(g) is the natural inclusion.

In the simplest nontrivial example, take g = sl2 with basis {e, f, h} and the standard relations [h,e] = 2e, [h,f] = −2f, [e,f] = h. Then U(g) is generated by e, f, h subject to those relations, and the PBW theorem guarantees a basis consisting of ordered monomials in h, e, f.

Representations and modules

A g-module is precisely the same data as a U(g)-module, with the action of g extended coherently to U(g) via the universal property. This correspondence means that many questions about representations of g can be studied within the framework of module theory over an associative algebra.
For finite-dimensional representations of semisimple g, the representation theory of g is reflected in the structure of U(g), including the action of central elements and the decomposition into weight spaces.
Central elements of U(g) (the center Z(U(g))) act as scalars on irreducible representations of a finite-dimensional semisimple g, and the Harish-Chandra isomorphism describes Z(U(g)) in terms of invariant polynomials on g. See Harish-Chandra for foundational work in this area.
The category of U(g)-modules often appears alongside geometric and categorical constructs such as D-modules and the category O. These connections illuminate how Lie theory interacts with algebraic geometry and representation theory.

Examples and special cases

For g = sl2, the finite-dimensional irreducible representations are classified by highest weight, and the action of U(g) on these modules encodes the familiar ladder structure of weight spaces. The center of U(g) in this case is generated by the Casimir element, which acts as a scalar on each irreducible module.
For g = gl_n or sl_n, U(g) provides a language for polynomial representations of the corresponding matrix Lie algebras. The interplay between U(g) and symmetric functions, Schur functors, and polynomial representations is a rich subject linking algebraic combinatorics with Lie theory.
In physics, U(g) serves as the algebra of observables for systems with symmetry g, with representations describing possible quantum states. The link to physics is deepened by deformations known as quantum groups, which arise from deforming the relations in U(g) and play a central role in integrable models and quantum geometry.

Center, invariants, and geometry

For semisimple g, the center Z(U(g)) is large enough to capture important invariants of representations. The Harish-Chandra isomorphism identifies Z(U(g)) with the invariants of the symmetric algebra S(g) under the adjoint action, tying algebraic structure to geometric invariant theory.
The associated graded picture provided by the PBW theorem connects U(g) to the commutative world of S(g), offering a bridge to geometric objects like the dual space g^* and varieties defined by invariants. See also symmetric algebra and invariant theory.

Variants and related constructions

In characteristic p > 0, one studies restricted enveloping algebras and related finite-dimensional quotients of U(g), which reflect features of the representation theory in modular settings.
Completions and topological versions of the enveloping algebra arise in analysis and mathematical physics, where one needs completed algebras to handle infinite-dimensional representations and formal power series.
The deformation perspective leads to quantum groups, which can be viewed as deformations of U(g) that depend on a parameter q and recover U(g) in the limit as q → 1. This perspective connects to integrable systems and knot theory.

History and further development

The construction of the universal enveloping algebra and the PBW property were developed in the early 20th century, culminating in the Poincaré–Birkhoff–Witt theorem, which formalizes how the associative structure encodes the Lie algebra. Over the following decades, the study of U(g) deepened through the work of Dixmier and others, linking enveloping algebras to representation theory, algebraic geometry, and mathematical physics.
The conceptual viewpoint—viewing representations of a Lie algebra as modules over a universal object—became a standard tool across many areas of mathematics, enabling a wide range of applications and generalizations.