Universal Enveloping Algebra - Further Description of Structure

Further Description of Structure

The fundamental Poincaré–Birkhoff–Witt theorem gives a precise description of U(L); the most important consequence is that L can be viewed as a linear subspace of U(L). More precisely: the canonical map h : L → U(L) is always injective. Furthermore, U(L) is generated as a unital associative algebra by L.

L acts on itself by the Lie algebra adjoint representation, and this action can be extended to a representation of L on U(L): L acts as an algebra of derivations on T(L), and this action respects the imposed relations, so it actually acts on U(L). (This is the purely infinitesimal way of looking at the invariant differential operators mentioned above.)

Under this representation, the elements of U(L) invariant under the action of L (i.e. such that any element of L acting on them gives zero) are called invariant elements. They are generated by the Casimir invariants.

As mentioned above, the construction of universal enveloping algebras is part of a pair of adjoint functors. U is a functor from the category of Lie algebras over K to the category of unital associative K-algebras. This functor is left adjoint to the functor which maps an algebra A to the Lie algebra A_L. The universal enveloping algebra construction is not exactly inverse to the formation of A_L: if we start with an associative algebra A, then U(A_L) is not equal to A; it is much bigger.

The facts about representation theory mentioned earlier can be made precise as follows: the abelian category of all representations of L is isomorphic to the abelian category of all left modules over U(L).

The construction of the group algebra for a given group is in many ways analogous to constructing the universal enveloping algebra for a given Lie algebra. Both constructions are universal and translate representation theory into module theory. Furthermore, both group algebras and universal enveloping algebras carry natural comultiplications which turn them into Hopf algebras.

The center of the universal enveloping algebra of a simple Lie algebra is described by the Harish-Chandra isomorphism.