Universal enveloping algebra

In mathematics, the universal enveloping algebra construction of abstract algebra is applied to a Lie algebra L in order to pass from a non-associative structure to a more familiar and associative algebra over a field U(L) while preserving the representation theory. That is, L is to be made into an algebra over the same field K, in such a way that L's representations become modules over U(L) in the usual sense.

Table of contents

1 General construction 2 Examples in particular cases 3 Further description of structure

General construction

Noting that any associative K-algebra becomes a Lie algebra with the bracket [a,b] = a.b-b.a, a construction and precise characterisation suggests itself on general grounds (as adjoint functors). Starting with the tensor algebra T(L) on the vector space underlying L, we should make U(L) be the quotient of T(L) made by imposing the relations like a'b-b'a = [a,b] for a and b in (the image in T(L)) of L, where on the RHS the bracket now means the given Lie algebra product. This means that U(L) can be said to exist, and computations in it carried out.

Examples in particular cases

In the case that L is abelian (that is, the bracket is always 0) this gives U(L) the polynomial algebra of L as vector space. In other cases it is interesting to identify it via a Lie group G giving rise to L. In that setting U(g) can be identified with the algebra of left-invariant differential operators on G; with L lying inside it as the left-invariant vector fields as first-order differential operators. For example if L is a vector space V as abelian Lie algebra, the left-invariant differential operators are the constant coefficient operators, which are indeed a polynomial algebra in the partial derivatives of first order.

Another characterisation in Lie group theory is of U(L) as the convolution algebra of distributions supported only at the identity element e. The center of U(L) is called Z(L) and consists of the left- and right- invariant operators; this in the case of G not commutative will not be generated by first-order operators (see for example Casimir operator).

The algebra of differential operators in n variables with polynomial coefficients may be obtained starting with the Lie algebra of the Heisenberg group.

Further description of structure

Since L acts on itself (by the Lie algebra adjoint representation), this means that L has a linear representation on U(L). We may see this as follows: insist that L acts as a derivation algebra on U(L), noting that it does so on T(L), and respects the imposed relations. (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 any element of L acting on them gives zero) are called invariant elements. They are generated by the Casimir invariants.

This algebra can also be turned into a Hopf algebra.