Universal Enveloping Algebra

In mathematics, for any Lie algebra L one can construct its universal enveloping algebra U(L). This construction passes from the non-associative structure L to a (more familiar, and possibly easier to handle) unital associative algebra which captures the important properties of L.

Any associative algebra A over the field K becomes a Lie algebra over K with the Lie bracket:

= ab − ba.

That is, from an associative product, one can construct a Lie bracket by taking the commutator with respect to that associative product. Denote this Lie algebra by A_L.

Construction of the universal enveloping algebra attempts to reverse this process: to a given Lie algebra L over K, find the "most general" unital associative K-algebra A such that the Lie algebra A_L contains L; this algebra A is U(L). The important constraint is to preserve the representation theory: the representations of L correspond in a one-to-one manner to the modules over U(L). In a typical context where L is acting by infinitesimal transformations, the elements of U(L) act like differential operators, of all orders.