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 AL.
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 AL 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.
Read more about Universal Enveloping Algebra: Motivation, Universal Property, Direct Construction, Examples in Particular Cases, Further Description of Structure
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