Editing Universal enveloping algebra (section)

{{Short description|Concept in mathematics}}
{{for|the universal enveloping W* algebra of a C* algebra|Sherman–Takeda theorem}}
In [[mathematics]], the '''universal enveloping algebra''' of a [[Lie algebra]] is the [[unital algebra|unital]] [[associative algebra|associative]] algebra whose [[Algebra representation|representations]] correspond precisely to the [[representation of a Lie algebra|representations]] of that Lie algebra.

Universal enveloping algebras are used in the [[representation theory]] of Lie groups and Lie algebras. For example, [[Verma module]]s can be constructed as quotients of the universal enveloping algebra.<ref>{{harvnb|Hall|2015}} Section 9.5</ref> In addition, the enveloping algebra gives a precise definition for the [[Casimir operator]]s. Because Casimir operators commute with all elements of a Lie algebra, they can be used to classify representations. The precise definition also allows the importation of Casimir operators into other areas of mathematics, specifically, those that have a [[differential algebra]]. They also play a central role in some recent developments in mathematics. In particular, their [[dual vector space|dual]] provides a commutative example of the objects studied in [[non-commutative geometry]], the [[quantum group]]s. This dual can be shown, by the [[Gelfand–Naimark theorem]], to contain the [[C-star algebra|C* algebra]] of the corresponding Lie group. This relationship generalizes to the idea of [[Tannaka–Krein duality]] between [[compact topological group]]s and their representations.

From an analytic viewpoint, the universal enveloping algebra of the Lie algebra of a Lie group may be identified with the algebra of [[Invariant_differential_operator|left-invariant differential operators]] on the group.