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Concept
Universal enveloping algebra
Summary
In mathematics, the universal enveloping algebra of a Lie algebra is the unital associative algebra whose representations correspond precisely to the representations of that Lie algebra.
Universal enveloping algebras are used in the representation theory of Lie groups and Lie algebras. For example, Verma modules can be constructed as quotients of the universal enveloping algebra. In addition, the enveloping algebra gives a precise definition for the Casimir operators. 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 provides a commutative example of the objects studied in non-commutative geometry, the quantum groups. This dual can be shown, by the Gelfand–Naimark theorem, to contain the C* algebra of the corresponding Lie group. This relationship generalizes to the idea of Tannaka–Krein duality between compact topological groups 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 left-invariant differential operators on the group.
The idea of the universal enveloping algebra is to embed a Lie algebra into an associative algebra with identity in such a way that the abstract bracket operation in corresponds to the commutator in and the algebra is generated by the elements of . There may be many ways to make such an embedding, but there is a unique "largest" such , called the universal enveloping algebra of .
Let be a Lie algebra, assumed finite-dimensional for simplicity, with basis . Let be the structure constants for this basis, so that
Then the universal enveloping algebra is the associative algebra (with identity) generated by elements subject to the relations
and no other relations.
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