Universal enveloping algebra

In mathematics, a universal enveloping algebra is the most general (unital, associative) algebra that contains all representations of a Lie algebra.

Universal enveloping algebras play a relatively minor role in the representation theory of Lie groups; the greatest utility is perhaps to give 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 which 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.

An intuitive idea of the algebra can be obtained as follows: imagine the space of all polynomials in one variable x. For some given polynomial p(x), substitute elements ${\displaystyle g\in {\mathfrak {g}}}$ of the Lie algebra ${\displaystyle {\mathfrak {g}}}$, to obtain the formal power series p(g). Next, apply any and all commutation relations appropriate for that Lie algebra, to identify as equal any other polynomials that arise. In this way, one obtains the universal enveloping algebra ${\displaystyle U({\mathfrak {g}})}$ as the space of all such formal power series. To get a better intuitive idea of the meaning of such a formal power series, suppose that one had a matrix representation ${\displaystyle R({\mathfrak {g}})}$ of ${\displaystyle {\mathfrak {g}}}$, which associates an ordinary matrix R(g) with each element ${\displaystyle g\in {\mathfrak {g}}}$. Substituting the matrix into the formal power series, it becomes well-defined, since the (ordinary) multiplication of matricies is well-defined. One gets some value simply by performing the needed matrix multiplications. This informal construction makes clear where the name "universal enveloping algebra" comes from: one has the glimmer that every Lie group G that one might ever obtain from ${\displaystyle {\mathfrak {g}}}$ is contained inside of it. This should be obvious, if one imagines that every Lie group corresponds to some matrix representation of ${\displaystyle {\mathfrak {g}}}$. It should certainly be clear that every representation of ${\displaystyle {\mathfrak {g}}}$ will be contained in the universal enveloping algebra.

...
Wikipedia