Editing Universal enveloping algebra (section)

==Representation theory==
The universal enveloping algebra preserves the representation theory: the [[representation of a Lie algebra|representations]] of <math>\mathfrak{g}</math> correspond in a one-to-one manner to the [[module (mathematics)|module]]s over <math>U(\mathfrak{g})</math>. In more abstract terms, the [[abelian category]] of all [[representation of a Lie algebra|representations]] of <math>\mathfrak{g}</math> is [[isomorphism of categories|isomorphic]] to the abelian category of all left modules over <math>U(\mathfrak{g})</math>.

The representation theory of [[semisimple Lie algebra]]s rests on the observation that there is an isomorphism, known as the [[Kronecker coefficient|Kronecker product]]:
:<math>U(\mathfrak{g}_1\oplus\mathfrak{g}_2)\cong U(\mathfrak{g}_1)\otimes U(\mathfrak{g}_2)</math>
for Lie algebras <math>\mathfrak{g}_1, \mathfrak{g}_2</math>. The isomorphism follows from a lifting of the embedding
:<math>i(\mathfrak{g}_1 \oplus \mathfrak{g}_2)
=i_1(\mathfrak{g}_1)\otimes 1 \oplus 1\otimes i_2(\mathfrak{g}_2)</math>
where
:<math>i:\mathfrak{g}\to U(\mathfrak{g})</math>
is just the canonical embedding (with subscripts, respectively for algebras one and two). It is straightforward to verify that this embedding lifts, given the prescription above. See, however, the discussion of the bialgebra structure in the article on [[tensor algebra]]s for a review of some of the finer points of doing so: in particular, the [[shuffle product]] employed there corresponds to the Wigner-Racah coefficients, i.e. the [[6j-symbol|6j]] and [[9j-symbol]]s, etc.

Also important is that the universal enveloping algebra of a [[free Lie algebra]] is isomorphic to the [[free associative algebra]].

Construction of representations typically proceeds by building the [[Verma module]]s of the [[highest weight]]s.

In a typical context where <math>\mathfrak{g}</math> is acting by ''[[infinitesimal transformation]]s'', the elements of <math>U(\mathfrak{g})</math> act like [[differential operator]]s, of all orders. (See, for example, the realization of the universal enveloping algebra as left-invariant differential operators on the associated group, as discussed above.)