Editing Universal enveloping algebra (section)

==Left-invariant differential operators==
Suppose <math>G</math> is a real Lie group with Lie algebra <math>\mathfrak{g}</math>. Following the modern approach, we may identify <math>\mathfrak{g}</math> with the space of left-invariant vector fields (i.e., first-order left-invariant differential operators). Specifically, if we initially think of <math>\mathfrak{g}</math> as the tangent space to <math>G</math> at the identity, then each vector in <math>\mathfrak{g}</math> has a unique left-invariant extension. We then identify the vector in the tangent space with the associated left-invariant vector field. Now, the commutator (as differential operators) of two left-invariant vector fields is again a vector field and again left-invariant. We can then define the bracket operation on <math>\mathfrak{g}</math> as the commutator on the associated left-invariant vector fields.<ref>E.g. {{harvnb|Helgason|2001}} Chapter II, Section 1</ref> This definition agrees with any other standard definition of the bracket structure on the Lie algebra of a Lie group.

We may then consider left-invariant differential operators of arbitrary order. Every such operator <math>A</math> can be expressed (non-uniquely) as a linear combination of products of left-invariant vector fields. The collection of all left-invariant differential operators on <math>G</math> forms an algebra, denoted <math>D(G)</math>. It can be shown that <math>D(G)</math> is isomorphic to the universal enveloping algebra <math>U(\mathfrak{g})</math>.<ref>{{harvnb|Helgason|2001}} Chapter II, Proposition 1.9</ref>

In the case that <math>\mathfrak{g}</math> arises as the Lie algebra of a real Lie group, one can use left-invariant differential operators to give an analytic proof of the [[Poincaré–Birkhoff–Witt theorem]]. Specifically, the algebra <math>D(G)</math> of left-invariant differential operators is generated by elements (the left-invariant vector fields) that satisfy the commutation relations of <math>\mathfrak{g}</math>. Thus, by the universal property of the enveloping algebra, <math>D(G)</math> is a quotient of <math>U(\mathfrak{g})</math>. Thus, if the PBW basis elements are linearly independent in <math>D(G)</math>—which one can establish analytically—they must certainly be linearly independent in <math>U(\mathfrak{g})</math>. (And, at this point, the isomorphism of <math>D(G)</math> with <math>U(\mathfrak{g})</math> is apparent.)