TITLE: Explicit isomorphism between distributions and universal enveloping algebra
QUESTION [7 upvotes]: The universal enveloping algebra of a Lie algebra $\mathfrak{g}$ is isomorphic to the algebra of distributions on the Lie group $G$ with support at the identity. The proof I have of this fact uses the universal property of the universal enveloping algebra and hence it is not constructive. I was wondering if there is an explicit map? For example what would happen for $\mathfrak{g} = \mathfrak{sl}(2, \mathbf{C})$ and $G = SL(2, \mathbf{C})$? For an explicit monomial in $\mathscr{U} \ \left ( \mathfrak{sl}(2, \mathbf{C}) \right )$ can we write the corresponding distribution and how it acts on functions on $SL(2, \mathbf{C})$?

REPLY [3 votes]: It seems to me the question is not fully correct because  there is NO canonical identification of U(g) and distribitions on G (sup. at e).
In Victor's answer you can take action LEFT invariant vec. fields and you can take RIGHT - the answer would be different.
It seems if you take "left" than U(g) will act as left invariant differential operator,
if you take "right" it will act as "right" invariant differential operator.
It seems to me that it is general fact which should be checked on R^n - take any differential operator D, consider distribution d= D\delta, then action of this distribution by convolution is exactly the same action of D.
The proof seems obvious - just integration by parts
I have not checked all this, hopefully it is nevertheless true :)