Given a complex Lie algebra $\mathfrak g$, we can form its universal enveloping algebra and interpret it as a noncommutative space.
Is this perspective useful? What does this space "look like"?
How is it related to the space $\mathbb C[G]$ where $G$ is an algebraic group with Lie algebra $\mathfrak g$?
How is it related to the flag variety (in the case of a semisimple Lie algebra)?
This question has also an incarnation which asks about
group algebras.
Best Answer
You can think of $Ug$ as the algebra of distributions on $G$ supported at 1. Alternatively, you can think of $Ug$ as differential operators, for instance as global sections of monodromic differential operators on the flag variety.
"Useful" is very subjective. IMHO, differential operator interpretations are useful as it brings new insights into $Ug$ via Beilinson-Bernstein Localization. In other interpretations, info flows in the opposite direction as $Ug$ helps to understand Poisson geometry of $g*$ or geometry of $G$.