Universal enveloping algebras, analytic functionals, and homological epimorphisms

We discuss homological properties of the algebra $\mathcal A(G)$ of analytic functionals on a complex Lie group $G$. We show that
(1) If $G$ is linearly complex reductive (for example, semisimple) and connected, then $\mathcal A(G)$ is homologically trivial.
(2) For each connected $G$ the Arens-Mchael envelope of $\mathcal A(G)$ is a homological epimorphism.
As a corollary, the solvability of a finite-dimensional complex Lie algebra $\mathfrak g$ is sufficient for the Arens-Michael envelope of the enveloping algebra $U(\mathfrak g)$ to be a homological epimorphism. (Pirkovskii proved earlier that this condition is necessary.)