Arens–Michael envelopes of nilpotent Lie algebras, holomorphic functions of exponential type and homological epimorphisms

Our aim is to give an explicit description of the Arens–Michael envelope for the universal enveloping algebra of a finite-dimensional nilpotent complex Lie algebra. It turns out that the Arens–Michael envelope belongs to a class of completions introduced by R. Goodman in 1970s. To find a precise form of this algebra we characterize preliminary the set of holomorphic functions of exponential type on a simply connected nilpotent complex Lie group. This approach leads to unexpected connections to Riemannian geometry and the theory of order and type for entire functions.
As a corollary, it is shown that the Arens–Michael envelope considered above is a homological epimorphism. So we get a positive answer to a question investigated earlier by Dosi and Pirkovskii. References: 36 entries.