Homological dimensions of algebras of analytic functionals and their completions

TL;DR

This paper investigates the homological dimensions of algebras of analytic functionals on complex Lie groups, revealing they depend only on the solvable part of the group's structure, not on reductive factors.

Contribution

It establishes that the homological dimensions of these algebras match the dimension of the solvable component in the group's decomposition, regardless of reductive factors.

Findings

Homological dimensions equal the dimension of the solvable factor.

Reductive factors do not influence the homological properties.

Results apply to the algebra and some of its completions.

Abstract

We show that the main homological dimensions of the algebra of analytic functionals on a connected complex Lie group, as well as some of its completions, coincide with the dimension of the simply connected solvable factor in the canonical decomposition of the linearization of this group. Thus, the possible nontriviality of a linearly complex reductive factor does not affect the homological properties of the algebras under consideration.