Homological dimensions of algebras of analytic functionals and their completions
Oleg Aristov
TL;DR
This work determines the main homological dimensions of algebras of analytic functionals on connected complex Lie groups and their Fréchet–Arens–Michael completions, tying them to the dimension of the solvable factor in the linearization: if $G/\Lambda = B\rtimes L$ with $B$ solvable and simply connected, then $\operatorname{dg}{\mathscr A}(G)=\operatorname{db}{\mathscr A}(G)=\dim B$, and the same holds for the completions ${\widehat{\mathscr A}}(G)$ and ${\widehat{\mathscr A}}(G)^{PI}$ as well as for the Arens–Michael completion $D={\mathscr A}_{\omega_{\max}^{\infty}}(G/\Lambda)$. The authors develop a framework based on analytic smash products, bar resolutions, and Tor functors in the topological setting, establishing a homological epimorphism ${\mathscr A}(G)\to D$ that preserves Tor and enabling a reduction to Lie-algebra cohomology $\mathrm{H}^{\mathrm{Lie}}_n(\mathfrak{b},M)$. This reduces the problem to the dimension of $B$ via nonvanishing cohomology at degree $\dim B$ and a corresponding length bound from $\bigwedge^\bullet \mathfrak{b}$. The results also extend to the PI and Fréchet-Arens–Michael envelopes, and the paper discusses the real-Lie-group analogue $\mathcal E'(G)$ as a parallel inquiry.
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.