Sobolev algebras on Lie groups and noncommutative geometry

TL;DR

This paper constructs quantum metric spaces from Sobolev algebras on Lie groups with subelliptic Laplacians, introduces associated noncommutative geometries, and relates spectral metrics to classical distances.

Contribution

It establishes the existence of quantum compact metric spaces for Sobolev algebras on Lie groups and introduces a canonical spectral triple linking noncommutative and classical geometries.

Findings

Quantum compact metric spaces exist for large enough p

Spectral dimension matches local dimension of the group

Spectral pseudo-metric recovers Carnot-Carathéodory distance

Abstract

We show that there exists a quantum compact metric space which underlies the setting of each Sobolev algebra associated to a subelliptic Laplacian $\Delta=-(X_1^2+\cdots+X_m^2)$ on a compact connected Lie group $G$ if $p$ is large enough, more precisely under the (sharp) condition $p > \frac{d}{\alpha}$ where $d$ is the local dimension of $(G,X)$ and where $0 < \alpha \leq 1$. We also provide locally compact variants of this result and generalizations for real second order subelliptic operators. We also introduce a compact spectral triple (=noncommutative manifold) canonically associated to each subelliptic Laplacian on a compact group. In addition, we show that its spectral dimension is equal to the local dimension of $(G,X)$. Finally, we prove that the Connes spectral pseudo-metric allows us to recover the Carnot-Carath\'eodory distance.