Curved Noncommutative Tori as Leibniz Quantum Compact Metric Spaces

TL;DR

This paper demonstrates that curved noncommutative tori are Leibniz quantum compact metric spaces and establishes their continuous variation over a specific group of matrices, enriching the understanding of noncommutative geometric structures.

Contribution

It proves that curved noncommutative tori are Leibniz quantum compact metric spaces and shows their continuous dependence on a matrix group with a natural length function.

Findings

Curved noncommutative tori are Leibniz quantum compact metric spaces.

They form a continuous family over a group of invertible matrices.

The family varies continuously with respect to a natural length function.

Abstract

We prove that curved noncommutative tori, introduced by Dabrowski and Sitarz, are Leibniz quantum compact metric spaces and that they form a continuous family over the group of invertible matrices with entries in the commutant of the quantum tori in the regular representation, when this group is endowed with a natural length function.