Equivariant isomorphism of Quantum Lens Spaces of low dimension

TL;DR

This paper classifies low-dimensional quantum lens spaces up to equivariant isomorphism using algebraic invariants, solving specific cases completely and providing insights into the underlying structure.

Contribution

It introduces the problem of equivariant isomorphism classification for quantum lens spaces and solves it for dimensions 3 and 5 in certain cases, expanding understanding of their structure.

Findings

Complete classification for dimension 3 quantum lens spaces.

Partial classification for dimension 5 in the prime case.

Identification of complex parameter dependencies indicating deep underlying structures.

Abstract

The quantum lens spaces form a natural and well-studied class of noncommutative spaces which can be subjected to classification using algebraic invariants by drawing on the fully developed classification theory of unital graph $C^*$-algebras. We introduce the problem of deciding when two quantum lens spaces are equivariantly isomorphic, and solve it in certain basic cases. As opposed to classification up to isomorphism, we can not appeal to a complete general classification theory in the equivariant case, but by combining existing partial results with an ad hoc analysis we can solve the case with dimension 3 completely, and the case with dimension 5 in the prime case. Our results can be formulated directly in terms of the parameters defining the quantum lens spaces, and here occasionally take on a rather complicated form which convinces us that there is a deep underlying explanation for our findings. We complement the fully established partial results with computer experiments that may indicate the way forward.