An invariant for homogeneous spaces of compact quantum groups

TL;DR

This paper introduces a new dimensional invariant for homogeneous spaces of compact quantum groups using spectral triples, providing a tool to classify and analyze quantum groups with explicit computations for several cases.

Contribution

It constructs a novel invariant for homogeneous spaces of compact quantum groups based on spectral triples, including explicit calculations for type A quantum groups.

Findings

Invariant computed for all type A quantum groups

Provides a classification tool for quantum groups

Demonstrates the invariant's effectiveness in various cases

Abstract

The central notion in Connes' formulation of non commutative geometry is that of a spectral triple. Given a homogeneous space of a compact quantum group, restricting our attention to all spectral triples that are `well behaved' with respect to the group action, we construct a certain dimensional invariant. In particular, taking the (quantum) group itself as the homogeneous space, this gives an invariant for a compact quantum group. Computations of this invariant in several cases, including all type A quantum groups, are given.