Quantum Isometry Group for Spectral Triples with Real Structure

TL;DR

This paper establishes the existence of a universal quantum isometry group for spectral triples with real structure, extending the concept to include real structures without fixing a volume form, thus broadening the framework of quantum symmetries.

Contribution

It proves the existence of a universal quantum isometry group for spectral triples with real structure, generalizing previous definitions to include real structures without fixing a volume form.

Findings

Existence of a universal quantum isometry group for spectral triples with real structure.

Extension of quantum isometry group concept to real spectral triples.

Provides a natural definition of quantum isometry group in this context.

Abstract

Given a spectral triple of compact type with a real structure in the sense of [Dabrowski L., J. Geom. Phys. 56 (2006), 86-107] (which is a modification of Connes' original definition to accommodate examples coming from quantum group theory) and references therein, we prove that there is always a universal object in the category of compact quantum group acting by orientation preserving isometries (in the sense of [Bhowmick J., Goswami D., J. Funct. Anal. 257 (2009), 2530-2572]) and also preserving the real structure of the spectral triple. This gives a natural definition of quantum isometry group in the context of real spectral triples without fixing a choice of 'volume form' as in [Bhowmick J., Goswami D., J. Funct. Anal. 257 (2009), 2530-2572].