Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 6 (2010), 007, 7 pages      arXiv:0811.3066      https://doi.org/10.3842/SIGMA.2010.007
Contribution to the Special Issue “Noncommutative Spaces and Fields”

Quantum Isometry Group for Spectral Triples with Real Structure

Debashish Goswami
Stat-Math Unit, Indian Statistical Institute, 203, B. T. Road, Kolkata 700108, India

Received November 06, 2009, in final form January 17, 2010; Published online January 20, 2010

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].

Key words: quantum isometry groups; spectral triples; real structures.

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