Jyotishman Bhowmick

Quantum isometry groups of 0-dimensional manifolds

Quantum isometry groups of spectral triples associated with approximately finite-dimensional C*-algebras are shown to arise as inductive limits of quantum symmetry groups of corresponding truncated Bratteli diagrams. This is used to determine explicitly the quantum isometry group of the natural spectral triple on the algebra of continuous functions on the middlethird Cantor set. It is also shown that the quantum symmetry groups of finite graphs or metric spaces coincide with the quantum isometry groups of the corresponding classical objects equipped with natural Laplacians.

Quantum isometry groups of noncommutative manifolds associated to group C∗-algebras

Abstract Let Γ be a finitely generated discrete group. The standard spectral triple on the group C ∗ -algebra C ∗ ( Γ ) is shown to admit the quantum group of orientation preserving isometries. This leads to new examples of compact quantum groups. In particular, the quantum isometry group of the C ∗ -algebra of the free group on n generators is computed and turns out to be a quantum group extension of the quantum permutation group A 2 n of Wang. The quantum groups of orientation and real structure preserving isometries are also considered and the construction of the Laplacian for the standard spectral triple on C ∗ ( Γ ) is discussed.

Quantum isometry group of the n-tori

We show that the quantum isometry group (introduced by Goswami) of the n-tori T n coincides with its classical isometry group; i.e. there does not exist any faithful isometric action on T n by a genuine (non-commutative as a C*-algebra) compact quantum group. Moreover, using an earlier result, we conclude that the quantum isometry group of the noncommutative n tori is a Rieffel deformation of the quantum isometry group of the commutative n-tori.

Quantum isometry groups

In the memory of my grandmother Acknowledgements I would like to start by expressing my deepest gratitudes to my supervisor Debashish Goswami who introduced me to the theories of noncommutative geometry and compact quantum groups and whose mere presence acted as a psychological support to an extent unknown even to him. I thank each and every faculty member of Stat Math unit, ISI Kolkata from each of whom I learnt something or the other. I am grateful to Prof. Shuzhou Wang, whose valuable comments and suggestions helped me to have a better understanding about the contents of this thesis and also led to an improvement of the work. I thank the National Board for Higher Mathematics, India for providing me with partial financial support. I should also mention the names of Max Planck Institute fur Mathematics of Bonn and Chern Institute of mathematics of Nankai University for allowing me to attend two workshop and conference on Non commutative Geometry and Quantum Groups hosted by them, from where I had valuable exposures about the subjects. I thank my parents, sister, aunt and uncle for their continuous support during the time of working on this thesis. I would have been unable to fix some critical latex problem unless my friend Rajat Subhra Hazra had spent his valuable time on it. I am grateful to Biswarup for the discussions I had with him on operator algebras. Many of my other friends including Koushik Saha, Abhijit da, Pusti da, Ashis da, Subhra, Subhajit and obviously Rajat were a continuous source of encouragement. I thank Subhajit for allowing me to use his room and providing me with a steady supply of 13 Tzameti et al. Lastly, I would like to mention the names of Subhra and Rajat again for bearing with my bhoyonkor and haayre during a critical part of my thesis work.

Some Counterexamples in the Theory of Quantum Isometry Groups

By considering spectral triples on

Quantum Isometry Groups of Discrete Quantum Spaces

{S^{2}_{\mu, c}\,\, (c >0 )}

Classical and Noncommutative Geometry

constructed by Chakraborty and Pal (Commun Math Phys 240(3):447–456, 2000), we show that in general the quantum group of volume and orientation preserving isometries (in the sense of Bhowmick and Goswami in J Funct Anal 257:2530–2572, 2009) for a spectral triple of compact type may not have a C*-action, and moreover, it can fail to be a matrix quantum group. It is also proved that the category with objects consisting of those volume and orientation preserving quantum isometries which induce C*-action on the C* algebra underlying the given spectral triple, may not have a universal object.

More Examples and Open Questions

We consider the construction of twisted tensor products in the category of C*-algebras equipped with orthogonal filtrations and under certain assumptions on the form of the twist compute the corresponding quantum symmetry group, which turns out to be the generalized Drinfeld double of the quantum symmetry groups of the original filtrations. We show how these results apply to a wide class of crossed products of C*-algebras by actions of discrete groups. We also discuss an example where the hypothesis of our main theorem is not satisfied and the quantum symmetry group is not a generalized Drinfeld double.