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Jyotishman Bhowmick

Publications and source records attributed to Jyotishman Bhowmick.

27 records · Page 2Linked to original sources
arXiv2010

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

Let G be a finitely generated discrete group. The standard spectral triple on the group C*-algebra C*(G) 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_{2n} of Wang. The quantum groups of orientation and real structure preserving isometries are also considered and construction of the Laplacian for the standard spectral triple on C*(G) discussed.

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arXiv2010

Quantum Isometry groups of the Podles Spheres

For $μ\in (0,1), c> 0,$ we identify the quantum group $SO_μ(3)$ as the universal object in the category of compact quantum groups acting by `orientation and volume preserving isometries' in the sense of \cite{goswami2} on the natural spectral triple on the Podles sphere $S^2_{μ, c}$ constructed by Dabrowski, D'Andrea, Landi and Wagner in \cite{Dabrowski_et_al}.

math.OA↗
arXiv2009

Some counterexamples in the theory of quantum isometry groups

By considering spectral triples on $S^{2}_{μ, c}$ ($c>0$) constructed by Chakraborty and Pal (\cite{chak_pal}), we show that in general the quantum group of volume and orientation preserving isometries (in the sense of \cite{goswami2}) 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.

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arXiv2009

Quantum symmetries of classical spaces

We give a general scheme for constructing faithful actions of genuine (noncommutative as $C^*$ algebra) compact quantum groups on classical topological spaces. Using this, we show that: (i) a compact connected classical space can have a faithful action by a genuine compact quantum group, and (ii) there exists a spectral triple on a classical connected compact space for which the quantum group of orientation and volume preserving isometries (in the sense of \cite{qorient}) is a genuine quantum group.

math.QA↗
arXiv2009

Quantum Isometry Group of the n tori

We show that the Quantum Isometry Group(as introduced in \cite{goswami}) of the n tori is the classical isometry group. Moreover, using a result in \cite{bhowmick goswami}, 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.

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arXiv2009

Quantum Isometry Groups: Examples and Computations

In this follow-up of the article: Quantum Group of Isometries in Classical and Noncommutative Geometry(arXiv:0704.0041) by Goswami, where quantum isometry group of a noncommutative manifold has been defined, we explicitly compute such quantum groups for a number of classical as well as noncommutative manifolds including the spheres and the tori. It is also proved that the quantum isometry group of an isospectral deformation of a (classical or noncommutative) manifold is a suitable deformation of the quantum isometry group of the original (undeformed) manifold.

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arXiv2009

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.

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arXiv2008

Quantum Group of Orientation preserving Riemannian Isometries

We formulate a quantum group analogue of the group of orinetation-preserving Riemannian isometries of a compact Riemannian spin manifold, more generally, of a (possibly $R$-twisted in the sense of a paper of one of the authors, and of compact type) spectral triple. The main advantage of this formulation, which is directly in terms of the Dirac operator, is that it does not need the existence of any `good ' Laplacian as in our previous works on quantum isometry groups. Several interesting examples, including those coming from Rieffel-type deformation as well as the equivariant spectral triples on $SU_μ(2)$ and $S^2_{μ0}$ are dicussed.

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