Jyotishman Bhowmick
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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This thesis contains the formulation and computation of quantum isometry groups.
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.
We give a new sufficient condition on a spectral triple to ensure that the quantum group of orientation and volume preserving isometries defined in \cite{qorient} has a $C^*$-action on the underlying $C^*$ algebra.
Quantum isometry groups of log-Laplacians on Cuntz--Krieger algebras
研究Cuntz-Krieger代数的量子等距群，结合拓扑马尔可夫链的Ahlfors正则结构，提出新的紧量子群家族，展现图的量子自同构群与易量子群的混合特性，并构造了Cantor空间上的量子 ergodic 行动。
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*…
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}.
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 gro…
We identify the quantum isometry groups of spectral triples built on the symmetric groups with length functions arising from the nearest-neighbor transpositions as generators. It turns out that they are isomorphic to certain "doubling" of the group algebras of the respective symmetric groups. We discuss the doubling procedure in the context of regular multiplier Hopf algebras. In the last section we study the dependence of the isometry group of S_n on the choice of generators in the case n=3. We show that two different choices of generators lead to non-isomorphic quantum isometry groups which …
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.
We prove the existence of a quantum isometry groups for new classes of metric spaces: (i) geodesic metrics for compact connected Riemannian manifolds (possibly with boundary) and (ii) metric spaces admitting a uniformly distributed probability measure. In the former case it also follows from recent results of the second author that the quantum isometry group is classical, i.e. the commutative $C^*$-algebra of continuous functions on the Riemannian isometry group.
It is proved that the (volume and orientation-preserving) quantum isometry group of a spectral triple obtained by deformation by some dual unitary 2-cocycle is isomorphic with a similar twist-deformation of the quantum isometry group of the original (undeformed) spectral triple. This result generalizes similar work by Bhowmick and Goswami for Rieffel-deformed spectral triples in [Comm. Math. Phys. 285 (2009), 421-444, arXiv:0707.2648].
In this article we have shown that the quantum isometry group of $C_r^*(\mathbb{Z})$, denoted by $\mathbb{Q}(\mathbb{Z},S)$ as in \cite{gos_man}, with respect to a symmetric generating set $S$ does not depend on the generating set $S$. Moreover, we have proved that the result is no longer true if the group $\mathbb{Z}$ is replaced by $\underbrace{\mathbb{Z} \times \mathbb{Z} \times\cdots \times \mathbb{Z}}_{n \ copies}$ for $n>1$.
In this paper, we propose a new procedure to deform spectral triples and their quantum isometry groups. The deformation data are a spectral triple $(\mathcal A,\mathcal H, D)$, a compact quantum group $\mathbb G$ acting algebraically and by orientation-preserving isometries on $(\mathcal A,\mathcal H,D)$ and a unitary fiber functor $ψ$ on $\mathbb G$. The deformation procedure is a genuine generalization of the cocycle deformation of Goswami and Joardar.
We study the quantum isometry groups of the noncommutative Riemannian manifolds associated to discrete group duals. The basic representation theory problem is to compute the law of the main character of the relevant quantum group, and our main result here is as follows: for the group Z_s^{*n}, with s>4 and n>1, half of the character follows the compound free Poisson law with respect to the measure $\underlineε$/2, where $ε$ is the uniform measure on the s-th roots of unity, and $ε\to\underlineε$ is the canonical projection map from complex to real measures. We discuss as well a number of techn…
We study quantum isometry groups, denoted by $\mathbb{Q}(Γ, S)$, of spectral triples on $C^*_r(Γ)$ for a finitely generated discrete group coming from the word-length metric with respect to a symmetric generating set $S$. We first prove a few general results about $\mathbb{Q}(Γ, S)$ including : \begin{itemize} \item For a group $Γ$ with polynomial growth property, the dual of $\mathbb{Q}(Γ, S)$ has polynomial growth property provided the action of $\mathbb{Q}(Γ,S)$ on $C^*_r(Γ)$ has full spectrum, \item $\mathbb{Q}(Γ, S) \cong QISO(\hatΓ, d)$ for any abelian $Γ$, where $d$ is a suitable metric…
We formulate a definition of isometric action of a compact quantum group (CQG) on a compact metric space, generalizing Banica's definition for finite metric spaces. For metric spaces $(X,d)$ which can be isometrically embedded in some Euclidean space, we prove the existence of a universal object in the category of the compact quantum groups acting isometrically on $(X,d)$. In fact, our existence theorem applies to a larger class, namely for any compact metric space $(X,d)$ which admits a one-to-one continuous map $f : X \raro \IR^n$ for some $n$ such that $d_0(f(x),f(y))=ϕ(d(x,y))$ (where $d_0…
As a contribution of the programme of Goswami and Mandal (2014), we carry out explicit computations of $\mathbb{Q}(Γ,S)$, the quantum isometry group of the canonical spectral triple on $C_{r}^{*}(Γ)$ coming from the word length function corresponding to a finite generating set S, for several interesting examples of $Γ$ not covered by the previous work Goswami and Mandal (2014). These include the braid group of 3 generators, $\mathbb{Z}_4^{*n}$ etc. Moreover, we give an alternative description of the quantum groups $H_s^{+}(n,0)$ and $K_n^{+}$ (studied in Banica and Skalski (2012), Banica and S…
Heat operators and isometry groups of Cuntz-Krieger algebras
本文利用谱非交换几何为拓扑马尔可夫链和Cuntz-Krieger代数构造热半群与谱三元组，穷尽奇K-同调，并显式计算Cuntz代数的热算子，进而刻画等距群并证明Voiculescu非交换拓扑熵在等距上消失。
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