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Debashish Goswami

Publications and source records attributed to Debashish Goswami.

At least 37 records · Page 2Linked to original sources
arXiv2019

Levi-Civita connections for a class of spectral triples

We give a new definition of Levi-Civita connection for a noncommutative pseudo-Riemannian metric on a noncommutative manifold given by a spectral triple. We prove the existence-uniqueness result for a class of modules of one forms over a large class of noncommutative manifolds, including the matrix geometry of the fuzzy 3-sphere, the quantum Heisenberg manifolds and Connes-Landi deformations of spectral triples on the Connes-Dubois Violette-Rieffel-deformation of a compact manifold equipped with a free toral action. It is interesting to note that in the example of the quantum Heisenberg manifold, the definition of metric compatibility given in the paper by Frolich et al failed to ensure the existence of a unique Levi-Civita connection. In the case of the matrix geometry, the Levi-Civita connection that we get coincides with the unique real torsion-less unitary connection obtained by Frolich et al.

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arXiv2019

Hopf coactions on odd spheres

We prove that the q-deformed unitary group, i.e., $U_q(N)$, is the universal compact quantum group in the category of (compact) quantum groups which coact on the q-deformed odd sphere $S_q^{2N-1}$ leaving the space spanned by the natural set of generators invariant and preserving the unique $SU_q(N)$ invariant functional on $S_q^{2N-1}$. Using this, we identify $U_q(N)$ as the quantum group of orientation preserving isometries (in the sense of Bhowmick and Goswami \cite{MR2555012}) for a natural spectral triple associated with $S_q^{2N-1}$ constructed by Chakraborty and Pal \cite{MR2458039}.

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arXiv2018

Non-existence of genuine (compact) quantum symmetries of compact, connected smooth manifolds

Suppose that a compact quantum group ${\mathcal Q}$ acts faithfully on a smooth, compact, connected manifold $M$, i.e. has a $C^{\ast}$ (co)-action $α$ on $C(M)$, such that $α(C^\infty(M)) \subseteq C^\infty(M, {\mathcal Q})$ and the linear span of $α(C^\infty(M))(1 \otimes {\mathcal Q})$ is dense in $C^\infty(M, {\mathcal Q})$ with respect to the Frechet topology. It was conjectured by the author quite a few years ago that ${\mathcal Q}$ must be commutative as a $C^{\ast}$ algebra i.e. ${\mathcal Q} \cong C(G)$ for some compact group $G$ acting smoothly on $M$. The goal of this paper is to prove the truth of this conjecture. A remarkable aspect of the proof is the use of probabilistic techniques involving Brownian stopping time.

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arXiv2018

Non-existence of faithful isometric action of compact quantum groups on compact, connected Riemannian manifolds

Suppose that a compact quantum group $\clq$ acts faithfully on a smooth, compact, connected manifold $M$, i.e. has a $C^*$ (co)-action $α$ on $C(M)$, such that the action $α$ is isometric in the sense of \cite{Goswami} for some Riemannian structure on $M$. We prove that $\clq$ must be commutative as a $C^{\ast}$ algebra i.e. $\clq\cong C(G)$ for some compact group $G$ acting smoothly on $M$. In particular, the quantum isometry group of $M$ (in the sense of \cite{Goswami}) coincides with $C(ISO(M))$.

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arXiv2016

Quantum Isometry group of dual of finitely generated discrete groups and quantum groups

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 on the dual compact abelian group $\hatΓ$. \end{itemize} We then carry out explicit computations of $\mathbb{Q}(Γ,S)$ for several classes of examples including free and direct product of cyclic groups, Baumslag-Solitar group, Coxeter groups etc. In particular, we have computed quantum isometry groups of all finitely generated abelian groups which do not have factors of the form $\mathbb{Z}_2^k$ or $\mathbb{Z}_4^l$ for some $k,l$ in the direct product decomposition into cyclic subgroups.

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arXiv2016

Hopf coactions on commutative algebras generated by a quadratically independent comodule

Let A be a commutative unital algebra over an algebraically closed field k of characteristic not equal to 2, whose generators form a finite-dimensional subspace V, with no nontrivial homogeneous quadratic relations. Let Q be a Hopf algebra that coacts on A inner-faithfully, while leaving V invariant. We prove that Q must be commutative when either: (i) the coaction preserves a non-degenerate bilinear form on V; or (ii) Q is co-semisimple, finite-dimensional, and char(k)=0.

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arXiv2015

Faithful actions of locally compact quantum groups on classical spaces

It is well-known that no non-Kac compact quantum group can faithfully act on $C(X)$ for a classical, compact Hausdorff space $X$. However, in this article we show that this is no longer true if we go to non-compact spaces and non-compact quantum groups, by exhibiting a large class of examples of locally compact quantum groups coming from bicrossed product construction, including non-Kac ones, which can faithfully and ergodically act on classical (non-compact) spaces. However, none of these actions can be isometric in the sense of Goswami, leading to the conjecture that the result obtained by Goswami and Joardar about non-existence of genuine quantum isometry of classical compact connected Riemannian manifolds may hold in the non-compact case as well.

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arXiv2015

Smooth actions of compact quantum groups on compact smooth manifolds

Definition of a smooth action of a CQG on a compact, smooth manifold is given and studied. It is shown that a smooth action is always injective. Furthermore A necessary and sufficient condition for a lift of the smooth action as a bimodule morphism on the bimodule of one forms has been deduced and it is also shown to be equivalent to the condition of preserving some Riemannian inner product on the manifold.

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arXiv2015

Existence and examples of quantum isometry group for a class of compact metric spaces

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 $d_0$ is the Euclidean metric) for some homeomorphism $ϕ$ of $\IR^+$. As concrete examples, we obtain Wang's quantum permutation group $\cls_n^+$ and also the free wreath product of $\IZ_2$ by $\cls_n^+$ as the quantum isometry groups for certain compact connected metric spaces constructed by taking topological joins of intervals in \cite{huang1}.

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arXiv2015

Quadratic independence of coordinate functions of certain homogeneous spaces and action of compact quantum groups

Let $G$ be one of the classical compact, simple, centre-less, connected Lie groups or rank $n$ with a maximal torus $T$, the Lie algebra $\clg$ and let $\{ E_i, F_i, H_i, i=1, \ldots, n \}$ be the standard set of generators corresponding to a basis of the root system. Consider the adjoint-orbit space $M=\{ {\rm Ad}_g(H_1),~g \in G \}$, identified with the homogeneous space $G/L$ where $L=\{ g \in G:~{\rm Ad}_g(H_1)=H_1\}$. We prove that the `coordinate functions' $\{ f_i, i=1, \ldots, n \}$, (where $f_i(g):=λ_i({\rm Ad}_g(H_1))$, $\{ λ_1, \ldots, λ_n\}$ is basis of $\clg^\prime$) are `quadratically independent' in the sense that they do not satisfy any nontrivial homogeneous quadratic relations among them. Using this, it is proved that there is no genuine compact quantum group which can act faithtully on $C(M)$ such that the action leaves invariant the linear span of the above cordinate functions. As a corollary, it is also shown that any compact quantum group having a faithful action on the noncommutative manifold obtained by Rieffel deformation of $M$ satisfying a similar `linearity' condition must be a Rieffel-Wang type deformation of some compact group.

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arXiv2015

An averaging trick for smooth actions of compact quantum groups on manifolds

We prove that, given any smooth action of a compact quantum group (in the sense of \cite{rigidity}) on a compact smooth manifold satisfying some more natural conditions, one can get a Riemannian structure on the manifold for which the corresponding $C^\infty(M)$-valued inner product on the space of one-forms is preserved by the action.

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arXiv2014

Rigidity of action of compact quantum groups II

Suppose that a compact quantum group Q acts faithfully and isomet- rically (in the sense of [10]) on a smooth compact, oriented, connected Riemannian manifold M . If the manifold is stably parallelizable then it is shown that the compact quantum group is necessarily commutative as a C \ast algebra i.e. Q = C(G) for some compact group G. Using this, it is also proved that the quantum isometry group of Rieffel deformation of such manifold M must be a Rieffel-Wang deformation of C(ISO(M))

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arXiv2014

Rigidity of action of compact quantum groups III: the general case

If a compact quantum group acts faithfully and smoothly (in the sense of Goswami 2009) on a smooth, compact, oriented, connected Riemannian manifold such that the action induces a natural bimodule morphism on the module of sections of the co-tangent bundle, then it is proved that the quantum group is necessarily commutative as a $C^{*}$ algebra i.e. isomorphic with $ C(G)$ for some compact group $G$. From this, we deduce that the quantum isometry group of such a manifold M coincides with $C(ISO(M))$ where $ISO(M) $ is the group of (classical) isometries, i.e. there is no genuine quantum isometry of such a manifold.

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arXiv2014

A note on geometric characterization of quantum isometries of classical manifolds

If a compact quantum group acts isometrically on a (possibly discon- nected) compact smooth Riemannian manifold such that the action commutes with the Laplacian then it is known that the differential of the action preserves Rieman- nian inner product on forms. In this note, we prove a partial converse to this, under the additional assumption that the manifold is ori- ented and the action preserves the orientation in a suitable sense. Using this an alternative line of arguments is given for proving that there is no quantum isometry for a compact, connected, Riemannian manifold.

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arXiv2014

Quantum Isometry Groups of Noncommutative Manifolds Obtained by Deformation Using Dual Unitary 2-Cocycles

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].

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arXiv2013

A homomorphism theorem and a Trotter product formula for quantum stochastic flows with unbounded coefficients

We give a new method for proving the homomorphic property of a quantum stochastic ow satisfying a quantum stochastic differential equation with unbounded coefficients, under some further hypotheses. As an application, we prove a Trotter product formula for quantum stochastic ows and obtain quantum stochastic dilations of a class of quantum dynamical semigroups generalizing results of [5]

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arXiv2011

Quantum Brownian Motion on noncommutative manifolds: construction, deformation and exit times

We begin with a review and analytical construction of quantum Gaussian process (and quantum Brownian motions) in the sense of [25],[10] and others, and then formulate and study in details (with a number of interesting examples) a definition of quantum Brownian motions on those noncommutative manifolds (a la Connes) which are quantum homogeneous spaces of their quantum isometry groups in the sense of [11]. We prove that bi-invariant quantum Brownian motion can be 'deformed' in a suitable sense. Moreover, we propose a noncommutative analogue of the well-known asymptotics of the exit time of classical Brownian motion. We explicitly analyze such asymptotics for a specific example on noncommutative two-torus Aθ, which seems to behave like a one-dimensional manifold, perhaps reminiscent of the fact that Aθ is a noncommutative model of the (locally one-dimensional) 'leaf-space' of the Kronecker foliation.

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