Searcharxiv⌕ Search

arXiv subjects

Partha Sarathi Chakraborty

Publications and source records attributed to Partha Sarathi Chakraborty.

At least 19 recordsLinked to original sources
arXiv2024

An approximate equivalence for the GNS representation of the Haar state of $SU_{q}(2)$

We use the crystallised $C^*$-algebra $C(SU_{q}(2))$ at $q=0$ to obtain a unitary that gives an approximate equivalence involving the GNS representation on the $L^{2}$ space of the Haar state of the quantum $SU(2)$ group and the direct integral of all the infinite dimensional irreducible representations of the $C^{*}$-algebra $C(SU_{q}(2))$ for nonzero values of the parameter $q$. This approximate equivalence gives a $KK$ class via the Cuntz picture in terms of quasihomomorphisms as well as a Fredholm representation of the dual quantum group $\widehat{SU_q(2)}$ with coefficients in a $C^*$-algebra in the sense of Mishchenko.

math.OA↗
arXiv2017

Gelfand-Kirillov dimension of some simple unitarizable modules

Let $\mathcal{O}_q(G)$ be the quantized algebra of regular functions on a semisimple simply connected compact Lie group $G$. Simple unitarizable left $\mathcal{O}_q(G)$-module are classified. In this article, we compute their Gelfand-Kirillov dimension where $G$ is of the type $A$, $C$ and $D$.

math.OA↗
arXiv2017

Comparison between two differential graded algebras in Noncommutative Geometry

Starting with a spectral triple one can associate two canonical differential graded algebras (dga) defined by Connes and Fröhlich et al. For the classical spectral triples associated with compact Riemannian spin manifolds both these dgas coincide with the de-Rham dga. Therefore, both are candidates for the noncommutative space of differential forms. Here we compare these two dgas and observe that in a very precise sense Connes' dga is more informative than that of Fröhlich et al.

math.OA↗
arXiv2017

Local index formula for the quantum double suspension

Our understanding of local index formula in noncommutative geometry is stalled for a while because we do not have more than one explicit computation, namely that of Connes for quantum SU(2) and do not understand the meaning of the various multilinear functionals involved in the formula. In such a situation further progress in understanding necessitates more explicit computations and here we execute the second explicit computation for the quantum double suspension, a construction inspired by the Toeplitz extension. More specifically we compute local index formula for the quantum double suspensions of $C(S^2)$ and the noncommutative $2$-torus.

math.KT↗
arXiv2014

An invariant for homogeneous spaces of compact quantum groups

The central notion in Connes' formulation of non commutative geometry is that of a spectral triple. Given a homogeneous space of a compact quantum group, restricting our attention to all spectral triples that are `well behaved' with respect to the group action, we construct a certain dimensional invariant. In particular, taking the (quantum) group itself as the homogeneous space, this gives an invariant for a compact quantum group. Computations of this invariant in several cases, including all type A quantum groups, are given.

math.QA↗
arXiv2014

Connes' calculus for The Quantum double suspension

Given a spectral triple $(\mathcal{A},\mathcal{H},D)\,$ Connes associated a canonical differential graded algebra $\,Ω_D^\bullet(\mathcal{A})$. However, so far this has been computed for very few special cases. We identify suitable hypotheses on a spectral triple that helps one to compute the associated Connes' calculus for its quantum double suspension. This allows one to compute $\,Ω_D^\bullet$ for spectral triples obtained by iterated quatum double suspension of the spectral triple associated with a first order differential operator on a compact smooth manifold. This gives the first systematic computation of Connes' calculus for a large family of spectral triples.

math.QA↗
arXiv2014

Multiplicativity of Connes' calculus

We consider the quadruples $\,(\mathcal{A},\mathbb{V},D,γ)$ where $\mathcal{A}$ is a unital, associative $\mathbb{K}\,$-algebra represented on the $\mathbb{K}\,$-vector space $\mathbb{V}$, $D\in \mathcal{E}nd(\mathbb{V})$, $γ\in\mathcal{E}nd(\mathbb{V})$ is a $\mathbb{Z}_2$-grading operator which commutes with $\mathcal{A}$ and anticommutes with $D$. We prove that the collection of such quadruples, denoted by $\,\widetilde{\mathcal{S}pec}\,$, is a monoidal category. We consider the monoidal subcategory $\,\widetilde{\mathcal{S}pec_{sub}}\,$ of objects of $\,\widetilde{\mathcal{S}pec}\,$ for which $γ\inπ(\mathcal{A})$. We show that there is a covariant functor $\,\mathcal{G}:\widetilde{\mathcal{S}pec}\longrightarrow\widetilde{\mathcal{S}pec_{sub}}\,$. Let $\,Ω_D^\bullet\,$ be the differential graded algebra defined by Connes ([Con2]) and $DGA$ denotes the category of differential graded algebras over the field $\mathbb{K}\,$. We show that $\mathcal{F}:\widetilde{\mathcal{S}pec_{sub}}\longrightarrow DGA\,$, given by $(\mathcal{A},\mathbb{V},D,γ)\longmapstoΩ_D^\bullet(\mathcal{A})$, is a monoidal functor. To show that $\,\mathcal{F}\circ\mathcal{G}\,$ is not trivial we explicitly compute it for the cases of compact manifold and the noncommutative torus along with the associated cohomologies.

math.QA↗
arXiv2013

Equivalence of Two Approaches to Yang-Mills on Non-commutative Torus

There are two notions of Yang-Mills action functional in noncommutative geometry. We show that for noncommutative n-torus both these notions agree. We also prove a structure theorem on the Hermitian structure of a finitely generated projective modules over spectrally invariant subalgebras of $C^*$-algebras.

math.OA↗
arXiv2013

Yang- Mills on Quantum Heisenberg Manifolds

In the noncommutative geometry program of Connes there are two variations of the concept of Yang-Mills action functional. We show that for the quantum Heisenberg manifolds they agree.

math.OA↗
arXiv2013

Record Maximum Oscillation Frequency in C-face Epitaxial Graphene Transistors

The maximum oscillation frequency (fmax) quantifies the practical upper bound for useful circuit operation. We report here an fmax of 70 GHz in transistors using epitaxial graphene grown on the C-face of SiC. This is a significant improvement over Si-face epitaxial graphene used in the prior high frequency transistor studies, exemplifying the superior electronics potential of C-face epitaxial graphene. Careful transistor design using a high κ dielectric T-gate and self-aligned contacts, further contributed to the record-breaking fmax.

cond-mat.mtrl-sci↗
arXiv2012

Application of artificial neural network in market segmentation: A review on recent trends

Despite the significance of Artificial Neural Network (ANN) algorithm to market segmentation, there is a need of a comprehensive literature review and a classification system for it towards identification of future trend of market segmentation research. The present work is the first identifiable academic literature review of the application of neural network based techniques to segmentation. Our study has provided an academic database of literature between the periods of 2000-2010 and proposed a classification scheme for the articles. One thousands (1000) articles have been identified, and around 100 relevant selected articles have been subsequently reviewed and classified based on the major focus of each paper. Findings of this study indicated that the research area of ANN based applications are receiving most research attention and self organizing map based applications are second in position to be used in segmentation. The commonly used models for market segmentation are data mining, intelligent system etc. Our analysis furnishes a roadmap to guide future research and aid knowledge accretion and establishment pertaining to the application of ANN based techniques in market segmentation. Thus the present work will significantly contribute to both the industry and academic research in business and marketing as a sustainable valuable knowledge source of market segmentation with the future trend of ANN application in segmentation.

nlin.AO↗
arXiv2010

The weak heat kernel asymptotic expansion and the quantum double suspension

In this paper we are concerned with the construction of a general principle that will allow us to produce regular spectral triples with finite and simple dimension spectrum. We introduce the notion of weak heat kernel asymptotic expansion (WHKAE) property of a spectral triple and show that the weak heat kernel asymptotic expansion allows one to conclude that the spectral triple is regular with finite simple dimension spectrum. The usual heat kernel expansion implies this property. Finally we show that WHKAE is stable under quantum double suspension, a notion introduced by Hong and Szymanski. Therefore quantum double suspending compact Riemannian spin manifolds iteratively we get many examples of regular spectral triples with finite simple dimension spectrum. This covers all the odd dimensional quantum spheres. Our methods also apply to the case of noncommutative torus.

math.OA↗
arXiv2010

K-groups of the quantum homogeneous space $SU_{q}(n)/SU_{q}(n-2)$

Quantum Steiffel manifolds were introduced by Vainerman and Podkolzin in \cite{VP}. They classified the irreducible representations of their underlying $C^*$-algebras. Here we compute the K groups of the quantum homogeneous spaces $SU_{q}(n)/SU_{q}(n-2), n\ge 3$. Specializing to the case $n=3$ we show that the fundamental unitary for quantum $SU(3)$ is nontrivial and is a unimodular element in $K_1$.

math.KT↗
arXiv2009

The geometry of determinant line bundles in noncommutative geometry

This paper is concerned with the study of the geometry of determinant line bundles associated to families of spectral triples parametrized by the moduli space of gauge equivalent classes of Hermitian connections on a Hermitian finite projective module. We illustrate our results with some examples that arise in noncommutative geometry.

hep-th↗
arXiv2008

Equivariant spectral triples and Poincaré duality for $SU_q(2)$

Let $\mathcal{A}$ be the $C^*$-algebra associated with $SU_q(2)$, $π$ be the representation by left multiplication on the $L_2$ space of the Haar state and let $D$ be the equivariant Dirac operator for this representation constructed by the authors earlier. We prove in this article that there is no operator other than the scalars in the commutant $π(\cla)'$ that has bounded commutator with $D$. This implies that the equivariant spectral triple under consideration does not admit a rational Poincaré dual in the sense of Moscovici, which in particular means that this spectral triple does not extend to a $K$-homology fundamental class for $SU_q(2)$. We also show that a minor modification of this equivariant spectral triple gives a fundamental class and thus implements Poincaré duality.

math.OA↗
arXiv2007

On equivariant Dirac operators for $SU_q(2)$

We explain the notion of minimality for an equivariant spectral triple and show that the triple for the quantum SU(2) group constructed by Chakraborty and Pal in \cite{c-p1} is minimal. We also give a decomposition of the spectral triple constructed by Dabrowski {\it et al} \cite{dlssv} in terms of the minimal triple constructed in \cite{c-p1}.

math.OA↗
arXiv2007

Torus equivariant spectral triples for odd dimensional quantum spheres coming from $C^*$-extensions

The torus group $(S^1)^{\ell+1}$ has a canonical action on the odd dimensional sphere $S_q^{2\ell+1}$. We take the natural Hilbert space representation where this action is implemented and characterize all odd spectral triples acting on that space and equivariant with respect to that action. This characterization gives a construction of an optimum family of equivariant spectral triples having nontrivial $K$-homology class thus generalizing our earlier results for $SU_q(2)$. We also relate the triple we construct with the $C^*$-extension \[ 0\longrightarrow \clk\otimes C(S^1)\longrightarrow C(S_q^{2\ell+3}) \longrightarrow C(S_q^{2\ell+1}) \longrightarrow 0. \]

math.KT↗