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Bipul Saurabh

Publications and source records attributed to Bipul Saurabh.

At least 19 recordsLinked to original sources
arXiv2026

Quantum Gromov-Hausdorff Convergence for Extensions of $C^*$-Algebras

We investigate the lifting of quantum Gromov-Hausdorff convergence through Toeplitz type $C^*$-algebra extensions by stable ideals in the framework of noncommutative metric geometry. Working with the spectral metric space construction of Hawkins and Zacharias (Comm. Math. Phys. 350 (2017), 475-506), we consider a sequence of complete sub-operator systems of the quotient or the unital $C^*$-algebra underlying the stable ideal, converging in the quantum Gromov-Hausdorff distance. We study whether this induces a corresponding convergent sequence of complete sub-operator systems of the extension. To address this problem, we construct complete sub-operator systems of the extension associated with those of the quotient and the unital $C^*$-algebra underlying the stable ideal. We also introduce the notion of unital $2$-contractive approximation together with its Toeplitz type refinement to provide the compatibility required by the commutator structure of the Dirac operator on the extension. We prove that, under this approximation hypothesis on the convergent sequence in the quotient or the unital $C^*$-algebra underlying the stable ideal, quantum Gromov-Hausdorff convergence lifts to the extension.

math.OA↗
arXiv2026

$C(SO_q(4)/SO_q(2))$ as a Groupoid $C^*$-algebra

In this paper, we prove that $C(SO_q(4)/SO_q(2))$ is isomorphic to the $C^*$-algebra of the tight groupoid $\mathcal{G}_{\mathrm{tight}}$ associated with the inverse semigroup generated by the standard generators of its classical limit $C(SO_0(4)/SO_0(2))$. We show that all four orbits of the unit space $\mathcal{G}_{\mathrm{tight}}^{(0)}$ under the natural action of $\mathcal{G}_{\mathrm{tight}}$ are locally closed, and that the associated isotropy groups are isomorphic to $\mathbb{Z}$. Consequently, every irreducible representation of $C^*(\mathcal{G}_{\mathrm{tight}})$ is induced from an irreducible representation of $C^*(\mathbb{Z})$, which are parametrized by $\mathbb{T}$. In this way, we obtain four families of irreducible representations parametrized by $\mathbb{T}$, and we explicitly construct their equivalence with the corresponding Soibelman irreducible representations of $C(SO_q(4)/SO_q(2))$.

math.OA↗
arXiv2026

$C(SO_q(2n+1)/SO_q(2n-1))$ as iterated torsioned quantum double suspensions of $C(\mathbb{T})$

Let $A$ be a unital $C^*$-algebra, and let $\Sigma^2_m A$ denote the $m$-torsioned quantum double suspension of $A$. For $q \in (0,1)$ and $n \geq 1$, we prove that the $C^*$-algebra corresponding to the quotient space $SO_q(2n+1)/SO_q(2n-1)$ is isomorphic to $\Sigma^{2(n-1)} \, \Sigma^2_2 \, \Sigma^{2(n-1)} C(\mathbb{T})$. It follows as a consequence that these spaces are independent of the deformation parameter $q$.

math.OA↗
arXiv2025

Spectral dimension of $p$-adic integers

The notion of spectral dimension was introduced by Chakraborty and Pal in \cite{cp}. In this paper, we show that the spectral dimension of the ring of $p$-adic integers, $\mathbb{Z}_p$, is equal to its manifold dimension, which is $0$. Finally, we determine the $K$-groups of $\mathbb{Z}_p$, and show that the generators of $K_0(\mathbb{Z}_p)$ can be expressed as finite span of the characters of $\mathbb{Z}_p$.

math.QA↗
arXiv2025

On the classification of $C^*$-algebras of twisted isometries with finite dimensional wandering spaces

Let \( m, n \in \mathbb{N}_0 \), and let \( X \) be a closed subset of \( \mathbb{T}^{\binom{m+n}{2}} \). We define \( C^{m,n}_X \) to be the universal \( C^* \)-algebra among those generated by \( m \) unitaries and \( n \) isometries satisfying doubly twisted commutation relations with respect to a twist \( \mathcal{U} = \{U_{ij}\}_{1 \leq i < j \leq m+n} \) of commuting unitaries having joint spectrum \( X \). We provide a complete list of the irreducible representations of \( C^{m,n}_X \) up to unitary equivalence and, under a denseness assumption on \( X \), explicitly construct a faithful representation of \( C^{m,n}_X \). Under the same assumption, we also give a necessary and sufficient condition on a fixed tuple \( \mathcal{U} \) of commuting unitaries with joint spectrum \( X \) for the existence of a universal tuple of \( \mathcal{U} \)-doubly twisted isometries. For \( X = \mathbb{T}^{\binom{m+n}{2}} \), we compute the \( K \)-groups of \( C^{m,n}_X \). We further classify the \( C^* \)-algebras generated by a pair of doubly twisted isometries with a fixed parameter \( θ\in \mathbb{R} \setminus \mathbb{Q} \), whose wandering spaces are finite-dimensional. Finally, for a fixed unitary \( U \), we classify all the \( C^* \)-algebras generated by a pair of \( U \)-doubly twisted isometries with finite-dimensional wandering spaces.

math.OA↗
arXiv2025

$K$-stability of $C^*$-algebras generated by isometries and unitaries with twisted commutation relations

In this article, we prove $K$-stability for a family of $C^*$-algebras, which are generated by a finite set of unitaries and isometries satisfying twisted commutation relations. This family includes the $C^*$-algebra of doubly non-commuting isometries and free twist of isometries. Next, we consider the $C^*$-algebra $A_{\mathcal{V}}$ generated by an $n$-tuple of $\mathcal{U}$-twisted isometries $\mathcal{V}$ with respect to a fixed $n\choose 2$-tuple $\mathcal{U}=\{U_{ij}:1\leq i<j \leq n\}$ of commuting unitaries (see \cite{NarJaySur-2022aa}). Under the assumption that the spectrum of the commutative $C^*$-algebra generated by $(\{U_{ij}:1\leq i<j \leq n\})$ does not contain any element of finite order in the torus group $\bbbt^{n\choose 2}$, we show that $A_{\mathcal{V}}$ is $K$-stable. Finally, we prove the same result for the $C^*$-algebra generated by a tuple of free $\mathcal{U}$-twisted isometries.

math.OA↗
arXiv2025

Vanishing of dimensions and nonexistence of spectral triples on compact Vilenkin groups

We compute the spectral dimension, the dimension of a symmetric random walk, and the Gelfand-Kirillov dimension for compact Vilenkin groups. As a result, we show that these dimensions are zero for any compact, totally disconnected, metrizable topological group. We provide an explicit description of the $K$-groups for compact Vilenkin groups. We express the generators of the $K_0$-groups in terms of the corresponding matrix coefficients for two specific examples: the group of $p$-adic integers and the $p$-adic Heisenberg group. Finally, we prove the nonexistence of a natural class of spectral triples on the group of $p$-adic integers.

math.OA↗
arXiv2024

Topological invariance of quantum homogeneous spaces of type $B$ and $D$

In this article, we study two families of quantum homogeneous spaces, namely, $SO_q(2n+1)/SO_q(2n-1)$, and $SO_q(2n)/SO_q(2n-2)$. By applying a two-step Zhelobenko branching rule, we show that the $C^*$-algebras $C(SO_q(2n+1)/SO_q(2n-1))$, and $C(SO_q(2n)/SO_q(2n-2))$ are generated by the entries of the first and the last rows of the fundamental matrix of the quantum groups $SO_q(2n+1)$, and $SO_q(2n)$, respectively. We then construct a chain of short exact sequences, and using that, we compute $K$-groups of these spaces with explicit generators. Invoking homogeneous $C^*$-extension theory, we show $q$-independence of some intermediate $C^*$-algebras arising as the middle $C^*$-algebra of these short exact sequences. As a consequence, we get the $q$-invariance of $SO_q(3)$, $SO_q(5)/SO_q(3)$, $SO_q(4)/SO_q(2)$, and $SO_q(6)/SO_q(4)$.

math.QA↗
arXiv2023

Computation of Gelfand-Kirillov dimension for $B$-type structures

Let $\mathcal{O}(\mbox{Spin}_{q^{1/2}}(2n+1))$ and $\mathcal{O}(SO_q(2n+1))$ be the quantized algebras of regular functions on the Lie groups $\mbox{Spin}(2n+1)$ and $SO(2n+1)$, respectively. In this article, we prove that the Gelfand-Kirillov dimension of a simple unitarizable $\mathcal{O}(\mbox{Spin}_{q^{1/2}}(2n+1))$-module $V_{t,w}^{\mbox{Spin}}$ is the same as the length of the Weyl word $w$. We show that the same result holds for the $\mathcal{O}(SO_q(2n+1))$-module $V_{t,w}$, which is obtained from $V_{t,w}^{\mbox{Spin}}$ by restricting the algebra action to the subalgebra $\mathcal{O}(SO_q(2n+1))$ of $\mathcal{O}(\mbox{Spin}_{q^{1/2}}(2n+1))$. Moreover, we consider the quantized algebras of regular functions on certain homogeneous spaces of $SO(2n+1)$ and $\mbox{Spin}(2n+1)$ and show that its Gelfand-Kirillov dimension is equal to the dimension of the homogeneous space as a real differentiable manifold.

math.RT↗
arXiv2022

Equivariant spectral triple for the quantum group $U_q(2)$ for complex deformation parameters

Let $q=|q|e^{iπθ},\,θ\in(-1,1],$ be a nonzero complex number such that $|q|\neq 1$ and consider the compact quantum group $U_q(2)$. For $θ\notin\mathbb{Q}\setminus\{0,1\}$, we obtain the $K$-theory of the $C^*$-algebra $C(U_q(2))$. We construct a spectral triple on $U_q(2)$ which is equivariant under its own comultiplication action. The spectral triple obtained here is even, $4^+$-summable, non-degenerate, and the Dirac operator acts on two copies of the $L^2$-space of $U_q(2)$. The $K$-homology class of the associated Fredholm module is shown to be nontrivial.

math.OA↗
arXiv2021

Representations and Classification of the compact quantum groups $U_q(2)$ for complex deformation parameters

In this article, we obtain a complete list of inequivalent irreducible representations of the compact quantum group $U_q(2)$ for non-zero complex deformation parameters $q$, which are not roots of unity. The matrix coefficients of these representations are described in terms of the little $q$-Jacobi polynomials. The Haar state is shown to be faithful and an orthonormal basis of $L^2(U_q(2))$ is obtained. Thus, we have an explicit description of the Peter-Weyl decomposition of $U_q(2)$. As an application, we discuss the Fourier transform and establish the Plancherel formula. We also describe the decomposition of the tensor product of two irreducible representations into irreducible components. Finally, we classify the compact quantum groups $U_q(2)$.

math.QA↗
arXiv2018

Spectral dimension of spheres

In this paper, we associate a growth graph and a length operator to a quotient space of a semisimple compact Lie group. Under certain assumptions, we show that the spectral dimension of a homogeneous space is greater than or equal to summability of the length operator. Using this, we compute spectral dimensions of spheres.

math.OA↗
arXiv2018

Spectral dimension of quaternion spheres

Employing ideas of noncommutative geometry, certain dimensional invariant for quantum homogeneous spaces has been proposed and here we take up its computation for quaternion spheres.

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

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↗
arXiv2016

Quantum Stiefel manifolds

Quantum analogs of Stiefel manifolds $SU_{q}(n)/SU_q(n-m)$ were introduced by Podkolzin \& Vainerman. The underlying $C^*$-algebra $C(SU_{q}(n)/SU_q(n-m))$ can be described as the $C^*$-subalgebra of $C(SU_q(n))$ generated by elements of last $m$ rows of the fundamental matrix of $SU_q(n)$. Using $R$-matrix of type $A_{n-1}$, one can find certain relations involving elements of last $m$ rows only. In this paper, by analyzing these relations and using a result of Neshveyev \& Tuset, we establish $C(SU_{q}(n)/SU_q(n-m))$ as a universal $C^*$-algbera given by finite sets of generators and relations.

math.OA↗
arXiv2015

$q$-invariance of quantum quaternion spheres

The $C^*$-algebra of continuous functions on the quantum quaternion sphere $H_q^{2n}$ can be identified with the quotient algebra $C(SP_q(2n)/SP_q(2n-2))$. In commutative case i.e. for $q=1$, the topological space $SP(2n)/SP(2n-2)$ is homeomorphic to the odd dimensional sphere $S^{4n-1}$. In this paper, we prove the noncommutative analogue of this result. Using homogeneous $C^*$-extension theory, we prove that the $C^*$-algebra $C(H_q^{2n})$ is isomorphic to the $C^*$-algebra $C(S_q^{4n-1})$. This further implies that for different values of $q \in [0,1)$, the $C^*$-algebras underlying the noncommutative space $H_q^{2n}$ are isomorphic.

math.OA↗