arXiv subjects

Debashish Goswami

Publications and source records attributed to Debashish Goswami.

At least 19 recordsLinked to original sources

arXiv2026

Projective corepresentations and cohomology of compact quantum groups

We study projective unitary (co)representations of compact quantum groups and the associated second cohomology theory. We introduce left/right/bi/strongly projective corepresentations and study them in details. In particular, we prove that given any compact quantum group $\q$, there are compact quantum groups $\tilde{\q_l}, \tilde{\q_r}, {\tilde \q}_{bi}, {\tilde \q}_{stp}$, each of which contains $\q$ as a Woronowicz subalgebra and every left/right/bi/strongly projective unitary corepresentation of $\q$ lifts to a linear corepresentation of these quantum groups respectively. We observe that the strongly projective corepresentations are associated with the second invariant ($S^1$-valued) cohomology $H^2_{uinv}(\cdot)$ of the quantum group. We define a suitable analogue of normalizer of a compact quantum group in a bigger compact quantum group and using this, associate a canonical discrete group $\Gamma_\q$ to a compact quantum group $\q$ which is an alternative generalization of the second group cohomology and we show by an example that $\Gamma_\q$ in general may be different from $H^2_{uinv}(\q,S^1) $.

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arXiv2026

Weak quantum hypergroups from finite index C*-inclusions

We study a finite index inclusion of simple unital C*-algebras and construct a canonical completely positive coproduct on the second relative commutant, thereby endowing it with a natural coalgebra structure. Motivated by this construction, we introduce the notion of a weak quantum hypergroup, a generalization of the quantum hypergroups of Chapovsky and Vainerman. We show that every finite index inclusion gives rise to such a weak quantum hypergroup, and that the corresponding weak quantum hypergroup possesses a Haar integral. In the irreducible case, this structure satisfies the axioms of a quantum hypergroup in the sense of Chapovsky and Vainerman, while in the depth 2 setting our framework yields the associated weak Hopf algebra constructed by Nikshych and Vainerman. These results provide a unified and intrinsically C*-algebraic framework for generalized quantum symmetries associated with finite index inclusions.

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arXiv2025

Second invariant cohomology of some finite-dimensional Hopf algebras

We use categorical description of the invariant 2-cohomology group of Hopf algebra to compute such cohomology for two finite dimensional Hopf algebras: the group ring of $Z_8\rtimes Aut(Z_8)$ and Kac-Paljutkin algebra. For the first of these two examples, our categorical approach helps to settle the problem of computing this cohomology, which was left open in by Guillot and Kassel (\cite{dtwist}), where only some partial information about this cohomology was obtained.

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arXiv2025

Linear coactions of discrete quantum groups on the circle

For a (unital) $C^*$-algebra $\cla$, we construct a $C^*$-algebraic discrete quantum group (DQG) $\clq_{\rm aut}(\cla)$, coacting on $\cla$, which is a quantum generalization of ${\text Aut}(\cla)$ in the framework of discrete quantum groups, in the sense that any other coaction of a DQG on $\cla$ factors through the above coaction of $\clq_{\rm aut}(\cla)$. We prove by an explicit calculation that if any Kac-type $C^*$-algebraic discrete quantum group $\mathcal{Q}$ has a `weakly faithful' coaction on $C(S^1)$ which is `linear' in the sense that it leaves the space spanned by $\{ Z, \overline{Z} \}$ invariant, then $\mathcal{Q}$ must be classical, i.e. isomorphic with $C_0(\Gamma)$ for some discrete group $\Gamma$. This parallels the well-known result of non-existence of genuine compact quantum group symmetry obtained by the first author and his collaborators ([GB16] and the references therein).

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arXiv2025

On outer automorphisms of certain graph $C^{*}$-algebras

Given a countable abelian group $A$, we construct a row finite directed graph $\Gamma(A)$ such that the $K_{0}$-group of the graph $\textrm{C}^{\ast}$-algebra $\textrm{C}^{\ast}(\Gamma(A))$ is canonically isomorphic to $A$. Moreover, each element of $\textrm {Aut}(A)$ is a lift of an automorphism of the graph $\textrm{C}^{\ast}$-algebra $\textrm{C}^{\ast}(\Gamma(A))$.

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arXiv2024

Complex structures on Three-point space

We discuss notions of almost complex, complex and Kähler structures in the realm of non-commutative geometry and investigate them for a class of finite dimensional spectral triples on the three-point space. We classify all the almost complex structures on this non-commutative manifold, which also turn out to be complex structures, but none of them are Kähler in our sense.

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arXiv2024

Quantum symmetry in multigraphs (part II)

This article is a continuation of "Quantum symmetry in multigraphs (part I)". In this article, we give an explicit construction of a non-Bichon type co-action on a multigraph that is, it preserves quantum symmetry of (V,E) in our sense but not always in Bichon's sense. This construction itself is motivated from automorphisms of quantum graphs.

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arXiv2024

Quantum symmetry in multigraphs (part I)

We introduce various notions of quantum symmetry in a directed or undirected multigraph with no isolated vertex and explore relations among them. If the multigraph is single edged (that is, a simple graph where loops are allowed), all our notions of quantum symmetry reduce to already existing notions of quantum symmetry provided by Bichon and Banica. Our constructions also show that any multigraph with at least two pairs of vertices with multiple edges among them possesses genuine quantum symmetry.

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arXiv2023

Quantum Symmetry on Potts Model

We formulate the notion of quantum group symmetry of the Hamiltonian corresponding to Potts model and compute it for few simple models. Our examples illustrate how a slight change of the model parameter may result in a drastic change of the quantum symmetry group, (in some cases, the classical symmetry group remains unaffected) signifying a case of phase transition.

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arXiv2022

Quantum Galois groups of subfactors

For a finite-index $\mathrm{II}_1$ subfactor $N \subset M$, we prove the existence of a universal Hopf $\ast$-algebra (or, a discrete quantum group in the analytic language) acting on $M$ in a trace-preserving fashion and fixing $N$ pointwise. We call this Hopf $\ast$-algebra the quantum Galois group for the subfactor and compute it in some examples of interest, notably for arbitrary irreducible finite-index depth-two subfactors. Along the way, we prove the existence of universal acting Hopf algebras for more general structures (tensors in enriched categories), in the spirit of recent work by Agore, Gordienko and Vercruysse.

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arXiv2021

Amenable fusion algebraic actions of discrete quantum groups on compact quantum spaces

In this paper, we introduce actions of fusion algebras on unital $C^*$-algebras, and define amenability for fusion algebraic actions. Motivated by S.\ Neshveyev et al.'s work, considering the co-representation ring of a compact quantum group as a fusion algebra, we define the canonical fusion algebraic (for short, CFA) form of a discrete quantum group action on a compact quantum space. Furthermore, through the CFA form, we define FA-amenability of discrete quantum group actions, and present some basic connections between FA-amenable actions and amenable discrete quantum groups. As an application, thinking of a state on a unital $C^*$-algebra as a "probability measure" on a compact quantum space, we show that amenability for a discrete quantum group is equivalent to both of FA-amenability for an action of this discrete quantum group on a compact quantum space and the existence of this kind of "probability measure" that is FA-invariant under this action.

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arXiv2021

Generalized symmetry in noncommutative (complex) geometry

We introduce Hopf algebroid covariance on Woronowicz's differential calculus. Using it, we develop quite a general framework of noncommutative complex geometry that subsumes the one in [2]. We present transverse complex and Kähler structures as examples and discuss several other examples. Relation with past literature is described.

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arXiv2021

A new look at Levi-Civita connection in noncommutative geometry

We prove the existence and uniqueness of Levi-Civita connections for strongly sigma-compatible pseudo-Riemannian metrics on tame differential calculi. Such pseudo-Riemannian metrics properly contain the classes of bilinear metrics as well as their conformal deformations. This extends the previous results in references 9 and 10. Star-compatibility of Levi-Civita connections for bilinear pseudo-Riemannian metrics are also discussed.

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arXiv2021

Levi-Civita connections for conformally deformed metrics on tame differential calculi

Given a tame differential calculus over a noncommutative algebra $\mathcal{A}$ and an $\mathcal{A}$-bilinear pseudo-Riemannian metric $g_0,$ consider the conformal deformation $ g = k. g_0, $ $k$ being an invertible element of $\mathcal{A}.$We prove that there exists a unique connection $\nabla$ on the bimodule of one-forms of the differential calculus which is torsionless and compatible with $g.$ We derive a concrete formula connecting $\nabla$ and the Levi-Civita connection for the pseudo-Riemannian metric $g_0.$ As an application, we compute the Ricci and scalar curvature for a general conformal perturbation of the canonical metric on the noncommutative $2$-torus as well as for a natural metric on the quantum Heisenberg manifold. For the latter, the scalar curvature turns out to be a negative constant.

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arXiv2020

Existence and rigidity of quantum isometry groups for compact metric spaces

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.

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arXiv2020

Levi-Civita connections and vector fields for noncommutative differential calculi

We study covariant derivatives on a class of centered bimodules $\mathcal{E}$ over an algebra A. We begin by identifying a $\mathbb{Z} ( A ) $-submodule $ \mathcal{X} ( A ) $ which can be viewed as the analogue of vector fields in this context; $ \mathcal{X} ( A ) $ is proven to be a Lie algebra. Connections on $\mathcal{E}$ are in one to one correspondence with covariant derivatives on $ \mathcal{X} ( A ). $ We recover the classical formulas of torsion and metric compatibility of a connection in the covariant derivative form. As a result, a Koszul formula for the Levi-Civita connection is also derived.

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arXiv2020

On the Koszul formula in noncommutative geometry

We prove a Koszul formula for the Levi-Civita connection for any pseudo-Riemannian bilinear metric on a class of centered bimodule of noncommutative one-forms. As an application to the Koszul formula, we show that our Levi-Civita connection is a bimodule connection. We construct a spectral triple on a fuzzy sphere and compute the scalar curvature for the Levi-Civita connection associated to a canonical metric.

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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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