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Jyotishman Bhowmick

Publications and source records attributed to Jyotishman Bhowmick.

At least 19 recordsLinked to original sources
arXiv2026

The Levi-Civita connection and Chern connections for cocycle deformations of Kähler manifolds

We consider unitary cocycle deformations of covariant $\ast$-differential calculi. We prove that complex structures, holomorphic bimodules and Chern connections on the deformed calculus are twists of their untwisted counterparts. Moreover, for cocycle deformations of a class of classical Kähler manifolds, the Levi-Civita connection on the space of one-forms of the deformed calculus is shown to be a direct sum of the Chern connections on the twisted holomorphic and the anti-holomorphic bimodules. Our class of examples also includes cocycle deformations of the Heckenberger-Kolb calculi.

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arXiv2025

On two notions of torsion and metric compatibility of connections in noncommutative geometry

We compare the notions of metric-compatibility and torsion of a connection in the frameworks of Beggs-Majid and Mesland-Rennie. It follows that for $\ast$-preserving connections, compatibility with a real metric in the sense of Beggs-Majid corresponds to Hermitian connections in the sense of Mesland-Rennie. If the calculus is quasi-tame, the torsion zero conditions are equivalent. A combination of these results proves the existence and uniqueness of Levi-Civita connections in the sense of Mesland-Rennie for unitary cocycle deformations of a large class of Riemannian manifolds as well as the Heckenberger-Kolb calculi on all quantized irreducible flag manifolds.

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arXiv2024

Levi-Civita connection on the irreducible quantum flag manifolds

We classify covariant metrics (in the sense of Beggs and Majid) on a class of quantum homogeneous spaces. In particular, our classification implies the existence of a unique (up to scalar) quantum symmetric covariant metric on the Heckenberger--Kolb calculi for the quantized irreducible flag manifolds. Moreover, we prove the existence and uniqueness of Levi-Civita connection for any real covariant metric for the Heckenberger--Kolb calculi. This generalizes Matassa's result for the quantum projective spaces.

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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

Covariant connections on bicovariant differential calculus

Given a bicovariant differential calculus $(\mathcal{E}, d)$ such that the braiding map is diagonalisable in a certain sense, the bimodule of two-tensors admits a direct sum decomposition into symmetric and anti-symmetric tensors. This is used to prove the existence of a bicovariant torsionless connection on $\mathcal{E}$. Following Heckenberger and Schm{ü}dgen, we study invariant metrics and the compatibility of covariant connections with such metrics. A sufficient condition for the existence and uniqueness of bicovariant Levi-Civita connections is derived. This condition is shown to hold for cocycle deformations of classical Lie groups.

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arXiv2020

Pseudo-Riemannian metrics on bicovariant bimodules

We study pseudo-Riemannian invariant metrics on bicovariant bimodules over Hopf algebras. We clarify some properties of such metrics and prove that pseudo-Riemannian invariant metrics on a bicovariant bimodule and its cocycle deformations are in one to one correspondence.

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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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arXiv2018

Quantum symmetries of the twisted tensor products of C*-algebras

We consider the construction of twisted tensor products in the category of C*-algebras equipped with orthogonal filtrations and under certain assumptions on the form of the twist compute the corresponding quantum symmetry group, which turns out to be the generalised Drinfeld double of the quantum symmetry groups of the original filtrations. We show how these results apply to a wide class of crossed products of C*-algebras by actions of discrete groups. We also discuss an example where the hypothesis of our main theorem is not satisfied and the quantum symmetry group is not a generalised Drinfeld double.

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arXiv2018

Tube representations and twisting of graded categories

We study deformation of tube algebra under twisting of graded monoidal categories. When a tensor category $\mathcal{C}$ is graded over a group $Γ$, a torus-valued 3-cocycle on $Γ$ can be used to deform the associator of $\mathcal{C}$. Based on a natural Fell bundle structure of the tube algebra over the action groupoid of the adjoint action of $Γ$, we show that the tube algebra of the twisted category is a 2-cocycle twisting of the original one.

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arXiv2014

Compact quantum metric spaces from quantum groups of rapid decay

We present a modified version of the definition of property RD for discrete quantum groups given by Vergnioux in order to accommodate examples of non-unimodular quantum groups. Moreover we extend the construction of spectral triples associated to discrete groups with length functions, originally due to Connes, to the setting of quantum groups. For quantum groups of rapid decay we study the resulting spectral triples from the point of view of compact quantum metric spaces in the sense of Rieffel.

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arXiv2013

Deformation of operator algebras by Borel cocycles

Assume that we are given a coaction δof a locally compact group G on a C*-algebra A and a T-valued Borel 2-cocycle ωon G. Motivated by the approach of Kasprzak to Rieffel's deformation we define a deformation A_ωof A. Among other properties of A_ωwe show that A_ω\otimes K(L^2(G)) is canonically isomorphic to A\rtimes_δ\hat G\rtimes_{\hatδ,ω}G. This, together with a slight extension of a result of Echterhoff et al., implies that for groups satisfying the Baum-Connes conjecture the K-theory of A_ωremains invariant under homotopies of omega.

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arXiv2013

Quantum gauge symmetries in Noncommutative Geometry

We discuss generalizations of the notion of i) the group of unitary elements of a (real or complex) finite dimensional C*-algebra, ii) gauge transformations and iii) (real) automorphisms, in the framework of compact quantum group theory and spectral triples. The quantum analogue of these groups are defined as universal (initial) objects in some natural categories. After proving the existence of the universal objects, we discuss several examples that are of interest to physics, as they appear in the noncommutative geometry approach to particle physics: in particular, the C*-algebras M_n(R), M_n(C) and M_n(H), describing the finite noncommutative space of the Einstein-Yang-Mills systems, and the algebras A_F=C+H+M_3(C) and A^{ev}=H+H+M_4(C), that appear in Chamseddine-Connes derivation of the Standard Model of particle physics minimally coupled to gravity. As a byproduct, we identify a "free" version of the symplectic group Sp(n) (quaternionic unitary group).

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arXiv2012

Quantum isometries and group dual subgroups

We study the discrete groups $Λ$ whose duals embed into a given compact quantum group, $\hatΛ\subset G$. In the matrix case $G\subset U_n^+$ the embedding condition is equivalent to having a quotient map $Γ_U\toΛ$, where $F=\{Γ_U|U\in U_n\}$ is a certain family of groups associated to $G$. We develop here a number of techniques for computing $F$, partly inspired from Bichon's classification of group dual subgroups $\hatΛ\subset S_n^+$. These results are motivated by Goswami's notion of quantum isometry group, because a compact connected Riemannian manifold cannot have non-abelian group dual isometries.

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arXiv2011

Quantum Isometries of the finite noncommutative geometry of the Standard Model

We compute the quantum isometry group of the finite noncommutative geometry F describing the internal degrees of freedom in the Standard Model of particle physics. We show that this provides genuine quantum symmetries of the spectral triple corresponding to M x F where M is a compact spin manifold. We also prove that the bosonic and fermionic part of the spectral action are preserved by these symmetries.

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