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Noncommutative metric geometry of quantum circle bundles

Jens Kaad

TL;DR

The paper develops a noncommutative metric framework for quantum circle bundles, treating the circle action as a vertical direction and pairing it with a horizontal spectral triple on the base. It introduces twisted Lipschitz triples and a modular lift to tame exponential growth, enabling the vertical and horizontal data to be combined into a quantum metric on the total algebra. Two main theorems establish conditions under which the total algebra becomes a compact quantum metric space and relate the construction to unbounded KK-theory, including a Kasparov product when a key parameter equals one. The framework is applied to the higher Vaksman-Soibelman quantum spheres and quantum projective spaces, yielding quantum metrics induced by $q$-geometric data and Drinfeld-Jimbo twists, and generalizing known results for quantum $SU(2)$.

Abstract

In this paper we investigate quantum circle bundles from the point of view of compact quantum metric spaces. The raw input data is a circle action on a unital $C^*$-algebra together with a quantum metric of spectral geometric origin on the fixed point algebra. Under a few extra conditions on the spectral subspaces we show that the spectral geometric data on the base algebra can be lifted to the total algebra. Notably, the lifted spectral geometry is independent of the choice of frames and is permitted to interact with the total algebra via a twisted derivation. Under these conditions, it is explained how to assemble our data into a quantum metric on the total algebra in a way which unifies and generalizes a couple of results in the literature relating to crossed products by the integers and to quantum $SU(2)$. We apply our ideas to the higher Vaksman-Soibelman quantum spheres and endow them with quantum metrics arising from $q$-geometric data. In this context, the twist is enforced by the structure of the Drinfeld-Jimbo deformation arising from the Lie algebra of the special unitary group.

Noncommutative metric geometry of quantum circle bundles

TL;DR

The paper develops a noncommutative metric framework for quantum circle bundles, treating the circle action as a vertical direction and pairing it with a horizontal spectral triple on the base. It introduces twisted Lipschitz triples and a modular lift to tame exponential growth, enabling the vertical and horizontal data to be combined into a quantum metric on the total algebra. Two main theorems establish conditions under which the total algebra becomes a compact quantum metric space and relate the construction to unbounded KK-theory, including a Kasparov product when a key parameter equals one. The framework is applied to the higher Vaksman-Soibelman quantum spheres and quantum projective spaces, yielding quantum metrics induced by -geometric data and Drinfeld-Jimbo twists, and generalizing known results for quantum .

Abstract

In this paper we investigate quantum circle bundles from the point of view of compact quantum metric spaces. The raw input data is a circle action on a unital -algebra together with a quantum metric of spectral geometric origin on the fixed point algebra. Under a few extra conditions on the spectral subspaces we show that the spectral geometric data on the base algebra can be lifted to the total algebra. Notably, the lifted spectral geometry is independent of the choice of frames and is permitted to interact with the total algebra via a twisted derivation. Under these conditions, it is explained how to assemble our data into a quantum metric on the total algebra in a way which unifies and generalizes a couple of results in the literature relating to crossed products by the integers and to quantum . We apply our ideas to the higher Vaksman-Soibelman quantum spheres and endow them with quantum metrics arising from -geometric data. In this context, the twist is enforced by the structure of the Drinfeld-Jimbo deformation arising from the Lie algebra of the special unitary group.
Paper Structure (24 sections, 41 theorems, 122 equations)

This paper contains 24 sections, 41 theorems, 122 equations.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. Statements and Declarations
  4. Conflict of interest
  5. Data availability
  6. Standing conventions
  7. Preliminaries on tensor products of $C^*$-modules
  8. Horizontal lifts of selfadjoint unbounded operators
  9. Preliminaries on compact quantum metric spaces
  10. Li's theorem for circle actions
  11. Finitely generated projective modules
  12. Preliminaries on Lipschitz operators
  13. Analytic operators
  14. Twisted Lipschitz operators and twisted Lipschitz triples
  15. First main theorem
  16. ...and 9 more sections

Key Result

Lemma 2.2

Suppose that $\mathcal{X} \subseteq X$ has a finite frame $\{ \zeta_j\}_{j = 1}^n$. It then holds that the inclusion maps induce an isomorphism of right $B$-modules $\mathcal{X} \otimes_{\mathcal{A}} Y \cong X \widehat{\otimes}_\rho Y$. Moreover, if $\mathcal{R} : \mathcal{X} \to X' \widehat{\otimes then there exists a unique $B$-linear bounded operator $c(R) : X \widehat{\otimes}_\rho Y \to X' \w

Theorems & Definitions (77)

  • Definition 2.1
  • Lemma 2.2
  • proof
  • Definition 2.3
  • Definition 2.4
  • Theorem 2.5
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Definition 3.4
  • ...and 67 more