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The finite dual coalgebra as a quantization of the maximal spectrum

Manuel L. Reyes

Abstract

In pursuit of a noncommutative spectrum functor, we argue that the Heyneman-Sweedler finite dual coalgebra can be viewed as a quantization of the maximal spectrum of a commutative affine algebra, integrating prior perspectives of Takeuchi, Batchelor, Kontsevich-Soibelman, and Le Bruyn. We introduce fully residually finite-dimensional algebras $A$ as those with enough finite-dimensional representations to let $A^\circ$ act as an appropriate depiction of the noncommutative maximal spectrum of $A$; importantly, this class includes affine noetherian PI algebras. In the case of prime affine algebras that are module-finite over their center, we describe how the Azumaya locus is represented in the finite dual. This is used to describe the finite dual of quantum planes at roots of unity as an endeavor to visualize the noncommutative space on which these algebras act as functions. Finally, we discuss how a similar analysis can be carried out for other maximal orders over surfaces.

The finite dual coalgebra as a quantization of the maximal spectrum

Abstract

In pursuit of a noncommutative spectrum functor, we argue that the Heyneman-Sweedler finite dual coalgebra can be viewed as a quantization of the maximal spectrum of a commutative affine algebra, integrating prior perspectives of Takeuchi, Batchelor, Kontsevich-Soibelman, and Le Bruyn. We introduce fully residually finite-dimensional algebras as those with enough finite-dimensional representations to let act as an appropriate depiction of the noncommutative maximal spectrum of ; importantly, this class includes affine noetherian PI algebras. In the case of prime affine algebras that are module-finite over their center, we describe how the Azumaya locus is represented in the finite dual. This is used to describe the finite dual of quantum planes at roots of unity as an endeavor to visualize the noncommutative space on which these algebras act as functions. Finally, we discuss how a similar analysis can be carried out for other maximal orders over surfaces.

Paper Structure

This paper contains 17 sections, 15 theorems, 95 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Prologue: noncommutative spectral theory
  3. Quantizing the maximal spectrum
  4. Outline of the paper
  5. Coalgebras as noncommutative discrete spaces
  6. Reminder on dual coalgebras
  7. Coalgebras as generalized sets
  8. Coalgebras of distributions on commutative schemes
  9. The finite dual as a quantized maximal spectrum
  10. Functoriality and the coradical
  11. Fully RFD algebras
  12. Morita equivalence and the quantized spectrum
  13. Affine algebras finite over their center
  14. The Azumaya locus in the dual coalgebra
  15. Picturing the quantum plane
  16. ...and 2 more sections

Key Result

Proposition 2.13

The restriction of the functor $\mathop{\mathrm{Dist}}\nolimits \colon \mathop{\mathrm{\mathsf{Sch}}}\nolimits_k \to \mathop{\mathrm{\mathsf{Coalg}}}\nolimits$ to the full subcategory of affine $k$-schemes is naturally isomorphic to the functor $X \mapsto \Gamma(X,\mathcal{O}_X)^\circ$.

Figures (2)

  • Figure 1: Illustration of the maps between quantum sets in \ref{['eq:qubit map']}
  • Figure 2: Depiction of $\mathcal{O}_q(\mathbb{C}^2)^\circ$ at $q = -1$

Theorems & Definitions (39)

  • Example 2.5
  • Remark 2.7
  • Remark 2.10
  • Definition 2.11
  • Proposition 2.13
  • proof
  • Example 2.15
  • Proposition 2.17
  • proof
  • Corollary 2.18
  • ...and 29 more