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The smallest quantum Mackey deformation

Yvann Gaudillot-Estrada

Abstract

When $G$ is a real semisimple group, there is a surprising interplay between its representation theory and that of its motion group $G_0$, known as the Mackey analogy. The present paper extends this analogy to the framework of $q$-deformations, for $G = \mathrm{SL}(2,\mathbb{R})$. In fact, we construct a deformation of $\mathrm{SL}(2,\mathbb{R})$ parametrized by $(q,t) \in \mathbb{R}_+^* \times \mathbb{R}$, where $q$ is the quantization parameter and $t$ is the Mackey parameter. We show how the representation theory varies along this deformation and we prove an analogue of the Connes-Kasparov isomorphism for the $q$-deformed reduced group C*-algebra.

The smallest quantum Mackey deformation

Abstract

When is a real semisimple group, there is a surprising interplay between its representation theory and that of its motion group , known as the Mackey analogy. The present paper extends this analogy to the framework of -deformations, for . In fact, we construct a deformation of parametrized by , where is the quantization parameter and is the Mackey parameter. We show how the representation theory varies along this deformation and we prove an analogue of the Connes-Kasparov isomorphism for the -deformed reduced group C*-algebra.
Paper Structure (16 sections, 28 theorems, 78 equations)

This paper contains 16 sections, 28 theorems, 78 equations.

Table of Contents

  1. Operator algebraic preliminaries
  2. Continuous fields of Hilbert spaces and C*-algebras
  3. The generalized Stone-Weierstrass theorem
  4. Representation theory of crossed products
  5. Deformed reduced C*-algebras of $\mathrm{SL}(2,\mathbb{R})$ and their duals
  6. Representation theory of $\mathrm{SL}(2,\mathbb{R})$
  7. The unitary dual of the motion group
  8. The quantization of $\mathrm{SL}(2,\mathbb{R})$
  9. The quantization of the motion group
  10. The continuous field of C*-algebras
  11. The algebraic quantum Mackey field
  12. Construction of the continuous field
  13. Geometric characterizations of the continuous field at $t = 0$ and $q= 1$
  14. The Mackey analogy for $q$-deformed $\mathrm{SL}(2,\mathbb{R})$
  15. The Mackey embedding and bijection
  16. ...and 1 more sections

Key Result

Proposition 1.1

There exists a unique continuous field of C*-algebras $f_\ast A$ over $Y$ such that

Theorems & Definitions (51)

  • Proposition 1.1
  • proof
  • Definition 1.2
  • Proposition 1.3
  • proof
  • Lemma 1.4
  • proof
  • Theorem 1.5
  • Lemma 1.6
  • proof
  • ...and 41 more