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Gaussian Generating functionals on easy quantum groups

Uwe Franz, Amaury Freslon, Adam Skalski

Abstract

We describe all Gaussian generating functionals on several easy quantum groups given by non-crossing partitions. This includes in particular the free unitary, orthogonal and symplectic quantum groups. We further characterize central Gaussian generating functionals and describe a centralization procedure yielding interesting (non-Gaussian) generating functionals.

Gaussian Generating functionals on easy quantum groups

Abstract

We describe all Gaussian generating functionals on several easy quantum groups given by non-crossing partitions. This includes in particular the free unitary, orthogonal and symplectic quantum groups. We further characterize central Gaussian generating functionals and describe a centralization procedure yielding interesting (non-Gaussian) generating functionals.

Paper Structure

This paper contains 9 sections, 21 theorems, 102 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Free unitary quantum groups
  4. Free orthogonal quantum groups
  5. The standard case
  6. The symplectic case
  7. Applications to central functionals
  8. Central Gaussian processes
  9. Centralizing Gaussian processes

Key Result

Theorem 1

Every Gaussian generating functional $\phi$ on the free unitary quantum group $U_{N}^{+}$ admits a unique decomposition into the sum of a "drift" part $D_{H}$ determined by an anti-hermitian matrix $H\in M_{N}(\mathbb{C})$ and a "diffusion" part $\Gamma_{W}$ determined by a matrix $W \in M_{N}(\math

Theorems & Definitions (54)

  • Theorem
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.4
  • Definition 2.5
  • Remark
  • Proposition 2.6
  • proof
  • Lemma 2.7
  • ...and 44 more