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Deformations and q-convolutions. Old and new results

Marek Bozejko, Wojciech Bozejko

Abstract

This paper is the survey of some of our results related to $q$-deformations of the Fock spaces and related to $q$-convolutions for probability measures on the real line $\mathbb{R}$. The main idea is done by the combinatorics of moments of the measures and related $q$-cumulants of different types. The main and interesting $q$-convolutions are related to classical continuous (discrete) $q$-Hermite polynomial. Among them are classical ($q=1$) convolutions, the case $q=0$, gives the free and Boolean relations, and the new class of $q$-analogue of classical convolutions done by Carnovole, Koornwinder, Biane, Anshelovich, and Kula. The paper contains many questions and problems related to the positivity of that class of $q$-convolutions. The main result is the construction of Brownian motion related to $q$-Discrete Hermite polynomial of type I.

Deformations and q-convolutions. Old and new results

Abstract

This paper is the survey of some of our results related to -deformations of the Fock spaces and related to -convolutions for probability measures on the real line . The main idea is done by the combinatorics of moments of the measures and related -cumulants of different types. The main and interesting -convolutions are related to classical continuous (discrete) -Hermite polynomial. Among them are classical () convolutions, the case , gives the free and Boolean relations, and the new class of -analogue of classical convolutions done by Carnovole, Koornwinder, Biane, Anshelovich, and Kula. The paper contains many questions and problems related to the positivity of that class of -convolutions. The main result is the construction of Brownian motion related to -Discrete Hermite polynomial of type I.
Paper Structure (8 sections, 10 theorems, 65 equations, 1 figure)

This paper contains 8 sections, 10 theorems, 65 equations, 1 figure.

Table of Contents

  1. Introduction
  2. $q$-CCR(CAR) for $|q|>1$, and q-continuous Hermite polynomials
  3. Combinatorial results on $P_2(2n)$
  4. $q$-Discrete Hermite polynomials, $0 \leq q \leq 1$
  5. Another q-analogues of classical convolutions
  6. Braided Hopf algebras of Kempf and Majid
  7. The construction of $q$-Discrete Fock space $\mathcal{F}_q^{disc}(\mathcal{H})$ and $q$-Discrete Brownian motions
  8. Matrix version of Khintchine inequalities

Key Result

Theorem 1

If $-1 \leq q \leq 1$, $s>0$, then there exist operators $A^\pm(f)=A^\pm_{q,s}(f)$, $g,f \in \mathbb{R}^N$, $N=\infty,1,2,\ldots$: where $\mathcal{H}^{\otimes 0} = \mathbb{C}\Omega$.

Figures (1)

  • Figure 1: Crossings

Theorems & Definitions (15)

  • Theorem 1: BY
  • Definition 1: q-conditions cummulants - Ph.Biane, M.Anshelevich
  • Remark 1
  • Theorem 2: Bożejko--YoshidaBY (Wick formula)
  • Theorem 3: Bozejko2007
  • Lemma 1
  • Theorem 4: Bozejko2007
  • Theorem 5
  • Theorem 6: Carnovale, Koornwinder CK
  • Definition 2
  • ...and 5 more