Noncommutative Independence in the Infinite Braid and Symmetric Group

Traffic Distributions and Independence II: Universal Constructions for Traffic Spaces

We investigate questions related to the notion of traffics introduced by the author C. Male as a noncommutative probability space with numerous additional operations and equipped with the notion of traffic independence. We prove that any sequence of unitarily invariant random matrices that converges in noncommutative distribution converges in distribution of traffics whenever it fulfills some factorization property. We provide an explicit description of the limit which allows to recover and extend some applications (on the freeness from the transposed ensembles by Mingo and Popa and the freene…

A transference method in quantum probability

Working with a rather general notion of independence, we provide a transference method which allows to compare the p-norm of sums of independent copies with the p-norm of sums of free copies. Our main technique is to construct explicit operator space Lp embeddings preserving independence to reduce the problem to L1, where some recent results by the first-named author can be used. We find applications for noncommutative Khincthine/Rosenthal type inequalities and for noncommutative Lp embedding theory.

循环单调独立与单调独立的矩阵模型

Matrix models for cyclic monotone and monotone independences

本文证明同一随机矩阵模型在适当状态下同时作为循环单调独立和单调独立的模型，从而为两种独立概念提供统一框架。

General de Finetti type theorems in noncommutative probability

We prove general de Finetti type theorems for classical and free independence. The de Finetti type theorems work for all non-easy quantum groups, which generalize a recent work of Banica, Curran and Speicher. We determine maximal distributional symmetries which means the corresponding de Finetti type theorem fails if a sequence of random variables satisfy more symmetry relations other than the maximal one. In addition, we define Boolean quantum semigroups in analogous to the easy quantum groups, by universal conditions on matrix coordinate generators and an orthogonal projection. Then, we show…

De Finetti Theorems for Braided Parafermions

The classical de Finetti theorem in probability theory relates symmetry under the permutation group with the independence of random variables. This result has application in quantum information. Here we study states that are invariant with respect to a natural action of the braid group, and we emphasize the pictorial formulation and interpretation of our results. We prove a new type of de Finetti theorem for the four-string, double-braid group acting on the parafermion algebra to braid qudits, a natural symmetry in the quon language for quantum information. We prove that a braid-invariant stat…

Quantum symmetries on noncommutative complex spheres with partial commutation relations

We introduce the notion of noncommutative complex spheres with partial commutation relations for the coordinates. We compute the corresponding quantum symmetry groups of these spheres, and this yields new quantum unitary groups with partial commutation relations. We also discuss some geometric aspects of the quantum orthogonal groups associated with the mixture of classical and free independence discovered by Speicher and Weber. We show that these quantum groups are quantum symmetry groups on some quantum spaces of spherical vectors with partial commutation relations.

C*-量子群的半直积：乘法酉算子方法

Semidirect products of C*-quantum groups: multiplicative unitaries approach

该研究针对带投影C*-量子群的分解问题，借助Radford定理的思路，在可管理乘法酉算子层面证明了带投影C*-量子群可唯一分解为普通与辫子C*-量子群的结论。

Quadratic independence of coordinate functions of certain homogeneous spaces and action of compact quantum groups

Let $G$ be one of the classical compact, simple, centre-less, connected Lie groups or rank $n$ with a maximal torus $T$, the Lie algebra $\clg$ and let $\{ E_i, F_i, H_i, i=1, \ldots, n \}$ be the standard set of generators corresponding to a basis of the root system. Consider the adjoint-orbit space $M=\{ {\rm Ad}_g(H_1),~g \in G \}$, identified with the homogeneous space $G/L$ where $L=\{ g \in G:~{\rm Ad}_g(H_1)=H_1\}$. We prove that the `coordinate functions' $\{ f_i, i=1, \ldots, n \}$, (where $f_i(g):=λ_i({\rm Ad}_g(H_1))$, $\{ λ_1, \ldots, λ_n\}$ is basis of $\clg^\prime$) are `quadrati…