COMPACT QUANTUM GROUPS GENERATED BY THEIR TORI

Associated to any closed subgroup $G\subset U_N^+$ is a family of toral subgroups $T_Q\subset G$, indexed by the unitary matrices $Q\in U_N$. The family $\{T_Q|Q\in U_N\}$ is expected to encode the main properties of $G$, and there are several conjectures in this sense. We verify here the generation conjecture, $G=$, for various classes of compact quantum groups. Our results generalize the previously known facts on the subject

COMPLEX HADAMARD MATRICES AND APPLICATIONS

A complex Hadamard matrix is a square matrix H ∈ M N (C) whose entries are on the unit circle, |H ij | = 1, and whose rows and pairwise orthogonal. The main example is the Fourier matrix, F N = (w ij) with w = e 2πi/N. We discuss here the basic theory of such matrices, with emphasis on geometric and analytic aspects. CONTENT

Methods of free probability

This is a joint introduction to classical and free probability, which are twin sisters. We first review the foundations of classical probability, notably with the main limiting theorems (CLT, CCLT, PLT, CPLT), and with a look into examples coming from Lie groups and random matrices. Then we present the foundations and main results of free probability, notably with free limiting theorems, and with a look into examples coming from quantum groups and random matrices. We discuss then a number of more advanced aspects, in relation on one hand with free geometry, and on the other hand with questions in operator algebras coming from subfactor theory.Comment: 400 pages. arXiv admin note: text overlap with arXiv:2208.0360