Quantum groups, from a functional analysis perspective

Abstract

It is well-known that any compact Lie group appears as closed subgroup of a unitary group, G ⊂ U N . The unitary group U N has a free analogue U + N , and the study of the closed quantum subgroups G ⊂ U + N is a problem of general interest. We review here the basic tools for dealing with such quantum groups, with all the needed preliminaries included, and we discuss as well a number of more advanced topics.2010 Mathematics Subject Classification. 46L65.

Quantum groups under very strong axioms

“…At a more advanced level, this condition appeared in [11], in connection with the Bercovici-Pata bijection [13] for the asymptotic laws of truncated characters, and also in [1], [10], in connection with various noncommutative geometry questions. See [3].…”

“…Classical face, easy slicing case. As explained in [3], when imposing the slicing condition on the lower face, which comes from the general noncommutative geometry considerations in [2], [6], some simplifications appear as well, the solutions being as follows:…”

Higher orbitals of quizzy quantum group actions

“…In this section we discuss the representation theory ofŌ N , and its connection with the representation theory of the hyperoctahedral group H N , and with some other related quantum groups. We will heavily rely on the Tannakian techniques introduced by Woronowicz in [36], further explained in [26], [28], and in the lecture notes [4].…”

“…In what follows we will rely on the proof from [9] of this result, with categorical input coming from [26]. See [4].…”

“…As explained in [9], there are only 4 such liberations, namely those of the groups O N , B N , S N , H N . We refer as well to [4] for this material.…”