Related papers: Easy quantum groups
Given a closed subgroup $G\subset U_N^+$ which is homogeneous, in the sense that we have $S_N\subset G\subset U_N^+$, the corresponding Tannakian category $C$ must satisfy $span(\mathcal{NC}_2)\subset C\subset span(P)$. Based on this…
Easy quantum groups have been studied intensively since the time they were introduced by Banica and Speicher in 2009. They arise as a subclass of ($C^*$-algebraic) compact matrix quantum groups in the sense of Woronowicz. Due to some…
This is an introduction to the quantum groups, or rather to the simplest quantum groups. The idea is that the unitary group $U_N$ has a free analogue $U_N^+$, whose standard coordinates $u_{ij}\in C(U_N^+)$ are allowed to be free, and the…
It is well-known that any compact Lie group appears as closed subgroup of a unitary group, $G\subset U_N$. The unitary group $U_N$ has a free analogue $U_N^+$, and the study of the closed quantum subgroups $G\subset U_N^+$ is a problem of…
We discuss the classification problem for the unitary easy quantum groups, under strong axioms, of noncommutative geometric nature. Our main results concern the intermediate easy quantum groups $O_N\subset G\subset U_N^+$. To any such…
The notion of simple compact quantum group is introduced. As non-trivial (noncommutative and noncocommutative) examples, the following families of compact quantum groups are shown to be simple: (a) The universal quantum groups $B_u(Q)$ for…
We propose a definition of partition quantum spaces. They are given by universal $C^*$-algebras whose relations come from partitions of sets. We ask for the maximal compact matrix quantum group acting on them. We show how those fit into the…
In 2009, Banica and Speicher began to study the compact quantum subgroups of the free orthogonal quantum group containing the symmetric group S_n. They focused on those whose intertwiner spaces are induced by some partitions. These…
We study the intermediate quantum groups $H_N\subset G\subset U_N^+$. The basic examples are $H_N,K_N,O_N,U_N,H_N^+,K_N^+,O_N^+,U_N^+$, which form a cube. Any other example $G$ sits inside the cube, and by using standard operations, namely…
We classify all compact quantum groups whose C*-algebra sits inside that of the free unitary quantum groups $U_{N}^{+}$. In other words, we classify all discrete quantum subgroups of $\widehat{U}_{N}^{+}$, thereby proving a quantum variant…
We consider compact matrix quantum groups whose fundamental corepresentation matrix has entries which are partial isometries with central support. We show that such quantum groups have a simple representation as semi-direct product quantum…
We investigate the "two-parameter" quantum symmetry groups that we previously constructed with Skalski, with the conclusion that some of these quantum groups, namely those without singletons, are "super-easy" in a suitable sense, that we…
Set partitions closed under certain operations form a tensor category. They give rise to certain subgroups of the free orthogonal quantum group $O_n^+$, the so called easy quantum groups, introduced by Banica and Speicher in 2009. This…
We discuss an extended easy quantum group formalism, with the Schur-Weyl theoretic Kronecker symbols being as general as possible, and allowed to take values in $\{-1,0,1\}$, and more generally in $\mathbb T\cup\{0\}$. Our study includes an…
We define new compact matrix quantum groups whose intertwiner spaces are dual to tensor categories of three-dimensional set partitions -- which we call spatial partitions. This extends substantially Banica and Speicher's approach of the so…
We introduce the notion of confined subalgebras in the context of the group von Neumann algebra. We also define Uniformly Recurrent States -- an operator-algebraic analog of Uniformly Recurrent Subgroups. Using this framework, we show that…
Easy quantum groups are compact matrix quantum groups, whose intertwiner spaces are given by the combinatorics of categories of partitions. This class contains the symmetric group and the orthogonal group as well as Wang's quantum…
We review the notion of simple compact quantum groups and examples, and discuss the problem of construction and classification of simple compact quantum groups. Several new quantum groups constructed by Banica, Curran and Speicher since the…
We propose a new approach to the half-liberation question, for the compact groups $T_N\subset G_N\subset U_N$, where $T_N=\mathbb Z_2^N$. Indeed, we can construct a quantum group $T_N^*\subset G_N^*\subset U_N^*$, simply by setting…
We study the orthogonal quantum groups satisfying the ``easiness'' assumption axiomatized in our previous paper, with the construction of some new examples, and with some partial classification results. The conjectural conclusion is that…