====== Free modified symmetric quantum group ======
By a **free modified symmetric quantum group** one means any element of the one-parameter sequence $(S_N^{\prime +})_{N\in \N}$ of [[compact matrix quantum group|compact matrix quantum groups]] defined by Banica and Speicher in [(:ref:BanSp09)]. Each $S_N^{\prime+}$ is a [[free_orthogonal_easy_quantum_groups|free]] counterpart of the [[modified symmetric group]] $S_N'$ of the corresponding dimension $N$. 

===== Definition =====

Given $N\in \N$, the **free modified symmetric quantum group** $S_N^{\prime +}$ is the [[compact matrix quantum group]] $(C(S_N^{\prime+}),u)$ where $u=(u_{i,j})_{i,j=1}^N$ organizes the generators $\{u_{i,j}\}_{i,j=1}^N$ of the (unital) [[wp>Universal_C*-algebra|universal C*-algebra]] 
$$C(S_N^{\prime +})\colon\hspace{-0.66em}= C^\ast_1\big\langle\{u_{i,j}\}_{i,j=1}^N\big\,\vert \,u=\overline u,\, uu^t=u^tu=I_N\otimes 1,\,\forall_{i,j,k=1}^N:i\neq j\Rightarrow u_{i,k}u_{j,k}=u_{k,i}u_{k,j}=0,  \, {\textstyle\sum_{l=1}^N} u_{i,l}={\textstyle\sum_{l=1}^N} u_{l,j}\big\rangle,$$
where $\overline u=(u^\ast_{i,j})_{i,j=1}^N$ is the complex conjugate of $u$ and $u^t=(u_{j,i})_{i,j=1}^N$ the transpose, where $I_N$ is the identity $N\!\times\!N$-matrix and where $1$ is the unit of the universal $C^\ast$-algebra.

The definition of $S_N^{\prime+}$ is often summarized by saying that it is the compact $N\!\times\!N$-matrix quantum group whose fundamental corepresentation matrix $u$ is **magic'**.

===== Basic Properties =====

The fundamental corepresentation matrix $u$ of $S_N^{\prime +}$ is in particular //orthogonal//. Hence, $S_N^{\prime +}$ is a compact quantum subgroup of the [[free orthogonal quantum group]] $O_N^+$.

If $I$ denotes the closed two-sided ideal of $C(S_N^{\prime +})$ generated by the relations $u_{i,j}u_{k,l}=u_{k,l}u_{i,j}$ for any $i,j,k,l=1,\ldots, N$, then $C(S_N^{\prime +})/I$ is isomorphic to the $C^\ast$-algebra $C(S_N')$ of continuous functions on the [[modified symmetric group]] $S_N'$, the latter interpreted as the subgroup $\{\pm P\,\vert\, P\in S_N\}$ of $\mathrm{GL}(N,\C)$ given by signed [[wp>permutation matrix|permutation matrices]]. Hence, $S_N^{\prime +}$ is a compact quantum supergroup of $S_N'$.

The free modified symmetric quantum groups $(S_N^{\prime +})_{N\in \N}$ are an [[easy_quantum_group|easy]] family of compact matrix quantum groups, i.e., the intertwiner spaces of their corepresentation categories are induced by a [[category of partitions]]. More precisely, it is the [[category of non-crossing partitions of even size]] that induces the corepresentation categories of $(S_N^{\prime +})_{N\in \N}$. Its canonical generating set is $\{\fourpart,\singleton\otimes \singleton\}$. 
===== Representation theory =====
===== Cohomology =====
===== Related quantum groups =====

===== References =====


[( :ref:BanSp09 >>
author:  Banica, Teodor and Speicher, Roland
title:   Liberation of orthogonal Lie groups
year:    2009
journal: Advances in Mathematics
volume:  222
issue:   4
pages:   1461--150
url:     https://doi.org/10.1016/j.aim.2009.06.009
archivePrefix: arXiv
eprint   :0808.2628
)]

[( :ref:Wang98 >>
author   : Shuzhou Wang
title    : Quantum Symmetry Groups of Finite Spaces
journal  : Communications in Mathematical Physics
year     : 1998
volume   : 195
number   : 1
pages    : 195--211
url      : http://dx.doi.org/10.1007/s002200050385
archivePrefix: arXiv
eprint   :0707.3168
)]