====== Freely modified bistochastic quantum group ======

The **freely modified bistochastic quantum groups** are the elements of a sequence $(B_N^{\#+})_{N\in \N}$ of [[compact matrix quantum group|compact matrix quantum groups]] introduced by Banica and Speicher in [(:ref:BanSp09)], although originally under the names $(B_N^{\prime +})_{N\in \N}$. Each $B_N^{\#+}$ can be seen as a [[free_orthogonal_easy_quantum_group|free]] counterpart of the [[bistochastic group]] $B_N$ of the corresponding dimension $N$. However, differently from the other matrix groups, $B_N$ actually has two free counterparts, the second being the [[free modified bistochastic quantum group]] $B_N^{\prime+}$, defined by Weber in [(:ref:Web12)]. 

===== Definition =====
Given $N\in \N$, the **freely modified bistochastic quantum group** $B_N^{\# +}$ is the [[compact matrix quantum group]] $(C(B_N^{\# +}),u)$ 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]] 
$$C(B_N^{\# +})\colon\hspace{-0.66em}= C^\ast_1\big\langle\{u_{i,j}\}_{i,j=1}^N\big\,\vert \,u=\overline u,\, uu^t=u^tu=I_N\otimes 1, \, \forall_{i,j=1}^n:{\textstyle\sum_{k=1}^N} u_{i,k}={\textstyle\sum_{l=1}^N} u_{l,j}\big\rangle,$$
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 can also be expressed by saying that the fundamental corpresentation matrix $u$ of $B_N^{\# +}$ is **bistochastic#**.

Note that $B_N^{\#+}$ is designated by $B_N^{\prime +}$ in [(:ref:BanSp09)]. It was later renamed by Weber in  [(:ref:Web12)] after the discovery of the discovery of the easy quantum group which is now called [[free modified bistochastic quantum group]] and commonly given the symbol $B_N^{\prime +}$ instead.

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

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

If $I$ denotes the closed two-sided ideal of $C(B_N^{\#+})$ 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 $C(B_N^{\# +})/I$ is isomorphic to the $C^\ast$-algebra $C(B_N')$ of continuous functions on the [[modified bistochastic group]] $B_N'$, the subgroup $\{\pm M\,\vert\, M\in B_N\}$ of $\mathrm{GL}(N,\C)$ given by signed [[wp>doubly stochastic matrix|bistochastic matrices]]. Hence, $B_N^{\# +}$ is a compact quantum supergroup of $B_N'$.

Moreover, if $J$ is the closed two-sided ideal of $C(B_N^{\#+})$ generated by the relations $u_{i,j}r=ru_{i,j}$ for any $i,j=1,\ldots,N$, where $r\hspace{-0.66em}\colon= \sum_{k=1}^Nu_{1,k}$, then $C(B_N^{\# +})/J$ is isomorphic to $C(B_N^{\prime +})$, the algebra of the [[free modified bistochastic quantum group]] $B_N^{\prime +}$. Thus, in particular, $B_N^{\#+}$ is a compact quantum supergroup of $B_N^{\prime+}$.


The freely modified bistochastic quantum groups $(B_N^{\# +})_{N\in \N}$ 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 all non-crossing partitions of even size with small blocks and even distances between legs]] that induces the corepresentation categories of $(B_N^{\#+})_{N\in \N}$. The partition $\singleton\otimes\singleton$ is its canonical generator. 


===== 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:Web12 >>
author:  Weber, Moritz
title:   On the classification of easy quantum groups
year:    2013
journal: Advances in Mathematics
volume:  245
pages:   500--533
url:     https://doi.org/10.1016/j.aim.2013.06.019
archivePrefix: arXiv
eprint   :1201.4723v2
)]