====== Free unitary quantum group ======

The **free unitary quantum groups** are the members of a sequence $(U_N^+)_{N\in \N}$ of [[compact matrix quantum group|compact matrix quantum groups]] introduced by Wang in [(:ref:Wang95)], Example 4.2. Each $U_N^+$ is a [[free_unitary_easy_quantum_group|free]] counterpart of the [[unitary group]] $U_N$ of the corresponding dimension $N$. 

===== Definition =====
Given $N\in \N$, the **free unitary quantum group** $U_N^+$ is the [[compact matrix quantum group]] $(C(U_N^+),u)$ 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]] 
$$C(U_N^+)\colon\hspace{-0.66em}= C^\ast_1\big\langle\{u_{i,j}\}_{i,j=1}^N\big\,\vert\, uu^\ast=u^\ast u=I_N\otimes 1\big\rangle,$$
where $u^\ast=(u^\ast_{j,i})_{i,j=1}^N$ is the complex conjugate transpose of $u=(u_{i,j})_{i,j=1}^N$, 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 $U_N^+$ is **unitary**.

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

If $I$ denotes the closed two-sided ideal of $C(U_N^+)$ 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 $C(U_N^+)/I$ is isomorphic to the $C^\ast$-algebra $C(U_N)$ of continuous functions on the [[unitary group]] $U_N$, the subgroup of $\mathrm{GL}(N,\C)$ given by all [[wp>unitary matrix|unitary matrices]]. Hence, $U_N^+$ is a compact quantum supergroup of $U_N$.

The free unitary quantum groups $(U_N^+)_{N\in \N}$ are a [[unitary_easy_quantum_group|(unitary) easy]] family of compact matrix quantum groups; i.e., the intertwiner spaces of their corepresentation categories are induced by a [[categories of two-colored partitions|category of (two-colored) partitions]]. More precisely, it is the [[category of non-crossing two-colored pair partitions with neutral blocks]] that induces the corepresentation categories of $(U_N^+)_{N\in \N}$. Its canonical generating set of partitions is $\emptyset$. 


===== Representation theory =====


===== Cohomology =====


===== Related quantum groups =====

===== References =====


[( :ref:Wang95 >>
author   : Shuzhou Wang
title    : Free products of compact quantum groups
journal  : Communications in Mathematical Physics
year     : 1995
volume   : 167
number   : 3
pages    : 671--692
url      : http://dx.doi.org/10.1007/BF02101540
)]