====== Unitary group ======

A **unitary group** is any member of a sequence $(U_N)_{N\in \N}$ of [[classical unitary matrix groups|classical matrix groups]].

===== Definition =====

For every $N\in \N$ the **unitary group** for dimension $N$ is the subgroup of the [[wp>general linear group]] $\mathrm{GL}(N,\C)$ given by all //unitary// $N\times N$-matrices, i.e., the set
$$U_N\colon\hspace{-0.6em}=\{ u\in \C^{N\times N}\,\vert\, uu^\ast=u^\ast u=I\},$$
where, if $u=(u_{i,j})_{i,j=1}^N$, then $u^\ast=(\overline{u}_{j,i})_{i,j=1}^N$ is the complex conjugate transpose of $u$ and where $I_N$ is the identity $N\!\times \!N$-matrix.


===== Basic properties =====

The unitary 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 two-colored pair partitions with neutral blocks]] that induces the corepresentation categories of $(U_N)_{N\in \N}$. Its canonical generating partition is the //crossing partition// $\Partition{\Pline (1,0) (2,1) \Pline (2,0) (1,1) \Ppoint 0 \Pw:1,2 \Ppoint 1 \Pw:1,2}$. 

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

===== Cohomology =====

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

===== References =====