====== Orthogonal group ======

An **orthogonal group** is any member of a sequence $(O_N)_{N\in \N}$ of [[classical orthogonal matrix groups]].

===== Definition =====

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


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

The orthogonal groups $(O_N)_{N\in \N}$ 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 pair partitions]] that induces the corepresentation categories of $(O_N)_{N\in \N}$. Its canonical generating partition is $\crosspart$. 

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

===== Cohomology =====

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

===== References =====