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orthogonal_group

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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 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 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

References

orthogonal_group.1583832315.txt.gz · Last modified: 2021/11/23 11:56 (external edit)

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