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unitary_group

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Unitary group

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

Definition

For every $N\in \N$ the unitary group for dimension $N$ is the subgroup of the 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 family of compact matrix quantum groups; i.e., the intertwiner spaces of their corepresentation categories are induced by a 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

References

unitary_group.txt · Last modified: 2021/11/23 11:56 (external edit)

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