# nLab special unitary group

group theory

### Cohomology and Extensions

The special unitary group is the subgroup of the unitary group on the elements with determinant equal to 1.

For $n$ a natural number, the special unitary group $\mathrm{SU}\left(n\right)$ is the group of isometries of the $n$-dimensional complex Hilbert space ${ℂ}^{n}$ which preserve the volume form on this space. It is the subgroup of the unitary group $U\left(n\right)$ consisting of the $n×n$ unitary matrices with determinant $1$.

More generally, for $V$ any complex vector space equipped with a nondegenerate Hermitian form? $Q$, $\mathrm{SU}\left(V,Q\right)$ is the group of isometries of $V$ which preserve the volume form derived from $Q$. One may write $\mathrm{SU}\left(V\right)$ if $Q$ is obvious, so that $\mathrm{SU}\left(n\right)$ is the same as $\mathrm{SU}\left({ℂ}^{n}\right)$. By $\mathrm{SU}\left(p,q\right)$, we mean $\mathrm{SU}\left({ℂ}^{p+q},Q\right)$, where $Q$ has $p$ positive eigenvalue?s and $q$ negative ones.

Revised on April 18, 2011 13:46:05 by Urs Schreiber (89.204.137.107)