nLab
special unitary group

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 $\mathrm{SU}(n)$ 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 $U(n)$ consisting of the $n\times n$ unitary matrices with determinant $1$.

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