= Codimension-one quantum dynamical algebra is special unitary
{title2=$\dim\mathfrak g=N^2-1\Longrightarrow\mathfrak g=\mathfrak{su}(N)$}
For $N\ge2$, let $\mathfrak g$ be a <Lie subalgebra> of $\mathfrak u(N)$ with codimension one. With inner product $-\operatorname{Tr}(XY)$ on skew-Hermitian matrices, its orthogonal complement is a line $\mathbb R Z$. Invariance of the inner product shows that $[X,Z]$ lies in that line for every $X\in\mathfrak g$; skew-adjointness of the adjoint action makes $[X,Z]$ orthogonal to $Z$, hence zero. A nonscalar skew-Hermitian $Z$ has a centralizer of dimension at most $(N-1)^2+1<N^2-1$, by its orthogonal eigenspace decomposition, and cannot centralize all of $\mathfrak g$. Thus $Z$ is scalar and $\mathfrak g=\mathfrak{su}(N)$. This gives <density operator controllability> and all determinant-one gates, but not the central phase direction needed for <unitary operator controllability>.
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