Cayley Charts on the Unitary Matrix Group

A skew-Hermitian matrix has purely imaginary diagonal entries, giving $N$ real parameters. For each $i<j$, its $(i,j)$ entry is an arbitrary complex number and the $(j,i)$ entry is then forced to be its negative conjugate, giving two real parameters. Thus $\mathfrak{u}(N)$ has real dimension $N+2\left(\genfrac{}{}{0px}{}{N}{2}\right)=N^2$.

If $(I-X)v=0$ for $X^{\ast }=-X$, then $Xv=v$. But $v^{\ast }Xv$ is purely imaginary because $X^{\ast }=-X$, while this equality gives $v^{\ast }Xv=v^{\ast }v>0$ for nonzero $v$. Hence $I-X$ is invertible. Direct multiplication, using that $I-X$ and $I+X$ commute, shows $(I+X)(I-X{)}^{-1}$ is unitary.

On the open set of unitary matrices $Q$ near $I$ where $Q+I$ is invertible, the inverse formula is $X=(Q-I)(Q+I{)}^{-1}$. The factors are polynomials in $Q$, so they commute; using $Q^{\ast }=Q^{-1}$ gives $X^{\ast }=-X$. The forward and inverse formulas are smooth rational matrix maps, and therefore define a chart of dimension $N^2$ near $I$.

Left translation by any fixed $Q_0\in U(N)$ sends a Cayley neighborhood of $I$ to a chart neighborhood of $Q_0$, so these translates cover $U(N)$. The group is closed, since $Q^{\ast }Q=I$ is a closed matrix equation, and bounded, since every column has norm one. Identifying $M_N(\mathbb{C})$ with ${\mathbb{R}}^{2N^2}$, [F1] makes $U(N)$ compact. Its subspace topology is Hausdorff and second-countable: the Euclidean metric restricts to a separating metric, and intersections with rational Euclidean balls form a countable base. On overlaps, translated Cayley transition maps are compositions of the forward and inverse rational formulas from step 3.1 with fixed matrix multiplication; matrix inversion is smooth on its open domain, so the transitions are smooth. Thus a finite subcover of these neighborhoods is a finite smooth atlas on a topological $N^2$-manifold, and [F2] supplies its smooth structure.

The Frobenius inner product is left invariant because $\mathrm{Re}\mathrm{tr}((Q_{0}A{)}^{\ast }{Q}_{0}B)=\mathrm{Re}\mathrm{tr}(A^{\ast }B)$. By [F3] it defines a Riemannian metric; [F4] and [F5] give its global positive smooth volume density. In each of the finitely many charts from step 4.1, take all rational coordinate balls whose closed balls lie in the chart domain. These balls form a countable cover without making pointwise choices. Compactness gives a finite subcover. On each selected closed ball the smooth density coefficient is bounded, so its integral is finite; every nonempty coordinate ball has positive integral. The total smooth volume is therefore finite and positive, so it can be normalized.

Under [A1], [F6] and [F7] make the associated Borel measure a regular left-invariant Radon measure; the normalization from step 5.1 makes it a probability. In each chart the density coefficient is smooth and strictly positive, so a Borel set is ${\mu }_F$-null exactly when its coordinate image is Lebesgue-null.

Under [A2], this normalized left-invariant probability on the compact group agrees with normalized Haar probability by uniqueness in [F8]. This proves only the optional Haar branch; the volume-null statement was proved in step 6.1.