Solution

Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 1 / 32A / d / Solution

For an anti-Hermitian matrix $A$, the real quadratic Hamiltonian
$$
I_A=\frac1{2i}\bar z^TAz
$$
satisfies
$$
\{I_A,I_B\}=-I_{[A,B]}.
$$
Thus $A\mapsto-I_A$ realizes the matrix commutator as a Poisson bracket; it is the quadratic moment-map construction for the unitary action.


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