##
Normal forms for linear Hamiltonian vector fields commuting with the action
of a compact Lie group

* Math. Proc. Camb. Philos. Soc. * **114** (1993) 235-268

** Ian Melbourne and Michael Dellnitz **

** Abstract **

We obtain normal forms for infinitesimally symplectic matrices (or linear
Hamiltonian vector fields) that commute
with the symplectic action of a compact Lie group of symmetries.
In doing so we extend Williamson's theorem on normal forms
when there is no symmetry present.

Using standard representation-theoretic results the symmetry can be factored
out and we reduce to finding normal forms over a real division ring.
There are three real division rings consisting of the real, complex and
quaternionic numbers. Of these, only the real case is covered in Williamson's
original work.

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