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

Published online by Cambridge University Press:  24 October 2008

Ian Melbourne
Affiliation:
Department of Mathematics, University of Houston, Houston, Texas 77204-3476, U.S.A.
Michael Dellnitz
Affiliation:
Institut für Angewandte Mathematik, Universität Hamburg, D-W-200 Hamburg 13, Germany

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.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 1993

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References

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