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Normal forms of involutive complex Hamiltonian matrices under the real symplectic group. (English) Zbl 0932.15006

Let \(S=S_1+iS_2\) be a Hamiltonian endomorphism of a \(2n\)-dimensional complex vector space, such that \(S^2=-I_{2n}\). The authors are interested in finding normal forms for the decomplexifications \(S_1, S_2\) of \(S\), under the conjugation with respect to the real symplectic Lie group \(Sp(n,{\mathbb R})\). They use some symplectic bases in the real symplectic space \(V\) and find three normal forms for the matrices of the Hamiltonian endomorphisms \(S_1,S_2\) in terms of three typical matrices \(N_n(\mu)\), \(Q_n\), \(\widetilde N_{2n}(\mu)\). The matrix \(N_n(\mu)\in {\mathcal M}_n({\mathbb R})\) is the usual Jordan block with \(\mu \in {\mathbb R} \) on the principal diagonal, the nonzero entries in \(Q_n\in {\mathcal M}_n({\mathbb R})\) are \(1\) on the second diagonal and \(\widetilde N_{2n}(\mu)\in {\mathcal M}_{2n}({\mathbb R})\) is the decomplexification of the Jordan block corresponding to \(\mu =\operatorname{Re} \mu+i \operatorname{Im} \mu\in {\mathbb C}\). The authors show that the pseudosymplectic bases are also useful in obtaining the normal forms of the corresponding \(S\).
Reviewer: V.Oproiu (Iaşi)

MSC:

15A21 Canonical forms, reductions, classification
22E60 Lie algebras of Lie groups
15B57 Hermitian, skew-Hermitian, and related matrices
37J05 Relations of dynamical systems with symplectic geometry and topology (MSC2010)
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