Müller, D.; Thiele, C. Normal forms of involutive complex Hamiltonian matrices under the real symplectic group. (English) Zbl 0932.15006 J. Reine Angew. Math. 513, 97-114 (1999). 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) Cited in 2 Documents 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) Keywords:complex Hamiltonian matrices; real symplectic group; normal form of a matrix; Lie group PDFBibTeX XMLCite \textit{D. Müller} and \textit{C. Thiele}, J. Reine Angew. Math. 513, 97--114 (1999; Zbl 0932.15006) Full Text: DOI