Linear Automorphisms that are Symplectomorphisms |
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Authors: | Janeczko Stanislaw; Jelonek Zbigniew |
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Institution: | Instytut Matematyczny PAN ul. niadeckich 8, 00-950 Warszawa, Poland
Wydzia Matematyki i Nauk Informacyjnych, Politechnika Warszawska Pl. Politechniki 1, 00-661 Warszawa, Poland, janeczko{at}ise.pw.edu.pl
Instytut Matematyczny PAN Polska Akademia Nauk, w. Tomasza 30, 31-027 Kraków, Poland
Max Planck Institut für Mathematik Vivatsgasse 7, 53111 Bonn, Germany najelone{at}cyf-kr.edu.pl |
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Abstract: | Let K be the field of real or complex numbers. Let (X K2n,) be a symplectic vector space and take 0 < k < n,N =. Let L1,...,LN X be 2k-dimensionallinear subspaces which are in a sufficiently general position.It is shown that if F : X X is a linear automorphism whichpreserves the form k on all subspaces L1,...,LN, then F is ank-symplectomorphism (that is, F* = k, where ). In particular, if K = R and k is odd then F mustbe a symplectomorphism. The unitary version of this theoremis proved as well. It is also observed that the set Al,2r ofall l-dimensional linear subspaces on which the form has rank 2r is linear in the Grassmannian G(l,2n), that is, there isa linear subspace L such that Al,2r = L G(l, 2n). In particular,the set Al,2r can be computed effectively. Finally, the notionof symplectic volume is introduced and it is proved that itis another strong invariant. |
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