## Linear Automorphisms that are Symplectomorphisms

### Authors:

- Stanisław Janeczko,
- Zbigniew Jelonek

### Abstract

Let K be the field of real or complex numbers. Let (X ≅ K2n, ω) be a symplectic vector space and take 0 \\textless\ k \\textless\ n,N = . Let L1,…,LN ⊂ X be 2k-dimensional linear subspaces which are in a sufficiently general position. It is shown that if F : X → X is a linear automorphism which preserves the form ωk on all subspaces L1,…,LN, then F is an εk-symplectomorphism (that is, F*ω = εkω, where ). In particular, if K = R and k is odd then F must be a symplectomorphism. The unitary version of this theorem is proved as well. It is also observed that the set Al,2r of all l-dimensional linear subspaces on which the form ω has rank ≤ 2r is linear in the Grassmannian G(l,2n), that is, there is a linear subspace L such that Al,2r = L ∩ G(l, 2n). In particular, the set Al,2r can be computed effectively. Finally, the notion of symplectic volume is introduced and it is proved that it is another strong invariant.

- Record ID
- WUT118567
- Author
- Journal series
- Journal of the London Mathematical Society, ISSN 0024-6107, 1469-7750
- Issue year
- 2004
- Vol
- 69
- No
- 2
- Pages
- 503-517
- DOI
- DOI:10.1112/S0024610703004952 Opening in a new tab
- URL
- http://jlms.oxfordjournals.org/content/69/2/503 Opening in a new tab
- Score (nominal)
- 0
- Score source
- journalList
- Publication indicators
- = 5; = 5; = 6
- Citation count
- 6

- Uniform Resource Identifier
- https://repo.pw.edu.pl/info/article/WUT118567/

- URN
`urn:pw-repo:WUT118567`

* presented citation count is obtained through Internet information analysis and it is close to the number calculated by the Publish or PerishOpening in a new tab system.