Symplectic Modules Over Colombeau-Generalized Numbers

Abstract

We study symplectic linear algebra over the ring of Colombeau generalized numbers. Due to the algebraic properties of it is possible to preserve a number of central results of classical symplectic linear algebra. In particular, we construct symplectic bases for any symplectic form on a free -module of finite rank. Further, we consider the general problem of eigenvalues for matrices over (𝕂 = ℝ or ℂ) and derive normal forms for Hermitian and skew-symmetric matrices. Our investigations are motivated by applications in non-smooth symplectic geometry and the theory of Fourier integral operators with non-smooth symbols.

Key Words:

• Colombeau generalized numbers
• Generalized eigenvalues
• Symplectic linear algebra
• Symplectic modules

2010 Mathematics Subject Classification:

• 15A63
• 15A18
• 15A21
• 46F30

ACKNOWLEDGMENT

We thank an anonymous referee for several suggestions that have substantially improved the presentation.

Notes

For example, c = e _S with S ⊆ I such that and would do.

If μ(1 − 2r) = 0, then μ = 2rμ, and hence μr = 2μr ² = 2μr, i.e., μr = 0 and thus μ = 0.

Communicated by I. Shestakov.