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.