Symplectic modules over Colombeau-generalized numbers

TL;DR

This paper explores symplectic linear algebra over Colombeau generalized numbers, establishing foundational results like symplectic bases and analyzing eigenvalues of matrices, with applications in non-smooth symplectic geometry.

Contribution

It extends classical symplectic linear algebra results to the setting of Colombeau generalized numbers, including construction of symplectic bases and eigenvalue analysis for matrices.

Findings

Constructed symplectic bases for free modules over Colombeau numbers

Derived normal forms for Hermitian and skew-symmetric matrices

Applied results to non-smooth symplectic geometry and Fourier integral operators

Abstract

We study symplectic linear algebra over the ring $\Rt$ of Colombeau generalized numbers. Due to the algebraic properties of $\Rt$ 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 $\Rt$-module of finite rank. Further, we consider the general problem of eigenvalues for matrices over $\Kt$ ($\K=\R$ 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.