The moduli spaces of presymplectic forms on almost abelian Lie algebras

TL;DR

This paper characterizes the existence and structure of presymplectic forms on almost abelian Lie algebras, revealing that their moduli space of symplectic forms is finite and structurally simple.

Contribution

It provides necessary and sufficient conditions for presymplectic forms and shows the finiteness and canonical form classification of the moduli space of symplectic forms.

Findings

Moduli space of symplectic forms is finite for all almost abelian Lie algebras.

All symplectic forms are permutations of a canonical 2-form.

Canonical representatives for matrix congruences are established.

Abstract

We obtain necessary and sufficient conditions to determine the existence of presymplectic forms of a given rank on all almost abelian Lie algebras. We also study the moduli space of presymplectic forms (this is the set of all closed 2-forms of a given rank under a certain natural equivalence relation) on almost abelian Lie algebras. Most importantly we show that for any almost abelian Lie algebra its moduli space of symplectic forms is finite. Moreover we show that up to such natural equivalence all symplectic forms are permutations of a canonical 2-form. The important step in the proof is obtaining canonical representatives for a certain congruence of matrices, which is of some interest for matrix theory on its own.