Alternate modules are subsymplectic

TL;DR

This paper introduces the concept of alternate modules as finite abelian groups with a bilinear form and proves that all such modules are subsymplectic, meaning they can be embedded into a standard symplectic module.

Contribution

It establishes that every alternate module can be embedded into a standard symplectic module, generalizing the structure of these modules.

Findings

Any alternate module is subsymplectic.

Existence of a Lagrangian of a given size implies embedding into a standard symplectic module.

Provides a structural classification of alternate modules.

Abstract

In this paper, an alternate module $(A,\phi)$ is a finite abelian group $A$ with a $\mathbb{Z}$-bilinear application $\phi:A\times A\rightarrow \mathbb{Q}/\mathbb{Z}$ which is alternate (i.e. zero on the diagonal). We shall prove that any alternate module is subsymplectic, i.e. if $(A,\phi)$ has a Lagrangian of cardinal $n$ then there exists an abelian group $B$ of order $n$ such that $(A,\phi)$ is a submodule of the standard symplectic module $B\times B^*$.