Locally compact abelian groups with symplectic self-duality

TL;DR

This paper investigates the structure of locally compact abelian groups with symplectic self-duality, providing conditions for their isomorphism to a product with their dual and exploring related subgroup structures.

Contribution

It offers new sufficient conditions for such groups to be isomorphic to a product with their dual and constructs counterexamples through subgroup analysis.

Findings

Identifies conditions under which groups are isomorphic to a product with their dual

Provides counterexamples to the general conjecture

Connects the structure of maximal isotropic subgroups to the main problem

Abstract

Is every locally compact abelian group which admits a symplectic self-duality isomorphic to the product of a locally compact abelian group and its Pontryagin dual? Several sufficient conditions, covering all the typical applications are found. Counterexamples are produced by studying a seemingly unrelated question about the structure of maximal isotropic subgroups of finite abelian groups with symplectic self-duality (where the original question always has an affirmative answer).