Decomposition of phase space and classification of Heisenberg groups

TL;DR

This paper investigates the structure of Heisenberg groups over locally compact abelian groups, establishing conditions under which these groups decompose into simpler components and classifying their isomorphism types.

Contribution

It proves that certain locally compact abelian groups with Heisenberg extensions decompose into a product of the group and its dual, and classifies these Heisenberg groups up to isomorphism.

Findings

Affirmative answer for common abelian groups with Heisenberg extensions

Decomposition of these groups into a product of the group and its dual

Classification of Heisenberg groups by the isomorphism class of the abelian group

Abstract

Is every locally compact abelian group which admits a Heisenberg central extension isomorphic to the product of a locally compact abelian group and its Pontryagin dual? An affirmative answer is obtained for all the commonly occurring types of abelian groups having Heisenberg central extensions, including Lie groups and certain finite Cartesian products of local fields and adeles. Furthermore, for these types of groups, it is found that the isomorphism class of the abelian group determines the Heisenberg group up to isomorphism, thereby providing a classification of such Heisenberg groups.