Symplectic structures on locally compact abelian groups and polarizations
type: Item
instance of: scholarly article
title: Symplectic structures on locally compact abelian groups and polarizations
zbMATH Open document ID: 0823.22006
DOI: 10.1007/BF02830885
author: R. Ranga Rao
published in: Proceedings of the Indian Academy of Sciences. Mathematical Sciences
publication date: 1995-10-22T00:00:00Z
review text: The goal of the note is the study of a family of locally compact abelian groups for which a nondegenerate symplectic structure admits a polarization. Let \(\mathcal L\) denote the class of locally compact Hausdorff, abelian and second countable groups. Let \({\mathcal L}_ 0\) be the class of groups \(X \in {\mathcal L}\) possessing the following properties: (i) The maximal compact connected subgroups of both \(X\) and its dual \(X^*\) are tori; (ii) \(\dim_{Q_ p} \text{hom}(X, Q_ p) < \infty\) for all primes \(p\); (iii) The subgroup \(\{x \in X : p.x = 0\}\) is finite for each prime \(p\). The main result of the present note is Theorem 1. If \(X \in {\mathcal L}_ 0\) and \(\omega\) is a nondegenerate symplectic structure on \(X\), then \((X,\omega)\) admits a polarization. Mention must be made of the fact that it is not known whether, for a general \(X \in {\mathcal L}\), a nondegenerate symplectic structure always admits a polarization.
Mathematics Subject Classification ID: 22B05
Mathematics Subject Classification ID: 53C15
Mathematics Subject Classification ID: 37J99
zbMATH DE Number: 562814
zbMATH Keywords: locally compact abelian groups
zbMATH Keywords: second countable groups
zbMATH Keywords: maximal compact connected subgroups
zbMATH Keywords: nondegenerate symplectic structures
zbMATH Keywords: polarizations
reviewed by: Olga Costinescu
MaRDI profile type: Publication
cites work: Q4748367
cites work: Sur les groupes topologiques localement compacts
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description: scientific article; zbMATH DE number 562814