Locally compact abelian groups with symplectic self-duality
Institute of Mathematical Sciences, Chennai.
A symplectic self-duality of a locally compact abelian group L is an isomorphism f from L onto its Pontryagin dual such that f(x)(x)=0 for all x in L. Groups admitting a symplectic self-duality arise in the context of Heisenberg groups.
The following question has been open so far: If L admits a symplectic self-duality then does L have to be isomorphic to the product of a locally compact abelian group and its Pontryagin dual?
1. We describe several classes of groups for which this question has an affirmative answer (covering all typical applications). 2. We study extensions of finite abelian groups to construct a topological torsion group which admits a symplectic self-duality but is not isomorphic to the product of a locally compact abelian group with its Pontryagin dual.