Transitive Subgroups of Transvections Acting on Some Symplectic Symmetric Spaces of Ricci Type

TL;DR

This paper investigates symmetric symplectic spaces of Ricci type, identifying conditions under which their transvection groups contain simply transitive subgroups, which are often one-dimensional extensions of the Heisenberg group.

Contribution

It classifies when these symmetric symplectic spaces admit simply transitive subgroups of their transvection groups, expanding understanding of their geometric and algebraic structure.

Findings

Many symmetric symplectic spaces of Ricci type admit simply transitive subgroups.

Such subgroups are typically one-dimensional extensions of the Heisenberg group.

The paper provides criteria for the existence of these subgroups.

Abstract

Symmetric symplectic spaces of Ricci type are a class of symmetric symplectic spaces which can be entirely described by reduction of certain quadratic Hamiltonian systems in a symplectic vector space. We determine, in a large number of cases, if such a space admits a subgroup of its transvection group acting simply transitively. We observe that the simply transitive subgroups obtained are one dimensional extensions of the Heisenberg group.