The generalized Heisenberg group arising from Weyl relations

Abstract

In this paper, we introduce the generalized Weyl operators canonically associated with the one-mode oscillator Lie algebra as unitary operators acting on the bosonic Fock space \(\Gamma ({\mathbb {C}})\). Next, we establish the generalized Weyl relations and deduce a group structure on the manifold \({\mathbb {R}}^2\times [-\pi , \pi [\times {\mathbb {R}}\) generalizing the well-known Heisenberg one in a natural way.