For every finite generating set on the integer Heisenberg group H(Z), we know from a fundamental result of Pansu on nilpotent groups that the wordmetric has the large-scale structure of a Carnot-Carathéodory Finsler metric on the real Heisenberg group H(R). We study the properties of those limit metrics and obtain results about the geometry of word metrics that reflect the dependence on generat...

1. Statement of results and outline of method. The purpose of this note is to announce results dealing with convolution operators on the Heisenberg group. As opposed to the well-known situation where the kernels are homogeneous and C°° away from the origin, the kernels we study are homogeneous but have singularities on a hyperplane. Convolution operators with such kernels arise in the study of ...

In this article we present a new deformation of SL(2,C) on the C∗-algebra level. The method of its construction is the Rieffel deformation. We give a detailed description of so obtained a quantum group (A,∆) in terms of affiliated elements α̂, β̂, γ̂, δ̂ ∈ A generating A. In order to construct β̂, we split A onto parts on which β̂ can easily be defined and then show that gluing procedure can be perfo...

If an integrable function f on the Heisenberg group is supported on the set B × R where B ⊂ Cn is compact and the group Fourier transform f̂(λ) is a finite rank operator for all λ ∈ R \ {0}, then f ≡ 0.

The general form of the operators commuting with the ground representation (appearing in many physical problems within single particle approximation) of the group is found. With help of the modified group projector technique, this result is applied to the system of identical particles with spin independent interaction, to derive the Dirac-Heisenberg hamiltonian and its effective space for arbit...

We investigate the relevance of admissibility criteria based on Plancherel measure for the characterization of tight Weyl-Heisenberg frames with integer oversampling. For this purpose we observe that functions giving rise to such Weyl-Heisenberg frames are admissible with regard to the action of a suitably defined type-I discrete group G. This allows to relate the construction of Weyl-Heisenber...

The orbit method of Kirillov is used to derive the p-mechanical brackets [25]. They generate the quantum (Moyal) and classical (Poisson) brackets on respective orbits corresponding to representations of the Heisenberg group. The extension of p-mechanics to field theory is made through the De Donder–Weyl Hamiltonian formulation. The principal step is the substitution of the Heisenberg group with...

Affine control systems on the generalized Heisenberg Lie groups are studied. Controllability of this kind of class of systems on the generalized Heisenberg Lie group is established by relating to their associated bilinear part. AMS Subject Classification: 22E20

A characterization of the maximal abelian sub-algebras of matrix algebras that are normalized by the canonical representation of a finite Heisenberg group is given. Examples are constructed using a classification result for finite Heisenberg groups.

The orbit method of Kirillov is used to derive the p-mechanical brackets [25]. They generate the quantum (Moyal) and classical (Poisson) brackets on respective orbits corresponding to representations of the Heisenberg group. The extension of p-mechanics to field theory is made through the De Donder–Weyl Hamiltonian formulation. The principal step is the substitution of the Heisenberg group with...