A negative answer to the Bernstein problem for entire H-perimeter minimizing intrinsic graphs is given in the setting of the first Heisenberg group H endowed with its Carnot-Carathéodory metric structure. Moreover, in all Heisenberg groups H an area formula for intrinsic graphs with Sobolev regularity is provided, together with the associated first and second variation formulae.

The goal of these notes is to introduce the reader to the Heisenberg group with its CarnotCarathéodory metric and to Pansu’s differentiation theorem. As they are very similar, we will first study Rademacher’s theorem about Lipschitz maps and then see how the same technique can be applied in the more complex setting of the Heisenberg group.

Curado and Rego-Monteiro introduced in [2] a new algebraic structure generalizing the Heisenberg algebra and containing also the q-deformed oscillator as a particular case. This algebra, called generalized Heisenberg algebra, depends on an analytical function f and the eigenvalues αn of the Hamiltonian are given by the one-step recurrence αn+1 = f(αn). This structure has been used in different ...

We perform Monte Carlo simulations to study the relaxation of single-domain nanoparticles that are located on a simple cubic lattice with anisotropy axes pointing in the z-direction, under the combined influence of anisotropy energy, dipolar interaction and ferromagnetic interaction of strength J. We compare the results of classical Heisenberg systems with three-dimensional magnetic moments [Fo...

Given q1, q2 ∈ C \ {0}, we construct a unital Banach algebra Bq1,q2 which contains a universal normalized solution to the (q1, q2)-deformed Heisenberg–Lie commutation relations in the following specific sense: (i) Bq1,q2 contains elements b1, b2, and b3 which satisfy the (q1, q2)-deformed Heisenberg–Lie commutation relations (that is, b1b2 − q1b2b1 = b3, q2b1b3 − b3b1 = 0, and b2b3 − q2b3b2 = 0...

In this article we construct a combinatorial counterpart of the action of a certain Heisenberg group on the space of conformal blocks which was studied by Andersen-Masbaum [1] and Blanchet-Habegger-Masbaum-Vogel [5]. It is a non-abelian analogue of the description of the Heisenberg action on the space of theta functions.

We prove results toward classifying compact Lorentz manifolds on which Heisenberg groups act isometrically. We give a general construction, leading to a new example, of codimension-one actions—those for which the dimension of the Heisenberg group is one less than the dimension of the manifold. The main result is a classification of codimension-one actions, under the assumption they are real-ana...

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...

The Segal–Shale–Weil representation associates to a symplectic transformation of the Heisenberg group an intertwining operator, called metaplectic operator. We develop an explicit construction of metaplectic operators for the Heisenberg group H(G) of a finite abelian group G, an important setting in finite time-frequency analysis. Our approach also yields a simple construction for the multivari...

We consider a class of one-dimensional Heisenberg spin models (plaquette chains) related to the recently found 1/5-depleted square-lattice Heisenberg system CaV4O9. A number of exact and exact-numerical results concerning the properties of the competing dimer and resonating plaquette phases are presented.