Heisenberg Group Research Articles

Regularity Theory for Nonlocal Equations with General Growth in the Heisenberg Group

Abstract We deal with a wide class of generalized nonlocal $p$-Laplace equations, so-called nonlocal $G$-Laplace equations, in the Heisenberg framework. Under natural hypotheses on the $N$-function $G$, we provide a unified approach to investigate in the spirit of De Giorgi-Nash-Moser theory, some local properties of weak solutions to such kind of problems, involving boundedness, Hölder continuity and Harnack inequality. To this end, an improved nonlocal Caccioppoli-type estimate as the main auxiliary ingredient is exploited several times.

Local C0,1-Regularity for the Parabolic p-Laplacian Equation on the Group SU(3)

In this article, when 2≤p≤4, we establish the Cloc0,1-regularity of weak solutions to the degenerate parabolic p-Laplacian equation ∂tu=−∑i=16Xi*(|∇Hu|p−2Xiu) on the group SU(3) granted with horizontal vector fields X1, …, X6. Compared to the Heisenberg group, Hn, we obtained the optimal range of p; that is, 2≤p≤4.

Conformally invariant differential operators on Heisenberg groups and minimal representations

Conformally invariant differential operators on Heisenberg groups and minimal representations

FR,□R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f\\left( R,\\square R\\right) $$\\end{document}-gravity and equivalency with the modified GUP Scalar field models

Inspired by the generalization of scalar field gravitational models with a minimum length we study the equivalent theory in modified theories of gravity. The quadratic generalized uncertainty principle (GUP) gives rise to a deformed Heisenberg algebra in the application, resulting in the emergence of additional degrees of freedom described by higher-order derivatives. The new degrees of freedom can be attributed to the introduction of a new scalar field, transforming the resulting theory into a two-scalar field theory. Thus, in order to describe all the degrees of freedom we investigate special forms of the sixth-order modify fR,□R-\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f\\left( R,\\square R\\right) -$$\\end{document}theory of gravity, where the gravitational Lagrangian has similar properties to that of the GUP scalar field theory. Finally, the cosmological applications are discussed, and we show that the de Sitter universe can be recovered without introducing a cosmological constant.

Biderivations of the mirror Heisenberg–Virasoro algebra

The mirror Heisenberg–Virasoro algebra [Formula: see text] is the semi-direct product of the Virasoro algebra and the twisted Heisenberg algebra. In this paper, all biderivations of the mirror Heisenberg–Virasoro algebra are determined by using the gradation of biderivations. As an application, we show that any linear commuting map [Formula: see text] on the mirror Heisenberg–Virasoro algebra is the form [Formula: see text], where [Formula: see text] is an element from the basic field and [Formula: see text] is a linear map having its range in the center of [Formula: see text].

Turning graphene into a lab for noncommutativity

It was recently shown that taking into account the granular structure of graphene lattice, the Dirac-like dynamics of its quasiparticles resists beyond the lowest energy approximation. This can be described in terms of new phase-space variables, (X→,P→), that enjoy generalized Heisenberg algebras. In this letter, we add to that picture the important case of noncommuting X→, for which [Xi,Xj]=iθij and we find that θij=ℓ2ϵij, with ℓ the lattice spacing. We close by giving both the general recipe and a possible specific kinematic setup for the practical implementation of this approach to test noncommutative theories in tabletop analog experiments on graphene.

Existence of a Minimizer for the Bianchi-Egnell Inequality on the Heisenberg Group

Existence of a Minimizer for the Bianchi-Egnell Inequality on the Heisenberg Group

Quantum Heisenberg enveloping algebras at roots of unity

In this article, the two-parameter quantum Heisenberg enveloping algebra, which serves as a model for certain quantum generalized Heisenberg algebras, has been studied at roots of unity. In this context, the quantum Heisenberg enveloping algebra becomes a polynomial identity algebra, and the dimension of simple modules is bounded by its PI degree. The PI degree, center, and complete classification of simple modules up to isomorphism are explicitly presented. We work over a field of arbitrary characteristic, although our results concerning the representations require that it is algebraically closed.

New cyclic groups based on the generalized order-k Pell sequences in the Heisenberg group and their application in cryptography

New cyclic groups based on the generalized order-k Pell sequences in the Heisenberg group and their application in cryptography

Isotopisms of nilpotent Leibniz algebras and Lie racks

In this paper we study the isotopism classes of two-step nilpotent algebras. We show that every nilpotent Leibniz algebra g with dim [ g , g ] = 1 is isotopic to the Heisenberg Lie algebra or to the Heisenberg algebra l 2 n + 1 J 1 , where J 1 is the n × n Jordan block of eigenvalue 1. We also prove that two such algebras are isotopic if and only if the Lie racks integrating them are isotopic. This gives the classification of Lie racks whose tangent space at the unit element is a nilpotent Leibniz algebra with one-dimensional commutator ideal. Eventually, we introduce new isotopism invariants for Leibniz algebras and Lie racks.

The hidden Lorentz covariance of quantum mechanics

This paper introduces a systematic algorithm for deriving a new unitary representation of the Lorentz algebra (so(1,3)) and an irreducible unitary representation of the extended (anti) de-Sitter algebra (so(2,4)) on L2(R3,1r). This representation is equivalent to a representation on L2(R3), and the corresponding similarity transformation is identified. An explicit representation in terms of differential operators is given, and it is shown that the inner product is Lorentz invariant. Ensuring Lorentz covariance demands a modification of the Heisenberg algebra, recognized as a phase space algebra at the interface of gravitational and quantum realms (IGQR), which we consider subordinate to Lorentz covariance. It is also demonstrated that time evolution can be cast in a manifestly covariant form. Each mass sector of the Hilbert space carries a representation of the Lorentz algebra, and the (anti) de-Sitter algebra on each mass sector contracts to the Poincare algebra in the flat configuration and momentum space limits. Finally, we show that three-dimensional fuzzy space also carries a unitary representation of these algebras, algebraically equivalent to the L2(R3,1r) representation but not necessarily equivalent as representations. Several outstanding issues are identified for future exploration.

Statistical properties of the 𝒜κ<0 Weyl–Heisenberg algebra coherent states

In this paper, we study the one-parameter [Formula: see text]-generalized oscillator algebra [Formula: see text] (which includes the case of the standard Weyl–Heisenberg algebra). This algebra admits representations of finite-dimensional for negative values of the parameter [Formula: see text]. We construct the associated Perelomov like coherent states. We discuss their statistical properties. In particular, we derive the expressions of the Mandel parameter and Husimi function and we discuss the squeezing of the [Formula: see text] coherent states.

Generalized Schrödinger operators on the Heisenberg group and Hardy spaces

Let L=−ΔHn+μ be a generalized Schrödinger operator on the Heisenberg group Hn, where ΔHn is the sub-Laplacian, and μ is a nonnegative Radon measure satisfying certain conditions. In this paper, we first establish some estimates of the fundamental solution and the heat kernel of L. Applying these estimates, we then study the Hardy spaces HL1(Hn) introduced in terms of the maximal function associated with the heat semigroup e−tL; in particular, we obtain an atomic decomposition of HL1(Hn), and prove the Riesz transform characterization of HL1(Hn). The dual space of HL1(Hn) is also studied.

A new class of weights associated with Schrödinger operator on Heisenberg groups and applications

A new class of weights associated with Schrödinger operator on Heisenberg groups and applications

Non-singular gravitational collapse through modified Heisenberg algebra

We study the effects of cut-off physics, in the form of a modified algebra inspired by Polymer Quantum Mechanics and by the Generalized Uncertainty Principle representation, on the collapse of a spherical dust cloud. We analyze both the Newtonian formulation, originally developed by Hunter, and the general relativistic formulation, that is the Oppenheimer–Snyder model; in both frameworks we find that the collapse is stabilized to an asymptotically static state above the horizon, and the singularity is removed. In the Newtonian case, by requiring the Newtonian approximation to be valid, we find lower bounds of the order of unity (in Planck units) for the deformation parameter of the modified algebra. We then study the behaviour of small perturbations on the non-singular collapsing backgrounds, and find that for certain range of the parameters (the polytropic index for the Newtonian case and the sound velocity in the relativistic setting) the collapse is stable to perturbations of all scales, and the non-singular super-Schwarzschild configurations have physical meaning.

The Heterotic-Ricci Flow and Its Three-Dimensional Solitons

We introduce a novel curvature flow, the Heterotic-Ricci flow, as the two-loop renormalization group flow of the Heterotic string common sector and study its three-dimensional compact solitons. The Heterotic-Ricci flow is a coupled curvature evolution flow, depending on a non-negative real parameter κ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\kappa $$\\end{document}, for a complete Riemannian metric and a three-form H on a manifold M. Its most salient feature is that it involves several terms quadratic in the curvature tensor of a metric connection with skew-symmetric torsion H. When κ=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\kappa = 0$$\\end{document} the Heterotic-Ricci flow reduces to the generalized Ricci flow and hence it can be understood as a modification of the latter via the second-order correction prescribed by Heterotic string theory, whereas when H=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H=0$$\\end{document} and κ>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\kappa >0$$\\end{document} the Heterotic-Ricci flow reduces to a constrained version of the RG-2 flow and hence it can be understood as a generalization of the latter via the introduction of the three-form H. Solutions of Heterotic supergravity with trivial gauge bundle, which we call Heterotic solitons, define a particular class of three-dimensional solitons for the Heterotic-Ricci flow and constitute our main object of study. We prove a number of structural results for three-dimensional Heterotic solitons, obtaining, in particular, the complete classification of compact three-dimensional strong Heterotic solitons as hyperbolic three-manifolds or quotients of the Heisenberg group equipped with a left-invariant metric. Furthermore, we prove that all Einstein three-dimensional Heterotic solitons have constant dilaton and leave as open the construction of a Heterotic soliton with non-constant dilaton. In this direction, we prove that Einstein Heterotic solitons with constant dilaton are rigid and therefore cannot be deformed into a solution with non-constant dilaton. This is, to the best of our knowledge, the first rigidity result for compact supergravity solutions in the literature.

Maps on the Mirror Heisenberg–Virasoro Algebra

Using the first cohomology from the mirror Heisenberg–Virasoro algebra to the twisted Heisenberg algebra (as the mirror Heisenberg–Virasoro algebra module), in this paper, we determined the derivations on the mirror Heisenberg–Virasoro algebra. Based on this result, we proved that any two-local derivation on the mirror Heisenberg–Virasoro algebra is a derivation. All half-derivations are described, and as corollaries, we have descriptions of transposed Poisson structures and local (two-local) half-derivations on the mirror Heisenberg–Virasoro algebra.

Scaling and instantaneous blow up

The main result is a simple proof of the Baras-Goldstein (1984) instantaneous blow up result for the heat equation with the inverse square potential. The proof relies heavily, indeed mainly, on scaling. Remarks are also given concerning the case when the underlying space ℝ N is replaced by the Heisenberg group ℍ N .

A Liouville theorem for superlinear parabolic equations on the Heisenberg group

Abstract We consider a parabolic nonlinear equation on the Heisenberg group. Applying the Gidas–Spruck type estimates, we prove that under suitable conditions, the equation does not have positive solutions. As an application of the nonexistence result, we provide optimal universal estimates for positive solutions.

High and low perturbations of the critical Choquard equation on the Heisenberg group

High and low perturbations of the critical Choquard equation on the Heisenberg group