Heisenberg Group Research Articles

C1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$C^{1}$\\end{document}-Regularity for subelliptic systems with drift in the Heisenberg group: the superquadratic controllable growth

We investigate the interior regularity to nonlinear subelliptic systems in divergence form with drift term for the case of superquadratic controllable structure conditions in the Heisenberg group. On the basis of a generalization of the A\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathcal{A}$\\end{document}-harmonic approximation technique, C1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$C^{1}$\\end{document}-regularity is established for horizontal gradients of vector-valued solutions to the subelliptic systems with drift term. Specially, our result is optimal in the sense that in the case of Hölder continuous coefficients we directly attain the optimal Hölder exponent for the horizontal gradients of weak solutions on the regular set.

Dimension two holomorphic distributions on four-dimensional projective space

We study two-dimensional holomorphic distributions on P4. We classify dimension two distributions, of degree at most 2, with either locally free tangent sheaf or locally free conormal sheaf and whose singular scheme has pure dimension one. We show that the corresponding sheaves are split. Next, we investigate the geometry of such distributions, studying from maximally non–integrable to integrable distributions. In the maximally non–integrable case, we show that the distribution is either of Lorentzian type or a push-forward by a rational map of the Cartan prolongation of a singular contact structure on a weighted projective 3-fold. We study distributions of dimension two in P4 whose conormal sheaves are the Horrocks–Mumford sheaves, describing the numerical invariants of their singular schemes which are smooth and connected. Such distributions are maximally non–integrable, uniquely determined by their singular schemes and invariant by a group H5⋊SL(2,Z5)⊂Sp(4,Q), where H5 is the Heisenberg group of level 5. We prove that the moduli spaces of Horrocks–Mumford distributions are irreducible quasi-projective varieties and we determine their dimensions. Finally, we observe that the space of codimension one distributions, of degree d≥6, on P4 has a family of degenerate flat holomorphic Riemannian metrics. Moreover, the degeneracy divisors of such metrics consist of codimension one distributions invariant by H5⋊SL(2,Z5) and singular along a degenerate abelian surface with (1,5)-polarization and level-5-structure.

Lipschitz regularity of sub-elliptic harmonic maps into CAT(0) space

Abstract We prove the local Lipschitz continuity of sub-elliptic harmonic maps between certain singular spaces, more specifically from the 𝑛-dimensional Heisenberg group into CAT ⁡ ( 0 ) \operatorname{CAT}(0) spaces. Our main theorem establishes that these maps have the desired Lipschitz regularity, extending the Hölder regularity in this setting proven in [Y. Gui, J. Jost and X. Li-Jost, Subelliptic harmonic maps with values in metric spaces of nonpositive curvature, Commun. Math. Res. 38 (2022), 4, 516–534] and obtaining same regularity as in [H.-C. Zhang and X.-P. Zhu, Lipschitz continuity of harmonic maps between Alexandrov spaces, Invent. Math. 211 (2018), 3, 863–934] for certain sub-Riemannian geometries; see also [N. Gigli, On the regularity of harmonic maps from RCD ⁢ ( K , N ) \mathrm{RCD}(K,N) to CAT ⁢ ( 0 ) \mathrm{CAT}(0) spaces and related results, preprint (2022), https://arxiv.org/abs/2204.04317; and A. Mondino and D. Semola, Lipschitz continuity and Bochner–Eells–Sampson inequality for harmonic maps from RCD ⁡ ( k , n ) \operatorname{RCD}(k,n) spaces to CAT ⁡ ( 0 ) \operatorname{CAT}(0) spaces, preprint (2022), https://arxiv.org/abs/2202.01590] for the generalisation to RCD spaces. The present result paves the way for a general regularity theory of sub-elliptic harmonic maps, providing a versatile approach applicable beyond the Heisenberg group.

Waring’s problem for locally nilpotent groups: The case of discrete Heisenberg groups

Kamke (Kamke, 1921) solved an analog of Waring’s problem with nth power monomials replaced by integer-valued polynomials. Larsen and Nguyen (Larsen and Nguyen, 2019) explored the view of algebraic groups as a natural setting for Waring’s problem. This paper applies the theory of polynomial maps and polynomial sequences in locally nilpotent groups developed in a previous work (Hu, 2024) to solve an analog of Waring’s problem for the general discrete Heisenberg group H2n+1(Z) for each integer n≥1.

Non-Compact Inaudibility of Weak Symmetry and Commutativity via Generalized Heisenberg Groups

Two Riemannian manifolds are said to be isospectral if there exists a unitary operator which intertwines their Laplace-Beltrami operator. In this paper, we prove in the non-compact setting the inaudibility of the weak symmetry property and the commutative property using an isospectral pair of 23 dimensional generalized Heisenberg groups.

Non-compact gauge groups, tensor fields and Yang-Mills-Einstein amplitudes

Scattering amplitudes in Yang-Mills-Einstein theories have been investigated mostly for compact gauge groups. While non-compact gauge groups are not physically viable in Yang-Mills theory, non-compact gaugings feature prominently in the supergravity literature, where any choice of perturbative vacuum spontaneously breaks the gauge group to a compact subgroup. In this paper, we formulate double-copy constructions for several five-dimensional N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2 supergravities with non-compact gauge groups. On one side of the double copy, we employ amplitudes from a super-Yang-Mills theory with a massive hypermultiplet. On the other, we use amplitudes from particular non-supersymmetric Yang-Mills-scalar theories with massive fermions, chosen to obey constraints coming from color/kinematics duality. Supergravities with massive self-dual tensors in five dimensions are also considered, showing that tensors are straightforwardly realized as double copies of gauge-theory fermions with suitable choices of signs in the corresponding solutions of the Dirac equation. We present several examples of these constructions, noting in particular the appearance of Heisenberg groups in the supergravity gauge symmetry and, in some cases, the possibility of exotic tensor-vector matter couplings.

The logarithmic Sobolev inequality on the Heisenberg group and applications to the uncertainty inequality and heat equation

We consider the logarithmic Sobolev inequality on the Heisenberg group. One can derive the logarithmic Sobolev inequality from the Sobolev inequality, and we consider an application to the uncertainty inequality on the Heisenberg group. Moreover, one can also obtain a dissipative estimate of a solution of the heat equation on the Heisenberg group from the logarithmic Sobolev inequality.

Normalized solutions for critical Schrödinger-Poisson system involving the [formula omitted]-subLaplacian in the Heisenberg group

This article deals with existence and multiplicity of normalized solutions for critical Schrödinger-Poisson system involving the p-subLaplacian in the Heisenberg group. Under appropriate assumptions, together with the truncation technique, the concentration-compactness principle, the genus theory, we prove that the existence and multiplicity of the normalized solutions in the Lp-subcritical case. As far as we know, this study seems to be the first contribution regarding existence of normalized solutions for the critical p-subLaplacian Schrödinger-Poisson system in the Heisenberg group. Moreover, the results of the paper are completely new even in the Euclidean case.

Compactness of commutators of Hardy operators on Heisenberg group

Compactness of commutators of Hardy operators on Heisenberg group

δ-(bi)derivations and transposed Poisson algebra structures on the n-th Schrödinger algebra

The [Formula: see text]-th Schrödinger algebra [Formula: see text] is the semidirect product of the Lie algebra [Formula: see text] with the [Formula: see text]-th Heisenberg Lie algebra [Formula: see text]. In this paper, we will show that [Formula: see text] has neither nontrivial [Formula: see text]-derivations nor nontrivial transposed Poisson algebra structures, and doesn’t have nonzero [Formula: see text]-biderivations. In addition, all [Formula: see text]-derivations and [Formula: see text]-biderivations on [Formula: see text] are all described.

Algebras of entire functions and representations of the twisted Heisenberg group

Algebras of entire functions and representations of the twisted Heisenberg group

On fractional inequalities on metric measure spaces with polar decomposition

Abstract In this paper, we prove the fractional Hardy inequality on polarisable metric measure spaces. The integral Hardy inequality for 1 < p ≤ q < ∞ 1<p\leq q<\infty is playing a key role in the proof. Moreover, we also prove the fractional Hardy–Sobolev type inequality on metric measure spaces. In addition, logarithmic Hardy–Sobolev and fractional Nash type inequalities on metric measure spaces are presented. In addition, we present applications on homogeneous groups and on the Heisenberg group.

Weakly q-deformed Heisenberg algebra and non-Hermitian Hamiltonians: Application in statistical physics

We have investigated a weakly q-deformed algebra, which leads to non-Hermitian operators. It has been explicitly shown that the harmonic oscillator Hamiltonian is quasi-Hermitian, and that the corresponding physical Hilbert-space metric Θ differs from that obtained in Ref. [1], by hermitizing the position operator. The reality of the Hamiltonian eigenvalues has been illustrated by analytically computing the first order correction to the energy spectrum of the harmonic oscillator (HO). Furthermore, we have shown that this q-deformation leads to a GUP with minimal uncertainties in both position and momentum measurements. Moreover, the physical implications of this q-deformation, have been investigated by studying the thermostatistics of a system of HOs and an ideal gas. For both systems, we computed the partition function, and then we derived some thermodynamic functions. In addition, for the ideal gas, a modified equation of state and a generalized Mayer's equation, consistent with the real behavior of gases, have been established. The obtained results show that the impact of this model would be significant at high temperatures. However, unlike earlier research, we observe that this q-deformation induced the similar corrections regardless of the system under study.

Complete group of central units in some group rings over Z and Z [ θ p ]

In the present work, we contribute with new results that enabled us to give an explicit description of the groups of central units of integral group rings and group rings with coefficients in Z [ θ ] , for certain groups G, where θ is a p − primitive root of unity and p is a prime number. In particular, we find a set of generators for the case where G is the Heisenberg Group.

A Note on the Entropy for Heisenberg Group Actions on the Torus

A Note on the Entropy for Heisenberg Group Actions on the Torus

Abstract ladder operators for non self-adjoint Hamiltonians, with applications

Ladder operators are useful, if not essential, in the analysis of some given physical system since they can be used to find easily eigenvalues and eigenvectors of its Hamiltonian. In this paper we extend our previous results on abstract ladder operators considering in many details what happens if the Hamiltonian of the system is not self-adjoint. Among other results, we give an existence criterion for coherent states constructed as eigenstates of our lowering operators. In the second part of the paper we discuss two different examples of our framework: pseudo-quons and a deformed generalized Heisenberg algebra. Incidentally, and interestingly enough, we show that pseudo-quons can be used to diagonalize an oscillator-like Hamiltonian written in terms of (non self-adjoint) position and momentum operators which obey a deformed commutation rule of the kind often considered in minimal length quantum mechanics.

On the cohomology of restricted Heisenberg Lie algebras

We show that the Heisenberg Lie algebras over a field F of characteristic p>0 admit a family of restricted Lie algebras, and we classify all such non-isomorphic restricted Lie algebra structures. We use the ordinary 1- and 2-cohomology spaces with trivial coefficients to compute the restricted 1- and 2-cohomology spaces of these restricted Heisenberg Lie algebras. We describe the restricted 1-dimensional central extensions, including explicit formulas for the Lie brackets and ⋅[p]-operators.

Loomis–Whitney inequalities on corank 1 Carnot groups

In this paper we provide another way to deduce the Loomis–Whitney inequality on higher dimensional Heisenberg groups \(\mathbb{H}^n\) based on the one on the first Heisenberg group \(\mathbb{H}^1\) and the known nonlinear Loomis–Whitney inequality (which has more projections than ours). Moreover, we generalize the result to the case of corank 1 Carnot groups and products of such groups. Our main tool is the modified equivalence between the Brascamp–Lieb inequality and the subadditivity of the entropy developed in Carlen and Cordero-Erausquin (2009).

A Stone–von Neumann equivalence of categories for smooth representations of the Heisenberg group

The classical Stone–von Neumann theorem relates the irreducible unitary representations of the Heisenberg group Hn to non-trivial unitary characters of its center Z, and plays a crucial role in the construction of the oscillator representation for the metaplectic group. In this paper we extend these ideas to non-unitary and non-irreducible representations, thereby obtaining an equivalence of categories between certain representations of Z and those of Hn. Our main result is a smooth equivalence, which involves the fundamental ideas of du Cloux on differentiable representations and smooth imprimitivity systems for Nash groups. We show how to extend the oscillator representation to the smooth setting and give an application to degenerate Whittaker models for representations of reductive groups. We also include an algebraic equivalence, which can be regarded as a generalization of Kashiwara’s lemma from the theory of D-modules.

The Integer Group Determinants for the Heisenberg Group of Order p3

The Integer Group Determinants for the Heisenberg Group of Order p3