Class Of Nilpotent Lie Groups Research Articles

Observability and controllability for the Schrödinger equation on quotients of groups of Heisenberg type

We give necessary and sufficient conditions for the controllability of a Schr\odinger equation involving the sub-Laplacian of a nilmanifold obtained by taking the quotient of a group of Heisenberg type by one of its discrete sub-groups. This class of nilpotent Lie groups is a major example of stratified Lie groups of step 2. The sub-Laplacian involved in these Schr\odinger equations is subelliptic, and, contrarily to what happens for the usual elliptic Schr\odinger equation for example on flat tori or on negatively curved manifolds, there exists a minimal time of controllability. The main tools used in the proofs are (operator-valued) semi-classical measures constructed by use of representation theory and a notion of semi-classical wave packets that we introduce here in the context of groups of Heisenberg type.

Best constants in Sobolev and Gagliardo\u2013Nirenberg inequalities on graded groups and ground states for higher order nonlinear subelliptic equations

In this paper the dependence of the best constants in Sobolev and Gagliardo–Nirenberg inequalities on the precise form of the Sobolev space norm is investigated. The analysis is carried out on general graded Lie groups, thus including the cases of mathbb {R}^n, Heisenberg, and general stratified Lie groups, in all these cases the results being new. The Sobolev norms may be defined in terms of Rockland operators, i.e. the hypoelliptic homogeneous left-invariant differential operators on the group. The best constants are expressed in the variational form as well as in terms of the ground state solutions of the corresponding nonlinear subelliptic equations. The orders of these equations can be high depending on the Sobolev space order in the Sobolev or Gagliardo–Nirenberg inequalities, or may be fractional. Applications are obtained also to equations with lower order terms given by different hypoelliptic operators. Already in the case of {mathbb {R}}^n, the obtained results extend the classical relations by Weinstein (Commun Math Phys 87(4):567–576 (1982/1983)) to a wide range of nonlinear elliptic equations of high orders with elliptic low order terms and a wide range of interpolation inequalities of Gagliardo–Nirenberg type. However, the proofs are different from those in Weinstein (Commun Math Phys 87(4):567–576 (1982/1983)) because of the impossibility of using the rearrangement inequalities already in the setting of the Heisenberg group. The considered class of graded groups is the most general class of nilpotent Lie groups where one can still consider hypoelliptic homogeneous invariant differential operators and the corresponding subelliptic differential equations.

Nilpotent Lie groups and hyperbolic automorphisms

A connected Lie group admitting an expansive automorphism is known to be nilpotent, but not all nilpotent Lie groups admit expansive automorphisms. In this article, we find sufficient conditions for a class of nilpotent Lie groups to admit expansive automorphisms.

Uncertainty relations on nilpotent Lie groups.

We give relations between main operators of quantum mechanics on one of most general classes of nilpotent Lie groups. Namely, we show relations between momentum and position operators as well as Euler and Coulomb potential operators on homogeneous groups. Homogeneous group analogues of some well-known inequalities such as Hardy's inequality, Heisenberg–Kennard type and Heisenberg–Pauli–Weyl type uncertainty inequalities, as well as Caffarelli–Kohn–Nirenberg inequality are derived, with best constants. The obtained relations yield new results already in the setting of both isotropic and anisotropic , and of the Heisenberg group. The proof demonstrates that the method of establishing equalities in sharper versions of such inequalities works well in both isotropic and anisotropic settings.

Spectral zeta function on pseudo H-type nilmanifolds

We explain the explicit integral form of the heat kernel for the sub-Laplacian on two step nilpotent Lie groups G based on the work of Beals, Gaveau and Greiner. Using such an integral form we study the heat trace of the sub-Laplacian on nilmanifolds L\G where L is a lattice. As an application a common property of the spectral zeta function for the sub-Laplacian on L\G is observed. In particular, we introduce a special class of nilpotent Lie groups, called pseudo H-type groups which are generalizations of groups previously considered by Kaplan. As is known such groups always admit lattices. Here we aim to explicitly calculate the heat trace and the spectrum of the (sub)-Laplacian on various low dimensional compact nilmanifolds including several pseudo H-type nilmanifolds L\G, i.e. where G is a pseudo H-type group. In an appendix we discuss a zeta function which typically appears as the Mellin transform for these heat traces.

Generalized analogs of the Heisenberg uncertainty inequality

We investigate locally compact topological groups for which a generalized analog of the Heisenberg uncertainty inequality hold. In particular, it is shown that this inequality holds for $\mathbb{R}^{n} \times K$ (where K is a separable unimodular locally compact group of type I), Euclidean motion group and several general classes of nilpotent Lie groups which include thread-like nilpotent Lie groups, 2-NPC nilpotent Lie groups and several low-dimensional nilpotent Lie groups.

Characterization of Shift-Invariant Spaces on a Class of Nilpotent Lie Groups with Applications

Given a simply connected nilpotent Lie group having unitary irreducible representations that are square-integrable modulo the center, we use operator-valued periodization to give a range-function type characterization of shift-invariant spaces of function on the group. We then give characterizations of frame and Riesz families for shift-invariant spaces.

Sinc-type functions on a class of nilpotent Lie groups

Let N be a simply connected, connected nilpotent Lie group with the following assumptions. Its Lie algebra 𝔫 is an n-dimensional vector space over the reals. Moreover, 𝔫=𝔷⊕𝔟⊕𝔞, 𝔷 is the center of 𝔫, 𝔷=ℝZn-2d⊕ℝZn-2d-1⊕⋯⊕ℝZ1, 𝔟=ℝYd⊕ℝYd-1⊕⋯⊕ℝY1, 𝔞=ℝXd⊕ℝXd-1⊕⋯⊕ℝX1. Next, assume 𝔷⊕𝔟 is a maximal commutative ideal of 𝔫, [𝔞,𝔟]⊆𝔷, and det ([Xi,Yj])1≤i,j≤d is a non-trivial homogeneous polynomial defined over the ideal [𝔫,𝔫]⊆𝔷. We do not assume that [𝔞,𝔞] is generally trivial. We obtain some precise description of band-limited spaces which are sampling subspaces of L2(N) with respect to some discrete set Γ. The set Γ is explicitly constructed by fixing a strong Malcev basis for 𝔫. We provide sufficient conditions for which a function f is determined from its sampled values on (f(γ))γ∈Γ. We also provide an explicit formula for the corresponding sinc-type functions. Several examples are also computed in the paper.

Quasisymmetric maps of boundaries of amenable hyperbolic groups

In this paper we show that if Y = N ×Qm is a metric space where N is a Carnot group endowed with the Carnot-Caratheodory metric then any quasisymmetric map of Y is actually bilipschitz. The key observation is that Y is the parabolic visual boundary of a mixed type locally compact amenable hyperbolic group. The same results also hold for a larger class of nilpotent Lie groups N . As part of the proof we also obtain partial quasi-isometric rigidity results for mixed type locally compact amenable hyperbolic groups. Finally we prove a rigidity result for uniform subgroups of bilipschitz maps of Y in the case of N = Rn.

Sub-semi-Riemannian geometry of general H-type groups

We study a special class of nilpotent Lie groups of step 2, that generalizes the class of the so-called H(eisenberg)-type groups, defined by A. Kaplan in 1980. We change the presence of an inner product to an arbitrary scalar product and relate the construction to the composition of quadratic forms. We present the geodesic equation for sub-semi-Riemannian metric on nilpotent Lie groups of step 2 and solve them for the case of general H-type groups. We also present some results on sectional curvature and the Ricci tensor of semi-Riemannian general H-type groups.

Explicit Ricci Solitons on Nilpotent Lie Groups

The primary purpose of this paper is to obtain explicit, coordinate-based descriptions of Ricci flow solutions—especially those corresponding to Ricci solitons—on two classes of nilpotent Lie groups. On the odd-dimensional classical Heisenberg groups, we determine the asymptotics of Ricci flow starting at any metric, and use Lott’s blowdown method to demonstrate convergence to soliton metrics. On the groups of real unitriangular matrices, which are more complicated, we describe the solitons and corresponding solutions using a suitable ansatz.

Nilpotent groups without exactly polynomial Dehn function

We prove super-quadratic lower bounds for the growth of the filling area function of a certain class of nilpotent Lie groups. This class contains groups for which it is known that their Dehn function grows no faster than n2log n. We therefore obtain the existence of (finitely generated) nilpotent groups whose Dehn functions do not have exactly polynomial growth and we thus answer a well-known question about the possible growth rate of Dehn functions of nilpotent groups.

The Lp –Lq version of Hardy's Theorem on nilpotent Lie groups

Let p, q be such that 2≤ p, q ≤ + ∞. We prove in this paper the Lp – Lq version of Hardy's Theorem for an arbitrary nilpotent Lie group G extending then earlier cases and the classical Hardy theorem proved recently by E. Kaniuth and A. Kumar. The case where 1 ≤ p, q ≤ + ∞ is studied for a restricted class of nilpotent Lie groups.

Analysis of restrictions of unitary representations of a nilpotent Lie group

Let G be a connected simply connected nilpotent Lie group, K an analytic subgroup of G and π an irreducible unitary representation of G. Let D π ( G ) K be the algebra of differential operators keeping invariant the space of C ∞ vectors of π and commuting with the action of K on that space. In this paper, we assume that the restriction of π to K has finite multiplicities and we show that D π ( G ) K is isomorphic to a subalgebra of the field of rational K-invariant functions on the co-adjoint orbit Ω ( π ) associated to π, and for some particular cases, that D π ( G ) K is even isomorphic to the algebra of polynomial K-invariant functions on Ω ( π ) . We prove also the Frobenius reciprocity for some restricted classes of nilpotent Lie groups, especially in the cases where K is normal or abelian.

A Geometric Criterion for Gelfand Pairs Associated with Nilpotent Lie Groups

In this paper, we give a proof to the orbit conjecture of Benson–Jenkins–Lipsman–Ratcliff and get a geometric criterion for Gelfand pairs associated with nilpotent Lie groups. Our proof is based on an analysis of the condition by using certain operators naturally attached to two step nilpotent Lie algebras provided with an inner product. A class of nilpotent Lie groups considered by Lauret is also treated as examples.

Müntz–Szasz Theorems for Nilpotent Lie Groups

The classic Müntz–Szasz theorem says that forf∈L2([0,1]) and {nk}∞k=1, a strictly increasing sequence of positive integers,∫10xnjf(x)dx=0∀j⇒f=0⇔∑j=1∞1nj=∞.We have generalized this theorem to compactly supported functions on Rnand to an interesting class of nilpotent Lie groups. On Rnwe rephrased the condition above on an integral against a monomial, as a condition on the derivative of the Fourier transformf. This transform, for compactly supportedf, has an entire extension to complexn-space and these derivatives are coefficients in a Taylor series expansion off. For nilpotent Lie groups, we have proven a Müntz–Szasz theorem for the matrix coefficients of the operator valued Fourier transform, on groups that have a fixed polarizer for the representations in general position. Our work here is inspired by recent work on Paley–Wiener theorems for nilpotent Lie groups by Moss (J. Funct. Anal.114(1993), 395–411) and Park (J. Funct. Anal.133(1995), 211–300), who have proven Paley–Wiener theorems on restricted classes of nilpotent Lie groups. Lipsman and Rosenberg (Trans. Amer. Math. Soc.348(1996), 1031–1050) have extended these results, for matrix coefficients, to any connected, simply connected nilpotent Lie group. As part of the proof of the Müntz–Szasz theorem for matrix coefficients, We construct a new basis in a nilpotent Lie algebra, which we call analmost strong Malcev basis. This new basis has many of the features of a strong Malcev basis, although it can be used to pass through subalgebras that are not ideals. Almost strong Malcev bases are unique up to a fixed strong Malcev basis. We will also show that, using almost strong Malcev bases, we can provide a partial answer to a question posed by Corwin and Greenleaf (“Representations of Nilpotent Lie Groups and Their Applications”, Cambridge Univ. Press, Cambridge, UK, 1990) on using additive coordinates for a cross-section.

Cut and conjugate loci in two-step nilpotent Lie groups

We show that cut and conjugate loci coincide for a large class of nilpotent Lie groups with left-invariant metric. One consequently obtains lower bounds for the injectivity radius of nonsingular 2-step nilmanifolds. The Jacobi equations are also used to classify Riemannian foliations of 3-dimensional nilmanifolds.

L2-spaces and Sub-Laplacians on homogeneous groups

In this paper, we introduce a special class of nilpotent Lie groups defined by hermitian maps, which includes all the groups of affine holomorphic automorphisims of Siegel domains of type II, in particular, the Heisenberg group. And we study harmonic analysis on these groups as spectral theory of the associated Sub-Laplacian instead of the group representation theory in usual way.

Zentrale Distributionen auf nilpotenten Liegruppen

We contribute some results to the following question discussed in several papers ([3], [9], [12]): To what extent the characters of the irreducible representations of a nilpotent Lie groupN determine all central tempered distributions onN. A hereditary property is proved, which enables us to give a negative answer for a large class of nilpotent Lie groups. However, we give a positive answer for a certain series of two-step nilpotent groups. Furthermore, proving a generalisation of the Schwartz Kernel Theorem we decide the question specially for all groups of dimension<5.

Fundamental Solutions for a Class of Hypoelliptic PDE Generated by Composition of Quadratic Forms

We introduce a class of nilpotent Lie groups which arise naturally from the notion of composition of quadratic forms, and show that their standard sublaplacians admit fundamental solutions analogous to that known for the Heisenberg group.