Nilpotent Lie Groups Research Articles

Weyl-Einstein structures on conformal solvmanifolds

A conformal Lie group is a conformal manifold (M, c) such that M has a Lie group structure and c is the conformal structure defined by a left-invariant metric g on M. We study Weyl-Einstein structures on conformal solvable Lie groups and on their compact quotients. In the compact case, we show that every conformal solvmanifold carrying a Weyl-Einstein structure is Einstein. We also show that there are no left-invariant Weyl-Einstein structures on non-abelian nilpotent conformal Lie groups, and classify them on conformal solvable Lie groups in the almost abelian case. Furthermore, we determine the precise list (up to automorphisms) of left-invariant metrics on simply connected solvable Lie groups of dimension 3 carrying left-invariant Weyl-Einstein structures.

Harmonic and anharmonic oscillators on the Heisenberg group

In this article, we present a notion of the harmonic oscillator on the Heisenberg group Hn, which, under a few reasonable assumptions, forms the natural analog of a harmonic oscillator on Rn: a negative sum of squares of operators on Hn, which is essentially self-adjoint on L2(Hn) with purely discrete spectrum and whose eigenvectors are Schwartz functions forming an orthonormal basis of L2(Hn). The differential operator in question is determined by the Dynin–Folland group—a stratified nilpotent Lie group—and its generic unitary irreducible representations, which naturally act on L2(Hn). As in the Euclidean case, our notion of harmonic oscillator on Hn extends to a whole class of so-called anharmonic oscillators, which involve left-invariant derivatives and polynomial potentials of order greater or equal 2. These operators, which enjoy similar properties as the harmonic oscillator, are in one-to-one correspondence with positive Rockland operators on the Dynin–Folland group. The latter part of this article is concerned with spectral multipliers. We obtain useful Lp-Lq-estimates for a large class of spectral multipliers of the sub-Laplacian LHn,2 and, in fact, of generic Rockland operators on graded groups. As a by-product, we obtain explicit hypoelliptic heat semigroup estimates and recover the continuous Sobolev embeddings on graded groups, provided 1 < p ≤ 2 ≤ q < ∞.

Integrability properties of quasi-regular representations ofNAgroups

Let $G = N \rtimes A$, where $N$ is a graded group and $A = \mathbb{R}^+$ acts on $N$ via homogeneous dilations. The quasi-regular representation $\pi = \mathrm{ind}_A^G (1)$ of $G$ can be realised to act on $L^2 (N)$. It is shown that for a class of analysing vectors the associated wavelet transform defines an isometry from $L^2 (N)$ into $L^2 (G)$ and that the integral kernel of the corresponding orthogonal projector has polynomial off-diagonal decay. The obtained reproducing formula is instrumental for proving decomposition theorems for function spaces on nilpotent Lie groups.

Poincaré profiles of Lie groups and a coarse geometric dichotomy

Poincaré profiles are analytically defined invariants, which provide obstructions to the existence of coarse embeddings between metric spaces. We calculate them for all connected unimodular Lie groups, Baumslag–Solitar groups and Thurston geometries, demonstrating two substantially different types of behaviour. For Lie groups, our dichotomy extends both the rank one versus higher rank dichotomy for semisimple Lie groups and the polynomial versus exponential growth dichotomy for solvable unimodular Lie groups. We provide equivalent algebraic, quasi-isometric and coarse geometric formulations of this dichotomy. As a consequence, we deduce that for groups of the form Ntimes S, where N is a connected nilpotent Lie group, and S is a rank one simple Lie group, both the growth exponent of N, and the conformal dimension of S are non-decreasing under coarse embeddings. These results are new even for quasi-isometric embeddings and give obstructions which in many cases improve those previously obtained by Buyalo–Schroeder.

Nilpotent Groups and Bi-Lipschitz Embeddings Into L1

Abstract We prove that if a simply connected nilpotent Lie group quasi-isometrically embeds into an $L^1$ space, then it is abelian. We reach this conclusion by proving that every Carnot group that bi-Lipschitz embeds into $L^1$ is abelian. Our proof follows the work of Cheeger and Kleiner, by considering the pull-back distance of a Lipschitz map into $L^1$ and representing it using a cut measure. We show that such cut measures, and the induced distances, can be blown up and the blown-up cut measure is supported on “generic” tangents of the original sets. By repeating such a blow-up procedure, one obtains a cut measure supported on half-spaces. This differentiation result then is used to prove that bi-Lipschitz embeddings can not exist in the non-abelian settings.

A Note on the Heat Kernel for the Rescaled Harmonic Oscillator from Two Step Nilpotent Lie Groups

A Note on the Heat Kernel for the Rescaled Harmonic Oscillator from Two Step Nilpotent Lie Groups

Nonnegative solutions of a fractional differential inequality on Grushin spaces and nilpotent Lie groups

Abstract In this paper, we investigate the nonnegative solutions of the differential inequality u p ≤ ℒ s ⁢ u u^{p}\leq\mathcal{L}^{s}u on the Grushin space 𝔾 α n {\mathbb{G}_{\alpha}^{n}} for ( p , s , α ) ∈ ( 1 , ∞ ) × ( 0 , 1 ) × ( 0 , ∞ ) {(p,s,\alpha)\in(1,\infty)\times(0,1)\times(0,\infty)} , where the ℒ s {\mathcal{L}^{s}} are the fractional powers of the Grushin operator ℒ {\mathcal{L}} . We show that any nonnegative solution of the fractional order differential inequality displayed above is zero if and only if p ≤ Q Q - 2 ⁢ s {p\leq\frac{Q}{Q-2s}} , where Q is the homogeneous dimension of 𝔾 α n {\mathbb{G}_{\alpha}^{n}} . Moreover, we also consider the similar problems of nonnegative weak solutions of the fractional sub-Laplacian differential inequality on nilpotent Lie groups.

Hermitian structures on a class of almost nilpotent solvmanifolds

In this paper we investigate the existence of invariant SKT, balanced and generalized Kähler structures on compact quotients Γ﹨G, where G is an almost nilpotent Lie group whose nilradical has one-dimensional commutator and Γ is a lattice of G. We first obtain a characterization of Hermitian almost nilpotent Lie algebras g whose nilradical n has one-dimensional commutator and a classification result in real dimension six. Then, we study the ones admitting SKT and balanced structures and we examine the behaviour of such structures under flows. In particular, we construct new examples of compact SKT manifolds.Finally, we prove some non-existence results for generalized Kähler structures in real dimension six. In higher dimension we construct the first examples of non-split generalized Kähler structures (i.e., such that the associated complex structures do not commute) on almost abelian Lie algebras. This leads to new compact (non-Kähler) manifolds admitting non-split generalized Kähler structures.

Homogeneous algebras via heat kernel estimates

We study homogeneous Besov and Triebel–Lizorkin spaces defined on doubling metric measure spaces in terms of a self-adjoint operator whose heat kernel satisfies Gaussian estimates together with its derivatives. When the measure space is a smooth manifold and such operator is a sum of squares of smooth vector fields, we prove that their intersection with L ∞ L^\infty is an algebra for pointwise multiplication. Our results apply to nilpotent Lie groups and Grushin settings.

Completeness of coherent state subsystems for nilpotent Lie groups

Let $G$ be a nilpotent Lie group and let $\pi$ be a coherent state representation of $G$. The interplay between the cyclicity of the restriction $\pi|_{\Gamma}$ to a lattice $\Gamma \leq G$ and the completeness of subsystems of coherent states based on a homogeneous $G$-space is considered. In particular, it is shown that necessary density conditions for Perelomov's completeness problem for the highest weight vector can be obtained via criteria for the cyclicity of $\pi|_{\Gamma}$.

A direct proof of the Brunn-Minkowski inequality in nilpotent Lie groups

The purpose of this work is to give a direct proof of the multiplicative Brunn-Minkowski inequality in nilpotent Lie groups based on Hadwiger-Ohmann's one of the classical Brunn-Minkowski inequality in Euclidean space.

A proof of the polynomial conjecture for restrictions of nilpotent lie groups representations

Let G G be a connected and simply connected nilpotent Lie group, K K an analytic subgroup of G G and π \pi an irreducible unitary representation of G G whose coadjoint orbit of G G is denoted by Ω ( π ) \Omega (\pi ) . Let U ( g ) \mathscr U(\mathfrak g) be the enveloping algebra of g C {\mathfrak g}_{\mathbb C} , g \mathfrak g designating the Lie algebra of G G . We consider the algebra D π ( G ) K ≃ ( U ( g ) / ker ⁡ ( π ) ) K D_{\pi }(G)^K \simeq \left (\mathscr U(\mathfrak g)/\operatorname {ker}(\pi )\right )^K of the K K -invariant elements of U ( g ) / ker ⁡ ( π ) \mathscr U(\mathfrak g)/\operatorname {ker}(\pi ) . It turns out that this algebra is commutative if and only if the restriction π | K \pi |_K of π \pi to K K has finite multiplicities (cf. Baklouti and Fujiwara [J. Math. Pures Appl. (9) 83 (2004), pp. 137-161]). In this article we suppose this eventuality and we provide a proof of the polynomial conjecture asserting that D π ( G ) K D_{\pi }(G)^K is isomorphic to the algebra C [ Ω ( π ) ] K \mathbb C[\Omega (\pi )]^K of K K -invariant polynomial functions on Ω ( π ) \Omega (\pi ) . The conjecture was partially solved in our previous works (Baklouti, Fujiwara, and Ludwig [Bull. Sci. Math. 129 (2005), pp. 187-209]; J. Lie Theory 29 (2019), pp. 311-341).

Smooth lattice orbits of nilpotent groups and strict comparison of projections

This paper provides sufficient density conditions for the existence of smooth vectors generating a frame or Riesz sequence in the lattice orbit of a square-integrable projective representation of a nilpotent Lie group. The conditions involve the product of lattice co-volume and formal dimension, and complement Balian–Low type theorems for the non-existence of smooth frames and Riesz sequences at the critical density. The proof hinges on a connection between smooth lattice orbits and generators for an explicitly constructed finitely generated Hilbert C⁎-module. An important ingredient in the approach is that twisted group C⁎-algebras associated to finitely generated nilpotent groups have finite decomposition rank, hence finite nuclear dimension, which allows us to deduce that any matrix algebra over such a simple C⁎-algebra has strict comparison of projections.

A non Ricci-flat Einstein pseudo-Riemannian metric on a 7-dimensional nilmanifold

We answer in the affirmative the question posed by Conti and Rossi [7,8] on the existence of nilpotent Lie algebras of dimension 7 with an Einstein pseudo-metric of nonzero scalar curvature. Indeed, we construct a left-invariant pseudo-Riemannian metric $g$ of signature $(3, 4)$ on a nilpotent Lie group of dimension 7, such that $g$ is Einstein and not Ricci-flat. We show that the pseudo-metric $g$ cannot be induced by any left-invariant closed $G_2^*$-structure on the Lie group. Moreover, some results on closed and harmonic $G_2^*$-structures on an arbitrary 7-manifold $M$ are given. In particular, we prove that the underlying pseudo-Riemannian metric of a closed and harmonic $G_2^*$-structure on $M$ is not necessarily Einstein, but if it is Einstein then it is Ricci-flat.

Pluriclosed and Strominger Kähler–like metrics compatible with abelian complex structures

We show that the existence of a left-invariant pluriclosed Hermitian metric on a unimodular Lie group with a left-invariant abelian complex structure forces the group to be 2-step nilpotent. Moreover, we prove that the pluriclosed flow starting from a left-invariant Hermitian metric on a 2-step nilpotent Lie group preserves the Strominger Kähler–like condition.

On the speed of convergence to the asymptotic cone for non-singular nilpotent groups

We study the speed of convergence to the asymptotic cone for a finitely generated nilpotent group endowed with a word metric. The first result on this theme is given by Burago who showed that an abelian group endowed with a word metric converges to the normed space with the speed \(O\left( \frac{1}{n}\right) \) in the sense of Gromov–Hausdorff distance. Later Krat showed the same statement for the Heisenberg group, and Breuillard and Le Donne constructed an example, the direct product of the \(\mathbb {Z}\) and the Heisenberg group with a specific word metric, whose speed of convergence is precisely \(O\left( \frac{1}{\sqrt{n}}\right) \). For 2-step nilpotent groups, we show that if the asymptotic cone is non-singular, then the speed of convergence is \(O\left( \frac{1}{n}\right) \) for any choice of generating set. Our argument can be applied to every nilpotent Lie group with a left-invariant sub-Finsler metric. In terms of sub-Finsler geometry, the condition being non-singular is equivalent to the strongly bracket generating condition, and also to absence of abnormal curves.

Left-invariant optimal control problems on Lie groups: classification and problems integrable by elementary functions

AbstractLeft-invariant optimal control problems on Lie groups are an important class of problems with a large symmetry group. They are theoretically interesting because they can often be investigated in full and general laws can be studied by using these model problems. In particular, problems on nilpotent Lie groups provide a fundamental nilpotent approximation to general problems. Also, left-invariant problems often arise in applications such as classical and quantum mechanics, geometry, robotics, visual perception models, and image processing.The aim of this paper is to present a survey of the main concepts, methods, and results pertaining to left-invariant optimal control problems on Lie groups that can be integrated by elementary functions. The focus is on describing extremal trajectories and their optimality, the cut time and cut locus, and optimal synthesis. Questions concerning the classification of left-invariant sub-Riemannian problems on Lie groups of dimension three and four are also addressed.Bibliography: 91 titles.

On the Signature of the Ricci Curvature on Nilmanifolds

We completely describe the signatures of the Ricci curvature of left-invariant Riemannian metrics on arbitrary real nilpotent Lie groups. The main idea in the proof is to exploit a link between the kernel of the Ricci endomorphism and closed orbits in a certain representation of the general linear group, which we prove using the ‘real GIT’ framework for the Ricci curvature of nilmanifolds.

Polynomial and horizontally polynomial functions on Lie groups

We generalize both the notion of polynomial functions on Lie groups and the notion of horizontally affine maps on Carnot groups. We fix a subset S of the algebra {mathfrak {g}} of left-invariant vector fields on a Lie group {mathbb {G}} and we assume that S Lie generates {mathfrak {g}}. We say that a function f:{mathbb {G}}rightarrow {mathbb {R}} (or more generally a distribution on {mathbb {G}}) is S-polynomial if for all Xin S there exists kin {mathbb {N}} such that the iterated derivative X^k f is zero in the sense of distributions. First, we show that all S-polynomial functions (as well as distributions) are represented by analytic functions and, if the exponent k in the previous definition is independent on Xin S, they form a finite-dimensional vector space. Second, if {mathbb {G}} is connected and nilpotent, we show that S-polynomial functions are polynomial functions in the sense of Leibman. The same result may not be true for non-nilpotent groups. Finally, we show that in connected nilpotent Lie groups, being polynomial in the sense of Leibman, being a polynomial in exponential chart, and the vanishing of mixed derivatives of some fixed degree along directions of {mathfrak {g}} are equivalent notions.

Towards semi-classical analysis for sub-elliptic operators

We discuss the recent developments of semi-classical and micro-local analysis in the context of nilpotent Lie groups and for sub-elliptic operators. In particular, we give an overview of pseudo-differential calculi recently defined on nilpotent Lie groups as well as of the notion of quantum limits in the Euclidean and nilpotent cases.