2-step Nilpotent Lie Groups Research Articles

Magnetic fields on non-singular 2-step nilpotent Lie groups

This work has a twofold purpose - the existence study of closed 2-forms, known as magnetic fields, on 2-step nilpotent Lie groups and generating examples for the non-singular family. At the Lie algebra level, while the existence of closed 2-forms for which the center is either nondegenerate or in the kernel of the 2-form, is always guaranteed, the existence of closed 2-forms for which the center is isotropic but not in the kernel of the 2-form, is a special situation. These 2-forms are called of type II. We obtain a strong obstruction for the existence on non-singular Lie algebras. Moreover, we prove that the only H-type Lie groups admitting left-invariant closed 2-forms of type II are the real, complex and quaternionic Heisenberg Lie groups of dimension three, six and seven, respectively. We also prove the non-existence of uniform magnetic fields under certain hypotheses. Finally we give a construction of non-singular Lie algebras, proving that in some families of these examples there are no closed 2-form of type II.

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.

Conformal Killing forms on 2-step nilpotent Riemannian Lie groups

Abstract We study left-invariant conformal Killing 2- or 3-forms on simply connected 2-step nilpotent Riemannian Lie groups. We show that if the center of the group is of dimension greater than or equal to 4, then every such form is automatically coclosed (i.e. it is a Killing form). In addition, we prove that the only Riemannian 2-step nilpotent Lie groups with center of dimension at most 3 and admitting left-invariant non-coclosed conformal Killing 2- and 3-forms are the following: The Heisenberg Lie groups and their trivial 1-dimensional extensions, endowed with any left-invariant metric, and the simply connected Lie group corresponding to the free 2-step nilpotent Lie algebra on 3 generators, with a particular 1-parameter family of metrics. The explicit description of the space of conformal Killing 2- and 3-forms is provided in each case.

Purely coclosed G$$_{\mathbf {2}}$$-structures on 2-step nilpotent Lie groups

We consider left-invariant (purely) coclosed G $$_2$$ -structures on 7-dimensional 2-step nilpotent Lie groups. According to the dimension of the commutator subgroup, we obtain various criteria characterizing the Riemannian metrics induced by left-invariant purely coclosed G $$_2$$ -structures. Then, we use them to determine the isomorphism classes of 2-step nilpotent Lie algebras admitting such type of structures. As an intermediate step, we show that every metric on a 2-step nilpotent Lie algebra admitting coclosed G $$_2$$ -structures is induced by one of them. Finally, we use our results to give the explicit description of the metrics induced by purely coclosed G $$_2$$ -structures on 2-step nilpotent Lie algebras with derived algebra of dimension at most two, up to automorphism.

Positive Hermitian curvature flow on complex 2-step nilpotent Lie groups

We study the positive Hermitian curvature flow of left-invariant metrics on complex 2-step nilpotent Lie groups. In this setting we completely characterize the long-time behaviour of the flow, showing that normalized solutions to the flow subconverge to a non-flat algebraic soliton, in Cheeger–Gromov topology. We also exhibit a uniqueness result for algebraic solitons on such Lie groups.

Killing–Yano 2-forms on 2-step nilpotent Lie groups

Fil: Andrada, Adrian Marcelo. Consejo Nacional de Investigaciones Cientificas y Tecnicas. Centro Cientifico Tecnologico Conicet - Cordoba. Centro de Investigacion y Estudios de Matematica. Universidad Nacional de Cordoba. Centro de Investigacion y Estudios de Matematica; Argentina. Universidad Nacional de Cordoba. Facultad de Matematica, Astronomia y Fisica; Argentina

Higher degree Killing forms on 2-step nilmanifolds

We study left-invariant Killing forms of arbitrary degree on simply connected 2-step nilpotent Lie groups endowed with left-invariant Riemannian metrics, and classify them when the center of the group is at most two-dimensional.

A Generalized Gelfand Pair Attached to a 3-Step Nilpotent Lie Group

Fil: Gallo, Andrea Lilen. Consejo Nacional de Investigaciones Cientificas y Tecnicas. Centro Cientifico Tecnologico Conicet - Cordoba. Centro de Investigacion y Estudios de Matematica. Universidad Nacional de Cordoba. Centro de Investigacion y Estudios de Matematica; Argentina. Universidad Nacional de Cordoba. Facultad de Matematica, Astronomia y Fisica; Argentina

Fundamental solutions of a class of ultra-hyperbolic operators on pseudo H-type groups

Pseudo H-type Lie groups Gr,s of signature (r,s) are defined via a module action of the Clifford algebra Cℓr,s on a vector space V≅R2n. They form a subclass of all 2-step nilpotent Lie groups. Based on their algebraic structure they can be equipped with a left-invariant pseudo-Riemannian metric. Let Nr,s denote the Lie algebra corresponding to Gr,s. In the case s>0 a choice of left-invariant vector fields [X1,…,X2n] which generate a complement of the center of Nr,s gives rise to a second order differential operatorΔr,s:=(X12+…+Xn2)−(Xn+12+…+X2n2), which we call ultra-hyperbolic. We prove that Δr,s is locally solvable if and only if r=0. In particular, it follows that Δr,s does not admit a fundamental solution in the space D′(Gr,s) of Schwartz distributions whenever r>0. In terms of classical special functions we present families of fundamental solutions of Δ0,s in the class of tempered distributions S′(G0,s) and study their properties.

AN UNCERTAINTY PRINCIPLE FOR SOLUTIONS OF THE SCHRÖDINGER EQUATION ON -TYPE GROUPS

In this paper we consider uncertainty principles for solutions of certain partial differential equations on $H$ -type groups. We first prove that, on $H$ -type groups, the heat kernel is an average of Gaussians in the central variable, so that it does not satisfy a certain reformulation of Hardy’s uncertainty principle. We then prove the analogue of Hardy’s uncertainty principle for solutions of the Schrödinger equation with potential on $H$ -type groups. This extends the free case considered by Ben Saïd et al. [‘Uniqueness of solutions to Schrödinger equations on H-type groups’, J. Aust. Math. Soc. (3)95 (2013), 297–314] and by Ludwig and Müller [‘Uniqueness of solutions to Schrödinger equations on 2-step nilpotent Lie groups’, Proc. Amer. Math. Soc.142 (2014), 2101–2118].

Symmetric Killing tensors on nilmanifolds

We study left-invariant symmetric Killing 2-tensors on 2-step nilpotent Lie groups endowed with a left-invariant Riemannian metric, and construct genuine examples, which are not linear combinations of parallel tensors and symmetric products of Killing vector fields.

Killing Forms on 2-Step Nilmanifolds

We study left-invariant Killing $k$-forms on simply connected $2$-step nilpotent Lie groups endowed with a left-invariant Riemannian metric. For $k=2,3$, we show that every left-invariant Killing $k$-form is a sum of Killing forms on the factors of the de Rham decomposition. Moreover, on each irreducible factor, non-zero Killing $2$-forms define (after some modification) a bi-invariant orthogonal complex structure and non-zero Killing $3$-forms arise only if the Riemannian Lie group is naturally reductive when viewed as a homogeneous space under the action of its isometry group. In both cases, $k=2$ or $k=3$, we show that the space of Killing $k$-forms of an irreducible Riemannian 2-step nilpotent Lie group is at most one-dimensional.

A support theorem for the X-ray transform on manifolds with plane covers

AbstractThis paper is concerned with support theorems of the X-ray transform on non-compact manifolds with conjugate points. In particular, we prove that all simply connected 2-step nilpotent Lie groups have a support theorem. Important ingredients of the proof are the concept of plane covers and a support theorem for simple manifolds by Krishnan. We also provide examples of non-homogeneous 3-dimensional simply connected manifolds with conjugate points which have support theorems.

The long‐time behavior of the homogeneous pluriclosed flow

We study the asymptotic behavior of the pluriclosed flow in the case of left-invariant Hermitian structures on Lie groups. We prove that solutions on 2-step nilpotent Lie groups and on almost-abelian Lie groups converge, after a suitable normalization, to self-similar solutions of the flow. Given that the spaces are solvmanifolds, an unexpected feature is that some of the limits are shrinking solitons. We also exhibit the first example of a homogeneous manifold on which a geometric flow has some solutions with finite extinction time and some that exist for all positive times.

English

We study a problem of isometric compact 2-step nilmanifolds ${M}/\Gamma $ using some information on their geodesic flows, where $M$ is a simply connected 2-step nilpotent Lie group with a left invariant metric and $\Gamma $ is a cocompact discrete subgroup of isometries of $M$. Among various works concerning this problem, we consider the algebraic aspect of it. In fact, isometry groups of simply connected Riemannian manifolds can be characterized in a purely algebraic way, namely by normalizers. So, suitable factorization of normalizers and expression of a vector bundle as an associated fiber bundle to a principal bundle, lead us to a general framework, namely groupoids. In this way, drawing upon advanced ingredients of Lie groupoids, normal subgroupoid systems and other notions, not only an answer in some sense to our rigidity problem has been given, but also the dependence between normalizers, automorphisms and especially almost inner automorphisms, has been clarified.

Graphs and metric 2-step nilpotent Lie algebras

Abstract Dani and Mainkar introduced a method for constructing a 2-step nilpotent Lie algebra 𝔫 G from a simple directed graph G in 2005. There is a natural inner product on 𝔫 G arising from the construction. We study geometric properties of the associated simply connected 2-step nilpotent Lie group N with Lie algebra 𝔫 g . We classify singularity properties of the Lie algebra 𝔫 g in terms of the graph G. A comprehensive description is given of graphs G which give rise to Heisenberg-like Lie algebras. Conditions are given on the graph G and on a lattice Γ ⊆ N for which the quotient Γ \ N, a compact nilmanifold, has a dense set of smoothly closed geodesics. This paper provides the first investigation connecting graph theory, 2-step nilpotent Lie algebras, and the density of closed geodesics property.

A remark on the Bismut–Ricci form on 2-step nilmanifolds

In this note, we observe that, on a 2-step nilpotent Lie group equipped with a left-invariant SKT structure, the (1,1)-part of the Bismut–Ricci form is seminegative definite. As an application, we give a simplified proof of the non-existence of invariant SKT static metrics on 2-step nilmanifolds and of the existence of a long-time solution to the pluriclosed flow in 2-step nilmanifolds.

Central conjugate locus of 2-step nilpotent Lie groups

We compute the conjugate locus with multiplicities of a geodesic that lies in the center of a 2-step nilpotent Lie group N with a left invariant metric. We compare the isometry types of two such nilpotent groups whose conjugate loci for central geodesics are the same in a suitable sense. This comparison is related to work of Gordon and Wilson (J Differ Geom 47(3):504–529, 1997).

Totally geodesic subalgebras in 2-step nilpotent Lie algebras

We describe totally geodesic subalgebras of a metric 2-step nilpotent Lie algebra $\n$. We prove that a totally geodesic subalgebra of $\n$ is either abelian and flat or can be decomposed as a direct sum determined by the curvature transformation. In addition, we give conditions under which a totally geodesic submanifold of a simply connected 2-step nilpotent Lie group is a totally geodesic subgroup. We follow Eberlein's 1994 paper in which he imposes the condition of nonsingularity on $\n$. We remove this restriction and illustrate the distinction between the nonsingular case and the unrestricted case.

Bandlimited spaces on some 2-step nilpotent Lie groups with one Parseval frame generator

Let N be a step two connected and simply connected non commutative nilpotent Lie group which is square-integrable modulo the center. Let Z be the center of N. Assume that N = P ⋊M such that P, and M are simply connected, connected abelian Lie groups, M acts non-trivially on P by automorphisms and dimP/Z = dimM. We study band-limited subspaces of L 2 (N) which admit Parseval frames generated by discrete translates of a single function. We also find characteristics of band-limited subspaces of L 2 (N) which do not admit a single Parseval frame. We also provide some conditions under which continuous wavelets transforms related to the left regular representation admit discretization, by some discrete set ⊂ N. Finally, we show some explicit examples in the last section.