Two-step Nilpotent Lie Group Research Articles

Ricci bi-conformal vector fields on Lorentzian five-dimensional two-step nilpotent Lie groups

In this paper, we completely classify Ricci bi-conformal vector fields on simply-connected five-dimensional two-step nilpotent Lie groups which are also connected and we show which of them are the Killing vector fields and gradient vector fields.

Complexity of non-abelian cut-and-project sets of polytopal type I: special homogeneous Lie groups

Abstract The aim of this paper is to determine the asymptotic growth rate of the complexity function of cut-and-project sets in the non-abelian case. In the case of model sets of polytopal type in homogeneous two-step nilpotent Lie groups, we can establish that the complexity function asymptotically behaves like $r^{{\mathrm {homdim}}(G) \dim (H)}$ . Further, we generalize the concept of acceptance domains to locally compact second countable groups.

Singular spherical maximal operators on a class of degenerate two-step nilpotent Lie groups

Singular spherical maximal operators on a class of degenerate two-step nilpotent Lie groups

On the classification of sub-Riemannian structures on a 5D two-step nilpotent Lie group

We classify the left-invariant sub-Riemannian structures on the unique five-dimensional simply connected two-step nilpotent Lie group with two-dimensional commutator subgroup; this 5D group is the first twostep nilpotent Lie group beyond the three-and five-dimensional Heisenberg groups. Alongside, we also present a classification, up to automorphism, of the subspaces of the associated Lie algebra (together with a complete set of invariants).

Diagram involutions and homogeneous Ricci-flat metrics

We introduce a combinatorial method to construct indefinite Ricci-flat metrics on nice nilpotent Lie groups. We prove that every nilpotent Lie group of dimension $\leq6$, every nice nilpotent Lie group of dimension $\leq7$ and every two-step nilpotent Lie group attached to a graph admits such a metric. We construct infinite families of Ricci-flat nilmanifolds associated to parabolic nilradicals in the simple Lie groups ${\rm SL}(n)$, ${\rm SO}(p,q)$, ${\rm Sp}(n,\mathbb R)$. Most of these metrics are shown not to be flat.

Yamabe Flow On Nilpotent Lie Groups

In this paper, we consider the Yamabe flow on Lie groups with left invariant metrics in particular case on the higher dimensional classical Heisenberg nilpotent Lie groups and the higher dimensional quaternion Lie groups. We construct a explicit solution of the Yamabe flow on each group. Then we investigate the some properties on these Lie groups under the Yamabe flow. At the end of paper, we study the deformation of some feature of compact nilmanifolds $$\Gamma {\setminus } N$$ under the Yamabe flow, where N is a simply connected two-step nilpotent Lie group with a left invariant metric, and $$\Gamma $$ is a discrete cocompact subgroup of N, in particular Heisenberg and quaternion Lie groups.

On the isomorphism problem for C∗-algebras of nilpotent Lie groups

We investigate to what extent a nilpotent Lie group is determined by its [Formula: see text]-algebra. We prove that, within the class of exponential Lie groups, direct products of Heisenberg groups with abelian Lie groups are uniquely determined even by their unitary dual, while nilpotent Lie groups of dimension [Formula: see text] are uniquely determined by the Morita equivalence class of their [Formula: see text]-algebras. We also find that this last property is shared by the filiform Lie groups and by the [Formula: see text]-dimensional free two-step nilpotent Lie group.

Rigidity for group actions on homogeneous spaces by affine transformations

We give a criterion for the rigidity of the action of a group of affine transformations of a homogeneous space of a real Lie group. Let $G$ be a real Lie group, $\unicode[STIX]{x1D6EC}$ a lattice in $G$, and $\unicode[STIX]{x1D6E4}$ a subgroup of the affine group $\text{Aff}(G)$ stabilizing $\unicode[STIX]{x1D6EC}$. Then the action of $\unicode[STIX]{x1D6E4}$ on $G/\unicode[STIX]{x1D6EC}$ has the rigidity property in the sense of Popa [On a class of type $\text{II}_{1}$ factors with Betti numbers invariants. Ann. of Math. (2)163(3) (2006), 809–899] if and only if the induced action of $\unicode[STIX]{x1D6E4}$ on $\mathbb{P}(\mathfrak{g})$ admits no $\unicode[STIX]{x1D6E4}$-invariant probability measure, where $\mathfrak{g}$ is the Lie algebra of $G$. This generalizes results of Burger [Kazhdan constants for $\text{SL}(3,\mathbf{Z})$. J. Reine Angew. Math.413 (1991), 36–67] and Ioana and Shalom [Rigidity for equivalence relations on homogeneous spaces. Groups Geom. Dyn.7(2) (2013), 403–417]. As an application, we establish rigidity for the action of a class of groups acting by automorphisms on nilmanifolds associated to two-step nilpotent Lie groups.

Leibniz’s rule on two-step nilpotent Lie groups

Let $\mathfrak{g}$ be a nilpotent Lie algebra which is also regarded as a homogeneous Lie group with the Campbell-Hausdorff multiplication. This allows to define a generalized multiplication $f \# g = (f^{\vee} * g^{\vee})^{\wedge}$ of two functions in the Schwartz class $\mathcal{S}(\mathfrak{g}^{*})$, where $\vee$ and $\wedge$ are the Abelian Fourier transforms on the Lie algebra $\mathfrak{g}$ and on the dual $\mathfrak{g}^{*}$. In the operator analysis on nilpotent Lie groups an important notion is the one of symbolic calculus which can be viewed as a higher order generalization of the Weyl calculus for pseudodifferential operators of Hormander. The idea of such a calculus consists in describing the product $f \# g$ for some classes of symbols. We find a formula for $D^{\alpha}(f \# g)$ for Schwartz functions $f,g$ in the case of two-step nilpotent Lie groups, that includes the Heisenberg group. We extend this formula to the class of functions $f,g$ such that $f^{\vee}, g^{\vee}$ are certain distributions acting by convolution on the Lie group, that includes usual classes of symbols. In the case of the Abelian group $R^{d}$ we have $f \# g = fg$, so $D^{\alpha}(f \# g)$ is given by the Leibniz rule.

The $$C^{*}$$ C ∗ -algebras of connected real two-step nilpotent Lie groups

Using the operator valued Fourier transform, the \(C^{*}\)-algebras of connected real two-step nilpotent Lie groups are characterized as algebras of operator fields defined over their spectra. In particular, it is shown by explicit computations, that the Fourier transform of such \(C^{*}\)-algebras fulfills the norm controlled dual limit property.

SHANNON-LIKE PARSEVAL FRAME WAVELETS ON SOME TWO STEP NILPOTENT LIE GROUPS

We construct Shannon-like Parseval frame wavelets on a class of non commutative two-step nilpotent Lie groups. Our work was inspired by a construction given by Azita Mayeli on the Heisenberg group. The tools used here are representation theoretic. However, a great deal of Gabor theory is used for the construction of the wavelets. The construction obtained here is very explicit, and we are even able to compute an upper bound for the L 2 norm

On Flag Curvature of Homogeneous Randers Spaces

AbstractIn this paper we give an explicit formula for the flag curvature of homogeneous Randers spaces of Douglas type and apply this formula to obtain some interesting results. We first deduce an explicit formula for the flag curvature of an arbitrary left invariant Randersmetric on a two-step nilpotent Lie group. Then we obtain a classification of negatively curved homogeneous Randers spaces of Douglas type. This results, in particular, in many examples of homogeneous non-Riemannian Finsler spaces with negative flag curvature. Finally, we prove a rigidity result that a homogeneous Randers space of Berwald type whose flag curvature is everywhere nonzero must be Riemannian.

Existence of isometric immersions into nilpotent Lie groups

We establish necessary and sufficient conditions for existence of isometric immersions of a simply connected Riemannian manifold into a two-step nilpotent Lie group. This comprises the case of immersions into H-type groups.

ON THE RANDERS METRICS ON TWO-STEP HOMOGENEOUS NILMANIFOLDS OF DIMENSION FIVE

In this paper we study the geometry of simply connected two-step nilpotent Lie groups of dimension five. We give the Levi–Civita connection, curvature tensor, sectional and scalar curvatures of these spaces and show that they have constant negative scalar curvature. Also we show that the only space which admits left-invariant Randers metric of Berwald type has three-dimensional center. In this case the explicit formula for computing flag curvature is obtained and it is shown that flag curvature and sectional curvature have the same sign.

Nilpotent Gelfand pairs and spherical transforms of Schwartz functions I: rank-one actions on the centre

The spectrum of a Gelfand pair of the form \({(K\ltimes N,K)}\), where N is a nilpotent group, can be embedded in a Euclidean space \({{\mathbb R}^d}\). The identification of the spherical transforms of K-invariant Schwartz functions on N with the restrictions to the spectrum of Schwartz functions on \({{\mathbb R}^d}\) has been proved already when N is a Heisenberg group and in the case where N = N 3,2 is the free two-step nilpotent Lie group with three generators, with K = SO3 (Astengo et al. in J Funct Anal 251:772–791, 2007; Astengo et al. in J Funct Anal 256:1565–1587, 2009; Fischer and Ricci in Ann Inst Fourier Gren 59:2143–2168, 2009). We prove that the same identification holds for all pairs in which the K-orbits in the centre of N are spheres. In the appendix, we produce bases of K-invariant polynomials on the Lie algebra \({{\mathfrak n}}\) of N for all Gelfand pairs \({(K\ltimes N,K)}\) in Vinberg’s list (Vinberg in Trans Moscow Math Soc 64:47–80, 2003; Yakimova in Transform Groups 11:305–335, 2006).

On the Moore Formula of Compact Nilmanifolds

Let G be a connected and simply connected two-step nilpotent Lie group and a lattice subgroup of G. In this note, we give a new multiplicity formula, according to the sense of Moore, of irreducible unitary representations involved in the decomposition of the quasi-regular representation Ind G (1). Extending then the Abelian case.

Length minimizing geodesics and the length spectrum of Riemannian two-step nilmanifolds

In the first part of this article, we prove an explicit lower bound on the distance to the cut point of an arbitrary geodesic in a simply connected two-step nilpotent Lie group G with a lieft invariant metric. As a result, we obtaine a lower bound on the injectivity radius of a simply connected two-step nilpotent Lie group with a left invariant metric. We use this lower bound to determine the form of certain length minimizing geodesics from the identity to elements in the center of G. We also give an example of a two-step nilpotent Lie group G such that along most geodesics in this group, the cut point and the first conjugate point do not coincide. In the second part of this article, we examine the relation between the Laplace spectrum and the length spectrum on nilmanifolds by showing that a method developed by Gordon and Wilson for constructing families of isospectral two-step nilmanifolds necessarily yields manifolds with the same length spectrum. As a consequence, all known methods for constructing families of isospectral two-step nilmanifolds necessarily yield manifolds with the same length spectrum.

Two-Step Nilpotent Lie Groups and Homogeneous Fiber Bundles

In this paper, we construct two-step nilpotent Lie groups from homogeneous fiber bundles over compact symmetric spaces. The structure of the constructed nilpotent groups is expressed in terms of the compact Lie groups involved in the fiber bundles. There are close relations between the geometric properties of the nilpotent groups and the total spaces of the fiber bundles. We will find new examples of nilpotent groups which are weakly symmetric and Riemannian geodesic orbit spaces.

Some two-step and three-step nilpotent Lie groups with small automorphism groups

We construct examples of two-step and three-step nilpotent Lie groups whose automorphism groups are small in the sense of either not having a dense orbit for the action on the Lie group, or being nilpotent (the latter being stronger). From the results we also get new examples of compact manifolds covered by two-step simply connected nilpotent Lie groups which do not admit Anosov automorphisms.

Hypergroups associated to harmonic NA groups

AbstractA harmonic NA group is a suitable solvable extension of a two-step nilpotent Lie group N of Heisenberg type by R+, which acts on N by anisotropic dilations. A hypergroup is a locally compact space for which the space of Borel measures has a convolution structure preserving the probability measures and satisfying suitable conditions. We describe a class of hypergroups associated to NA groups.