Two-step Nilpotent Research Articles

Some properties of relative Rota–Baxter operators on groups

We find connection between relative Rota–Baxter operators and usual Rota–Baxter operators. We prove that any relative Rota–Baxter operator on a group H with respect to ( G , Ψ ) defines a Rota–Baxter operator on the semi-direct product H ⋊ Ψ G . On the other side, we give condition under which a Rota–Baxter operator on the semi-direct product H ⋊ Ψ G defines a relative Rota–Baxter operator on H with respect to ( G , Ψ ) . We introduce homomorphic post-groups and find their connection with λ-homomorphic skew left braces. Further, we construct post-group on arbitrary group and a family of post-groups which depends on integer parameter on any two-step nilpotent group. We find all verbal solutions of the quantum Yang-Baxter equation on two-step nilpotent group.

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.

The holomorphic discrete series contribution to the generalized Whittaker Plancherel formula II. Non-tube type groups

For every simple Hermitian Lie group G, we consider a certain maximal parabolic subgroup whose unipotent radical N is either abelian (if G is of tube type) or two-step nilpotent (if G is of non-tube type). By the generalized Whittaker Plancherel formula we mean the Plancherel decomposition of L2(G/N,ω), the space of square-integrable sections of the homogeneous vector bundle over G/N associated with an irreducible unitary representation ω of N. Assuming that the central character of ω is contained in a certain cone, we construct embeddings of all holomorphic discrete series representations of G into L2(G/N,ω) and show that the multiplicities are equal to the dimensions of the lowest K-types. The construction is in terms of a kernel function which can be explicitly defined using certain projections inside a complexification of G. This kernel function carries all information about the holomorphic discrete series embedding, the lowest K-type as functions on G/N, as well as the associated Whittaker vectors.

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.

Isotopisms of nilpotent Leibniz algebras and Lie racks

In this paper we study the isotopism classes of two-step nilpotent algebras. We show that every nilpotent Leibniz algebra g with dim [ g , g ] = 1 is isotopic to the Heisenberg Lie algebra or to the Heisenberg algebra l 2 n + 1 J 1 , where J 1 is the n × n Jordan block of eigenvalue 1. We also prove that two such algebras are isotopic if and only if the Lie racks integrating them are isotopic. This gives the classification of Lie racks whose tangent space at the unit element is a nilpotent Leibniz algebra with one-dimensional commutator ideal. Eventually, we introduce new isotopism invariants for Leibniz algebras and Lie racks.

Real non-degenerate two-step nilpotent Lie algebras of dimension eight

We classify the non-degenerate two-step nilpotent Lie algebras of dimension 8 over the field of real numbers, using known results over complex numbers. We write explicit structure constants for these real Lie algebras.

Уравнения от одной переменной над двуступенно нильпотентными группами и цепочки централизаторов

In the article we prove the equivalence of the concepts of equationally Noetherianity of a torsion-free two-step nilpotent group and the existence in it of an infinite chain of strictly decreasing centralizers and present the idea of a two-way connection between equationally Noetherian partially commutative two-step nilpotent groups and equationally Noetherian graphs with loops.

Extra-special Leibniz superalgebras

Abstract A two-step nilpotent Leibniz superalgebra L is called extra-special if the center and the derived subalgebra of L are equal and both are of dimension one. In this paper, the structure, the capability and the second homology of extra-special Leibniz superalgebras are determined.

Derivations of two-step nilpotent algebras

In this paper we study the Lie algebras of derivations of two-step nilpotent algebras. We obtain a class of Lie algebras with trivial center and abelian ideal of inner derivations. Among these, the relations between the complex and the real case of the indecomposable Heisenberg Leibniz algebras are thoroughly described. Finally we show that every almost inner derivation of a complex nilpotent Leibniz algebra with one-dimensional commutator ideal, with three exceptions, is an inner derivation.

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).

The Addition Theorem for two-step nilpotent torsion groups

Abstract The Addition Theorem for the algebraic entropy of group endomorphisms of torsion abelian groups was proved in [D. Dikranjan, B. Goldsmith, L. Salce and P. Zanardo, Algebraic entropy for abelian groups, Trans. Amer. Math. Soc. 361 (2009), 7, 3401–3434]. Later, this result was extended to all abelian groups [D. Dikranjan and A. Giordano Bruno, Entropy on abelian groups, Adv. Math. 298 (2016), 612–653] and, recently, to all torsion finitely quasihamiltonian groups [A. Giordano Bruno and F. Salizzoni, Additivity of the algebraic entropy for locally finite groups with permutable finite subgroups, J. Group Theory 23 (2020), 5, 831–846]. In contrast, when it comes to metabelian groups, the additivity of the algebraic entropy fails [A. Giordano Bruno and P. Spiga, Some properties of the growth and of the algebraic entropy of group endomorphisms, J. Group Theory 20 (2017), 4, 763–774]. Continuing the research within the class of locally finite groups, we prove that the Addition Theorem holds for two-step nilpotent torsion groups.

Vacuum Einstein field equations in smooth metric measure spaces: the isotropic case* *Supported by projects PID2019-105138GB-C21/AEI/10.13039/501100011033 (Spain) and ED431C 2019/10 (Xunta de Galicia, Spain).

On a smooth metric measure spacetime (M, g, e −f dvol g ), we define a weighted Einstein tensor. It is given in terms of the Bakry–Émery Ricci tensor as a tensor which is symmetric, divergence-free, concomitant of the metric and the density function. We consider the associated vacuum weighted Einstein field equations and show that isotropic solutions have nilpotent Ricci operator. Moreover, the underlying manifold is a Brinkmann wave if it is two-step nilpotent and a Kundt spacetime if it is three-step nilpotent. More specific results are obtained in dimension 3, where all isotropic solutions are given in local coordinates as plane waves or Kundt spacetimes.

Two-step nilpotent Leibniz algebras

In this paper we give a complete classification of two-step nilpotent Leibniz algebras in terms of Kronecker modules associated with pairs of bilinear forms. Among these, we describe the complex and the real case of the indecomposable Heisenberg Leibniz algebras as a generalization of the classical (2n+1)-dimensional Heisenberg Lie algebra h2n+1. Then we use the Leibniz algebras - Lie local racks correspondence proposed by S. Covez to show that nilpotent real Leibniz algebras have always a global integration. As an application, we integrate the indecomposable nilpotent real Leibniz algebras with one-dimensional commutator ideal. Finally we show that every Lie quandle integrating a Leibniz algebra is induced by the conjugation of a Lie group and the Leibniz algebra is the Lie algebra of that Lie group.

Local Root Numbers for Heisenberg-Representations — Some Explicit Results

Heisenberg representations [Formula: see text] of (pro-)finite groups [Formula: see text] are by definition irreducible representations of the two-step nilpotent factor group [Formula: see text] Better known are Heisenberg groups which can be understood as allowing faithful Heisenberg representations. A special feature is that [Formula: see text] will be induced by characters [Formula: see text] of subgroups in multiple ways, where the pairs [Formula: see text] can be interpreted as maximal isotropic pairs. If [Formula: see text] is a [Formula: see text]-adic number field and [Formula: see text] the absolute Galois group then maximal isotropic pairs rewrite as [Formula: see text] where [Formula: see text] is an abelian extension and [Formula: see text] a character. We will consider the extended local Artin-root-number [Formula: see text] for those [Formula: see text] which are essentially tame and express it by a formula not depending on the various maximal isotropic pairs [Formula: see text] for [Formula: see text]

HOMOGENEOUS RIEMANNIAN MANIFOLDS WITH NON-TRIVIAL NULLITY

We develop a general theory for irreducible homogeneous spaces M = G/H, in relation to the nullity distribution ν of their curvature tensor. We construct natural invariant (different and increasing) distributions associated with the nullity, that give a deep insight of such spaces. In particular, there must exist an order-two transvection, not in the nullity, with null Jacobi operator. This fact was very important for finding out the first homogeneous examples with non-trivial nullity, i.e., where the nullity distribution is not parallel. Moreover, we construct irreducible examples of conullity k = 3, the smallest possible, in any dimension. None of our examples admit a quotient of finite volume. We also proved that H is trivial and G is solvable if k = 3. Another of our main results is that the leaves, i.e., the integral manifolds, of the nullity are closed (we used a rather delicate argument). This implies that M is a Euclidean affine bundle over the quotient by the leaves of ν. Moreover, we prove that ν⊥ defines a metric connection on this bundle with transitive holonomy or, equivalently, ν⊥ is completely non-integrable (this is not in general true for an arbitrary autoparallel and at invariant distribution). We also found some general obstruction for the existence of non-trivial nullity: e.g., if G is reductive (in particular, if M is compact), or if G is two-step nilpotent.

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.

Periodic Time-Optimal Controls on Two-Step Free-Nilpotent Lie Groups

For two-step free nilpotent Lie algebras, we describe symplectic foliations and Casimir functions. A left-invariant time-optimal problem is considered in which the set of admissible controls is given by a strictly convex compact set in the first layer of the Lie algebra that contains the origin in its interior. We describe integrals for the vertical subsystem of the Hamiltonian system of the Pontryagin maximum principle. The properties of solutions to this system for low ranks of the Poisson bivector are described.

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.