Solvable Lie Groups Research Articles

Compatibility of Balanced and SKT Metrics on Two-Step Solvable Lie Groups

AbstractIt has been conjectured by Fino and Vezzoni that a compact complex manifold admitting both a compatible SKT and a compatible balanced metric also admits a compatible Kähler metric. Using the shear construction and classification results for two-step solvable SKT Lie algebras from our previous work, we prove this conjecture for compact two-step solvmanifolds endowed with an invariant complex structure which is either (a) of pure type or (b) of dimension six. In contrast, we provide two counterexamples for a natural generalisation of this conjecture in the homogeneous invariant setting. As part of the work, we obtain further classification results for invariant SKT, balanced and Kähler structures on two-step solvable Lie groups. In particular, we give the full classification of left-invariant SKT structures on two-step solvable Lie groups in dimension six.

Coherence of augmented Iwasawa algebras

The augmented Iwasawa algebra of a p-adic Lie group is a generalisation of the Iwasawa algebra of a compact p-adic Lie group. We prove that a split-semisimple group over a p-adic field has a coherent augmented Iwasawa algebra if and only if its root system is of rank one. We deduce that the general linear group of degree n has a coherent augmented Iwasawa algebra precisely when n is at most two. We also characterise when certain solvable p-adic Lie groups have a coherent augmented Iwasawa algebra.

FRÖLICHER SPECTRAL SEQUENCE AND HODGE STRUCTURES ON THE COHOMOLOGY OF COMPLEX PARALLELISABLE MANIFOLDS

For complex parallelisable manifolds Γ\\G, with G a solvable or semisimple complex Lie group, the Frölicher spectral sequence degenerates at the second page. In the solvable case, the de Rham cohomology carries a pure Hodge structure. In contrast, in the semisimple case, purity depends on the lattice, but there is always a direct summand of the de Rham cohomology which does carry a pure Hodge structure and is independent of the lattice.

Critical metrics for quadratic curvature functionals on some solvmanifolds

We prove the existence of four-dimensional compact manifolds admitting some non-Einstein Lorentzian metrics, which are critical points for all quadratic curvature functionals. For this purpose, we consider left-invariant semi-direct extensions G_{mathcal S}=H rtimes exp ({mathbb {R}}S) of the Heisenberg Lie group H, for any mathcal S in {mathfrak {s}}{mathfrak {p}}(1,mathbb R), equipped with a family g_a of left-invariant metrics. After showing the existence of lattices in all these four-dimensional solvable Lie groups, we completely determine when g_a is a critical point for some quadratic curvature functionals. In particular, some four-dimensional solvmanifolds raising from these solvable Lie groups admit non-Einstein Lorentzian metrics, which are critical for all quadratic curvature functionals.

Almost co-Kähler manifolds and [formula omitted]-quasi-Einstein solitons

The present paper aims to investigate (m,ρ)-quasi-Einstein metrics on almost co-Kähler manifolds M. It is proven that if a (κ,μ)-almost co-Kähler manifold with κ<0 is (m,ρ)-quasi-Einstein manifold, then M represents a N(κ)-almost co-Kähler manifold and the manifold is locally isomorphic to a solvable non-nilpotent Lie group. Next, we study the three dimensional case and get the above mentioned result along with the manifold M3 becoming an η-Einstein manifold. We also show that there does not exist (m,ρ)-quasi-Einstein structure on a compact (κ,μ)-almost co-Kähler manifold of dimension greater than three with κ<0. Further, we prove that an almost co-Kähler manifold satisfying η-Einstein condition with constant coefficients reduces to a K-almost co-Kähler manifold, provided ma1≠(2n−1)b1 and m≠1. We also characterize perfect fluid spacetime whose Lorentzian metric is equipped with (m,ρ)-quasi Einstein solitons and acquired that the perfect fluid spacetime has vanishing vorticity, or it represents dark energy era under certain restriction on the potential function. Finally, we construct an example of an almost co-Kähler manifold with (m,ρ)-quasi-Einstein solitons.

Almost Kähler structures on complex hyperbolic space

Complex hyperbolic space has the structure of a solvable Lie group. Based on the recent classification of Riemannian left-invariant metrics we describe all left-invariant almost K?hler structures on that group. The only K?hler structure is the standard structure of the complex hyperbolic space and all others are strictly almost K?hler. We calculated the Chern connection and obtained the characterization by its torsion tensor. It is proved that the only SCF-soliton is the K?hler one.

The spheres of Sol

Let Sol be the three-dimensional solvable Lie group equipped with its standard left-invariant Riemannian metric. We give a precise description of the cut locus of the identity, and a maximal domain in the Lie algebra on which the Riemannian exponential map is a diffeomorphism. As a consequence, we prove that the metric spheres in Sol are topological spheres, and we characterize their singular points almost exactly.

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.

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.

Classification of Left Invariant Riemannian Metrics on Complex Hyperbolic Space

It is well known that \({\mathbb {C}}H^n\) has the structure of a solvable Lie group with left invariant metric of constant holomorphic sectional curvature. In this paper we give the full classification of all possible left invariant Riemannian metrics on this Lie group. We prove that each of those metrics is of constant negative scalar curvature, only one of them being Einstein (up to isometry and scaling).

Continuity Criterion for Locally Bounded Finite-Dimensional Representations of Simply Connected Solvable Lie Groups

Continuity Criterion for Locally Bounded Finite-Dimensional Representations of Simply Connected Solvable Lie Groups

Left invariant Riemannian metrics with harmonic curvature are Ricci-parallel in solvable Lie groups and Lie groups of dimension ≤6

We show that any left invariant metric with harmonic curvature on a solvable Lie group is Ricci-parallel. We show the same result for any Lie group of dimension ≤ 6.

Classification of 6-dimensional splittable flat solvmanifolds

A flat solvmanifold is a compact quotient $$\Gamma \backslash G$$ where G is a simply-connected solvable Lie group endowed with a flat left invariant metric and $$\Gamma $$ is a lattice of G. Any such Lie group can be written as $$G={\mathbb {R}}^k < imes _{\phi } {\mathbb {R}}^m$$ with $${\mathbb {R}}^m$$ the nilradical. In this article we focus on 6-dimensional splittable flat solvmanifolds, which are obtained quotienting G by a lattice $$\Gamma $$ that can be decomposed as $$\Gamma =\Gamma _1 < imes _{\phi }\Gamma _2$$ , where $$\Gamma _1$$ and $$\Gamma _2$$ are lattices of $${\mathbb {R}}^k$$ and $${\mathbb {R}}^m$$ , respectively. We analyze the relation between these lattices and the conjugacy classes of finite abelian subgroups of $$\mathsf{GL}(n,{\mathbb {Z}})$$ , which is known up to $$n\le 6$$ . From this we obtain the classification of 6-dimensional splittable flat solvmanifolds.

Linear control systems on the homogeneous spaces of the 2D Lie group

In this paper, we classify all the possible linear control systems on the homogeneous spaces of the 2D solvable Lie group and study their controllability and control sets.

On LCK solvmanifolds with a property of Vaisman solvmanifolds

Abstract The purpose in this paper is to determine a locally conformal Kähler solvmanifold such that the nilradical of the solvable Lie group is constructed by a Heisenberg Lie group.

Foliations Formed by Generic Coadjoint Orbits of a Class of Real Seven-Dimensional Solvable Lie Groups

In this paper, we consider exponential, connected and simply connected Lie groups which are corresponding to seven-dimensional Lie algebras such that their nilradical is a five-dimensional nilpotent Lie algebra $\mathfrak{g}_{5,2}$ given in Table~\ref{tab1}. In particular, we give a description of the geometry of the generic orbits in the coadjoint representation of some considered Lie groups. We prove that, for each considered group, the family of the generic coadjoint orbits forms a measurable foliation in the sense of Connes. The topological classification of these foliations is also provided.

Topological Loops with Decomposable Solvable Multiplication Groups

In this paper we deal with the class mathcal {C} of decomposable solvable Lie groups having dimension six. We determine those Lie groups in mathcal {C} and their subgroups which are the multiplication groups Mult(L) and the inner mapping groups Inn(L) for three-dimensional connected simply connected topological loops L. This result completes the classification of the at most 6-dimensional solvable multiplication Lie groups of the loops L. Moreover, we obtain that every at most 3-dimensional connected topological proper loop having a solvable Lie group of dimension at most six as its multiplication group is centrally nilpotent of class two.

C-minimal topological groups

Abstract In this paper, we study topological groups having all closed subgroups (totally) minimal and we call such groups c-(totally) minimal. We show that a locally compact c-minimal connected group is compact. Using a well-known theorem of [P. Hall and C. R. Kulatilaka, A property of locally finite groups, J. Lond. Math. Soc. 39 1964, 235–239] and a characterization of a certain class of Lie groups, due to [S. K. Grosser and W. N. Herfort, Abelian subgroups of topological groups, Trans. Amer. Math. Soc. 283 1984, 1, 211–223], we prove that a c-minimal locally solvable Lie group is compact. It is shown that a topological group G is c-(totally) minimal if and only if G has a compact normal subgroup N such that G / N G/N is c-(totally) minimal. Applying this result, we prove that a locally compact group G is c-totally minimal if and only if its connected component c ⁢ ( G ) c(G) is compact and G / c ⁢ ( G ) G/c(G) is c-totally minimal. Moreover, a c-totally minimal group that is either complete solvable or strongly compactly covered must be compact. Negatively answering [D. Dikranjan and M. Megrelishvili, Minimality conditions in topological groups, Recent Progress in General Topology. III, Atlantis Press, Paris 2014, 229–327, Question 3.10 (b)], we find, in contrast, a totally minimal solvable (even metabelian) Lie group that is not compact.

An Interpolation From Sol to Hyperbolic Space

We study a one-parameter family of nonisomorphic solvable Lie groups, which, when equipped with canonical left-invariant metrics, becomes an interpolation from a model of the Sol geometry to a model of Hyperbolic Space, with a stop at . These Lie groups are also Bianchi groups of Type VI with orthogonal coordinates. As a continuation of joint work with Richard Schwartz on Sol, we primarily analyze those Lie groups in our interpolation with some positive sectional curvature. Our main result is a characterization of the cut locus at the identity of the group that maximizes scalar curvature.

Simply transitive NIL-affine actions of solvable Lie groups

Abstract Every simply connected and connected solvable Lie group 𝐺 admits a simply transitive action on a nilpotent Lie group 𝐻 via affine transformations. Although the existence is guaranteed, not much is known about which Lie groups 𝐺 can act simply transitively on which Lie groups 𝐻. So far, the focus was mainly on the case where 𝐺 is also nilpotent, leading to a characterization depending only on the corresponding Lie algebras and related to the notion of post-Lie algebra structures. This paper studies two different aspects of this problem. First, we give a method to check whether a given action ρ : G → Aff ⁡ ( H ) \rho\colon G\to\operatorname{Aff}(H) is simply transitive by looking only at the induced morphism φ : g → aff ⁡ ( h ) \varphi\colon\mathfrak{g}\to\operatorname{aff}(\mathfrak{h}) between the corresponding Lie algebras. Secondly, we show how to check whether a given solvable Lie group 𝐺 acts simply transitively on a given nilpotent Lie group 𝐻, again by studying properties of the corresponding Lie algebras. The main tool for both methods is the semisimple splitting of a solvable Lie algebra and its relation to the algebraic hull, which we also define on the level of Lie algebras. As an application, we give a full description of the possibilities for simply transitive actions up to dimension 4.