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Flat left-invariant pseudo-Riemannian metrics on quadratic Lie groups
Abstract
A flat quadratic Lie algebra (g,〈,〉,k) is a Lie algebra g endowed with a flat pseudo-Euclidean metric 〈,〉 and a quadratic structure k. In geometrical terms, it is a Lie algebra of a Lie group endowed with both a flat left-invariant pseudo-Riemannian metric and a bi-invariant metric. It is known that if a Lie algebra admits a symplectic form and a quadratic structure then it is a flat quadratic Lie algebra. In this paper, we show that there exist non-abelian Lie algebras which admit a flat Lorentzian metric and a quadratic structure. If g is such Lie algebra, then we show that g is a trivial central extension of R⋊H2k+1 and [g,g]≃H2k+1, where H2k+1 is the Heisenberg Lie algebra and k≥2. In particular, we prove that g is 3-step solvable, not nilpotent and hence admits no symplectic structure. We show that there exists only one non-abelian Lie algebra of dimension n≤6 which admits a flat Lorentzian metric and a quadratic structure. Using a variant of the double extension method, which conserves both the flat and the quadratic metrics, we prove that any Lie algebra which admits a flat Lorentzian metric and a quadratic structure, is a double extension of an abelian Lie algebra and we construct a large class of such (non-abelian) Lie algebras in higher dimensions. Oscillator Lie algebras denoted by gλ constitute an important class of quadratic Lie algebras which are of the form R⋊H2k+1. It is known that gλ admits no flat Lorentzian metric. We show in this paper that gλ admits a Ricci-flat Lorentzian metric. The paper also study the class of quadratic 2-step nilpotent Lie algebras (g,k) which constitute examples of flat quadratic Lie algebras since k is also flat in this case.
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In this paper we give the classification of the irreducible non solvable Lie
algebras of dimensions with nondegenerate, symmetric and invariant
bilinear forms.
A Lorentzian flat Lie group is a Lie group G with a flat left invariant
metric with signature . The Lie algebra
of G endowed with is called flat Lorentzian Lie
algebra. It is known that the metric of a flat Lorentzian Lie group is
geodesically complete if and only if its Lie algebra is unimodular. In this
paper, we determine all nonunimodular flat Lorentzian Lie algebras leading to
the determination of all incomplete Lorentzian flat Lie groups.
The main purpose of this paper is to prove that there are no closed timelike geodesics in a (compact or noncompact) flat Lorentz 2-step nilmanifold N / Γ , N/\Gamma , where N N is a simply connected 2-step nilpotent Lie group with a flat left-invariant Lorentz metric, and Γ \Gamma a discrete subgroup of N N acting on N N by left translations. For this purpose, we shall first show that if N N is a 2-step nilpotent Lie group endowed with a flat left-invariant Lorentz metric g , g, then the restriction of g g to the center Z Z of N N is degenerate. We shall then determine all 2-step nilpotent Lie groups that can admit a flat left-invariant Lorentz metric. We show that they are trivial central extensions of the three-dimensional Heisenberg Lie group H 3 H_{3} . If ( N , g ) \left ( N,g\right ) is one such group, we prove that no timelike geodesic in ( N , g ) \left ( N,g\right ) can be translated by an element of N . N. By the way, we rediscover that the Heisenberg Lie group H 2 k + 1 H_{2k+1} admits a flat left-invariant Lorentz metric if and only if k = 1. k=1.
A flat pseudo-Euclidean Lie algebra is a left-symmetric algebra endowed with a non-degenerate symmetric bilinear form such that left multiplications are skew-symmetric. We observe that, in many classes of flat pseudo-Euclidean Lie algebras, the left-symmetric product is actually of Novikov. This leads us to study pseudo-Euclidean Novikov algebras (A,〈,〉), that is, Novikov algebra A endowed with a non-degenerate symmetric bilinear form such that left multiplications are skew-symmetric. We show that a Lorentzian Novikov algebra (A,〈,〉) must be transitive. This implies that the underlying Lie algebra AL must be unimodular. In geometrical terms, the left-invariant metric on a corresponding Lie group is geodesically complete. In this case, we show that the restriction of the product to [AL,AL]⊥ is trivial. Using the double extension process, we prove that (A,〈,〉) is a pseudo-Euclidean Novikov algebra such that [AL,AL] is Lorentzian if and only if AL splits as AL=[AL,AL]⊥⊕[AL,AL] where [AL,AL] and [AL,AL]⊥ are abelian, [AL,AL] is Lorentzian and adx is skew-symmetric for any x∈[AL,AL]⊥. This also implies, in this case, that A is transitive, AL is unimodular and two-solvable. This result solves the problem for Lorentzian Novikov algebras such that [AL,AL] is non-degenerate. If [AL,AL] is degenerate, we show that such Lorentzian Novikov algebra is obtained by the double extension process from a flat Euclidean algebra, and we give some applications in low dimensions.
In this paper, we study Lorentzian left invariant Einstein metrics on nilpotent Lie groups. We show that if the center of such Lie groups is degenerate then they are Ricci-flat and their Lie algebras can be obtained by the double extension process from an abelian Euclidean Lie algebra. We show that all nilpotent Lie groups up to dimension 5 endowed with a Lorentzian Einstein left invariant metric have degenerate center and we use this fact to give a complete classification of these metrics. We show that if g is the Lie algebra of a nilpotent Lie group endowed with a Lorentzian left invariant Einstein metric with non zero scalar curvature then the center Z(g) of g is nondegenerate Euclidean, the derived ideal [g,g] is nondegenerate Lorentzian and Z(g)⊂[g,g]. We give the first examples of Ricci-flat Lorentzian nilpotent Lie algebra with nondegenerate center.
A flat pseudo-Euclidean Lie algebra is a real Lie algebra with a non degenerate symmetric bilinear form and a left symmetric product whose the commutator is the Lie bracket and such that the left multiplications are skew-symmetric. We show that the center of a flat pseudo-Euclidean nilpotent Lie algebra of signature must be degenerate and all flat pseudo-Euclidean nilpotent Lie algebras of signature can be obtained by using the double extension process from flat Lorentzian nilpotent Lie algebras. We show also that the center of a flat pseudo-Euclidean 2-step nilpotent Lie algebra is degenerate and all these Lie algebras are obtained by using a sequence of double extension from an abelian Lie algebra. In particular, we determine all flat pseudo-Euclidean 2-step nilpotent Lie algebras of signature . The paper contains also some examples in low dimension.
We consider the four-dimensional oscillator group, equipped with a well-known one-parameter family of left-invariant Lorentzian metrics, which includes the bi-invariant one (Gadea & Oubiña, 1999). In a suitable system of global coordinates, the Ricci soliton equation for these metrics translates into a system of partial differential equations. Solving such system, we prove that all these metrics are Ricci solitons. In particular, the bi-invariant metric on the oscillator group gives rise to infinitely many Ricci solitons (and so, also to Yamabe solitons).
Un groupe de Lie G admet une structure symplectique invariante s’il existe sur G une 2-forme différentielle fermée invariante à gauche dont le rang est égal à la dimension de G. Un tel groupe sera appelé par abus de langage, symplectique et son algèbre de Lie sera dite symplectique. Le principal résultat de ce travail est de fournir une classification des groupes symplectiques nilpotents par leurs algèbres de Lie. L’idée centrale dans cette classification est la notion de double extension (section 2) d’une algèbre symplectique: grosso modo en additionnant un plan symplectique à une algèbre symplectique on obtient une algèbre symplectique. Cette notion est l’analogue symplectique de la notion de double extension des algèbres de Lie orthogonales, que nous avons introduite dans [Me-Re 1].Nous montrons que toute algèbre symplectique nilpotente s’obtient par une suite de doubles extensions à partir de l’algèbre réduite à zéro (théorème 2.5). Dire que le groupe de Lie symplectique (G,ω) est double extension du groupe (H, Ω) veut dire que ce dernier est une variété réduite de Marsden-Weinstein de (G,ω). Toute nilvariété symplectique étant quotient d’un groupe nilpotent symplectique par un sous-groupe discret co-compact, [Be-Go] la double extension permet d’obtenir toutes ces variétés.
We study automorphisms and derivations of a Lie algebra L of finite dimension satisfying certain centrality conditions. As a consequence, we obtain that every nilpotent normal subgroup of the automorphism group of L is unipotent for a very large class of Lie algebras. This result extends one of Leger and Luks. We show that the automorphism group of a nilpotent Lie algebra can have trivial center and have yet a unipotent identity component.
Medina and Revoy gave in (Medina and Revoy, Manuscripta Math. 52 (1985) 81–95) a rich class of examples of Lorentzian homogeneous, compact manifolds. These manifolds, the oscillator manifolds, are quotients of the so-called oscillator groups by lattices. Except for the nice quotients of SL(2, ), these manifolds are roughly the only Lorentzian, homogeneous manifolds of finite volume, with non-compact group of isometries (Adams and Stuck, Invent. Math. 129 (1997) 239–261; Zhegib, Sur les espaces-temps homoge`nes, cole Normale, Lyon, 1995). In this paper we exhibit new locally symmetric structures on the oscillator manifolds. These structures are obtained from explicit pseudo-Riemannian, locally symmetric, left-invariant metrics and from affine structures on the oscillator groups. Other examples of locally symmetric manifolds are provided by the determination of the group of isometries of the oscillator groups.
We study quadratic Lie algebras over a field K of null characteristic which admit, at the same time, a symplectic structure. We see that if K is algebraically closed every such Lie algebra may be constructed as the T∗-extension of a nilpotent Lie algebra admitting an invertible derivation and also as the double extension of another quadratic symplectic Lie algebra by the one-dimensional Lie algebra. Finally, we prove that every symplectic quadratic Lie algebra is a special symplectic Manin algebra and we give an inductive description in terms of symplectic quadratic double extensions.
We give a precise description of compact flat spacetimes and use this to draw conclusions about the existence of closed timelike and null geodesics. Other related results and comments are also given.
Nilpotent 6-dimensional Lie algebras over any field of characteristic not 2 are classified. The proof of this classification is essentially constructive: for a given 6-dimensional nilpotent Lie algebra L, following the steps of the proof, it is possible to find a Lie algebra M that occurs in the classification, and an isomorphism L -> M. In the proof a method due to Skjelbred and Sund is used, along with a method based on Grobner bases to find isomorphisms. (c) 2006 Elsevier Inc. All rights reserved.
This article outlines what is known to the author about the Riemannian geometry of a Lie group which has been provided with a Riemannian metric invariant under left translation.
This paper provides a study of algebraic Ricci solitons in the
pseudo-Riemannian case. In the Riemannian case, all nontrivial homogeneous
algebraic Ricci solitons are expanding algebraic Ricci solitons. In this paper,
we obtain a steady algebraic Ricci soliton and a shrinking algebraic Ricci
soliton in the Lorentzian setting.
We call the Lie algebra of a Lie group with a left invariant
pseudo-Riemannian flat metric pseudo-Riemannian flat Lie algebra. We give a new
proof of a classical result of Milnor on Riemannian flat Lie algebras. We
reduce the study of Lorentzian flat Lie algebras to those with trivial center
or those with degenerate center. We show that the double extension process can
be used to construct all Lorentzian flat Lie algebras with degenerate center
generalizing a result of Aubert-Medina on Lorentzian flat nilpotent Lie
algebras. Finally, we give the list of Lorentzian flat Lie algebras with
degenerate center up to dimension 6.
The determination of affine Lie groups (i.e., which carry a left-invariant affine structure) is an open problem. In this work we begin the study of Lie groups with a left-invariant, flat pseudo-Riemannian metric (flat pseudo-Riemannian groups). We show that in such groups the left-invariant affine structure defined by the Levi-Civita connection is geodesically complete if and only if the group is unimodular. We also show that the cotangent manifold of an affine Lie group is endowed with an affine Lie group structure and a left-invariant, flat hyperbolic metric. We describe a double extension process which allows us to construct all nilpotent, flat Lorentzian groups. We give examples and prove that the only Heisenberg group which carries a left invariant, flat pseudo-Riemannian metric is the three dimensional one.