Classification of Einstein Lorentzian 3-nilpotent Lie groups with 1-dimensional nondegenerate center
We give a complete classification of Einstein Lorentzian 3-nilpotent simply connected Lie groups with 1-dimensional nondegenerate center.
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arXiv subjects
Mohamed Boucetta
Publications and source records attributed to Mohamed Boucetta.
Left invariant generalized complex and Kähler structures on simply connected four dimensional Lie groups: classification and invariant cohomologies
We give a complete classification of left invariant generalized complex structures of type 1 on four dimensional simply connected Lie groups and we compute for each class its invariant generalized Dolbeault cohomology, its invariant generalized Bott-Chern cohomology and its invariant generalized Aeppli cohomology. We classify also left invariant generalized Kähler structures on four dimensional simply connected Lie groups.
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Poisson algebras and symmetric Leibniz bialgebra structures on oscillator Lie algebras
Oscillator Lie algebras are the only non commutative solvable Lie algebras which carry a bi-invariant Lorentzian metric. In this paper, we determine all the Poisson structures, and in particular, all symmetric Leibniz algebra structures whose underlying Lie algebra is an oscillator Lie algebra. We give also all the symmetric Leibniz bialgebra structures whose underlying Lie bialgebra structure is a Lie bialgebra structure on an oscillator Lie algebra. We derive some geometric consequences on oscillator Lie groups.
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Contravariant Pseudo-Hessian manifolds and their associated Poisson structures
A contravariant pseudo-Hessian manifold is a manifold $M$ endowed with a pair $(\nabla,h)$ where $\nabla$ is a flat connection and $h$ is a symmetric bivector field satisfying a contravariant Codazzi equation. When $h$ is invertible we recover the known notion of pseudo-Hessian manifold. Contravariant pseudo-Hessian manifolds have properties similar to Poisson manifolds and, in fact, to any contravariant pseudo-Hessian manifold $(M,\nabla,h)$ we associate naturally a Poisson tensor on $TM$. We investigate these properties and we study in details many classes of such structures in order to highlight the richness of the geometry of these manifolds.
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On Einstein Lorentzian nilpotent Lie groups
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 $\mathfrak{g}$ 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 $Z(\mathfrak{g})$ of $\mathfrak{g}$ is nondegenerate Euclidean, the derived ideal $[\mathfrak{g},\mathfrak{g}]$ is nondegenerate Lorentzian and $Z(\mathfrak{g})\subset[\mathfrak{g},\mathfrak{g}]$. We give the first examples of Ricci-flat Lorentzian nilpotent Lie algebra with nondegenerate center.
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Analytic Linear Lie rack Structures on Leibniz Algebras
A linear Lie rack structure on a finite dimensional vector space $V$ is a Lie rack operation $(x,y)\mapsto x\rhd y$ pointed at the origin and such that for any $x$, the left translation $\mathrm{L}_x:y\mapsto \mathrm{L}_x(y)= x\rhd y$ is linear. A linear Lie rack operation $\rhd$ is called analytic if for any $x,y\in V$, \[ x\rhd y=y+\sum_{n=1}^\infty A_{n,1}(x,\ldots,x,y), \]where $A_{n,1}:V\times\ldots\times V\Leftarrow V$ is an $n+1$-multilinear map symmetric in the $n$ first arguments. In this case, $A_{1,1}$ is exactly the left Leibniz product associated to $\rhd$. Any left Leibniz algebra $(\mathfrak{h},[\;,\;])$ has a canonical analytic linear Lie rack structure given by $x\stackrel{c}{\rhd} y=\exp(\mathrm{ad}_x)(y)$, where $\mathrm{ad}_x(y)=[x,y]$. In this paper, we show that a sequence $(A_{n,1})_{n\geq1}$ of $n+1$-multilinear maps on a vector space $V$ defines an analytic linear Lie rack structure if and only if $[\;,\;]:=A_{1,1}$ is a left Leibniz bracket, the $A_{n,1}$ are invariant for $(V,[\;,\;]:)$ and satisfy a sequence of multilinear equations. Some of these equations have a cohomological interpretation and can be solved when the zero and the 1-cohomology of the left Leibniz algebra $(V,[\;,\;])$ are trivial. On the other hand, given a left Leibniz algebra $(\mathfrak{h},[\;,\;])$, we show that there is a large class of (analytic) linear Lie rack structures on $(\mathfrak{h},[\;,\;])$ which can be built from the canonical one and invariant multilinear symmetric maps on $\mathfrak{h}$. A left Leibniz algebra on which all the analytic linear Lie rack structures are build in this way will be called rigid. We use our characterizations of analytic linear Lie rack structures to show that $\mathfrak{sl}_2(\mathbb{R})$ and $\mathfrak{so}(3)$ are rigid. We conjecture that any simple Lie algebra is rigid as a left Leibniz algebra.
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On Riemann-Poisson Lie groups
A Riemann-Poisson Lie group is a Lie group endowed with a left invariant Riemannian metric and a left invariant Poisson tensor which are compatible in the sense introduced in C.R. Acad. Sci. Paris sér. {\bf I 333} (2001) 763-768. We study these Lie groups and we give a characterization of their Lie algebras. We give also a way of building these Lie algebras and we give the list of such Lie algebras up to dimension 5.
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Lorentzian left invariant metrics on three dimensional unimodular Lie groups and their curvatures
There are five unimodular simply connected three dimensional unimodular non abelian Lie groups: the nilpotent Lie group $\mathrm{Nil}$, the special unitary group $\mathrm{SU}(2)$, the universal covering group $\widetilde{\mathrm{PSL}}(2,\mathbb{R})$ of the special linear group, the solvable Lie group $\mathrm{Sol}$ and the universal covering group $\widetilde{\mathrm{E}_0}(2)$ of the connected component of the Euclidean group. For each $G$ among these Lie groups, we give explicitly the list of all Lorentzian left invariant metrics on $G$, up to un automorphism of $G$. Moreover, for any Lorentzian left invariant metric in this list we give its Ricci curvature, scalar curvature, the signature of the Ricci curvature and we exhibit some special features of these curvatures. Namely, we give all the metrics with constant curvature, semi-symmetric non locally symmetric metrics and the Ricci solitons.
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The geometry of the Sasaki metric on the sphere bundle of Euclidean Atiyah vector bundles
Let $(M,\langle,\rangle_{TM})$ be a Riemannian manifold. It is well-known that the Sasaki metric on $TM$ is very rigid but it has nice properties when restricted to $T^{(r)}M=\{u\in TM,|u|=r \}$. In this paper, we consider a general situation where we replace $TM$ by a vector bundle $E\longrightarrow M$ endowed with a Euclidean product $\langle,\rangle_E$ and a connection $\nabla^E$ which preserves $\langle,\rangle_E$. We define the Sasaki metric on $E$ and we consider its restriction $h$ to $E^{(r)}=\{a\in E,\langle a,a\rangle_E=r^2 \}$. We study the Riemannian geometry of $(E^{(r)},h)$ generalizing many results first obtained on $T^{(r)}M$ and establishing new ones. We apply the results obtained in this general setting to the class of Euclidean Atiyah vector bundles introduced by the authors in arXiv preprint arXiv:1808.01254 (2018). Finally, we prove that any unimodular three dimensional Lie group $G$ carries a left invariant Riemannian metric such that $(T^{(1)}G,h)$ has a positive scalar curvature.
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The geometry of generalized Cheeger-Gromoll metrics on the total space of transitive Euclidean Lie algebroids
Natural metrics (Sasaki metric, Cheeger-Gromoll metric, Kaluza-Klein metrics etc.. ) on the tangent bundle of a Riemannian manifold is a central topic in Riemannian geometry. Generalized Cheeger-Gromoll metrics is a family of natural metrics $h_{p,q}$ depending on two parameters with $p\in\mathbb{R}$ and $q\geq0$. This family has been introduced recently and possesses interesting geometric properties. If $p=q=0$ we recover the Sasaki metric and when $p=q=1$ we recover the classical Cheeger-Gromoll metric. A transitive Euclidean Lie algebroid is a transitive Lie algebroid with an Euclidean product on its total space. In this paper, we show that natural metrics can be built in a natural way on the total space of transitive Euclidean Lie algebroids. Then we study the properties of generalized Cheeger-Gromoll metrics on this new context. We show a rigidity result of this metrics which generalizes so far all rigidity results known in the case of the tangent bundle. We show also that considering natural metrics on the total space of transitive Euclidean Lie algebroids opens new interesting horizons. For instance, Atiyah Lie algebroids constitute an important class of transitive Lie algebroids and we will show that natural metrics on the total space of Atiyah Euclidean Lie algebroids have interesting properties. In particular, if $M$ is a Riemannian manifold of dimension $n$, then the Atiyah Lie algebroid associated to the $\mathrm{O}(n)$-principal bundle of orthonormal frames over $M$ possesses a family depending on a parameter $k>0$ of transitive Euclidean Lie algebroids structures say $AO(M,k)$. When $M$ is a space form of constant curvature $c$, we show that there exists two constants $C_n<0$ and $K(n,c)>0$ such that $(AO(M,k),h_{1,1})$ is a Riemannian manifold with positive scalar curvature if and only if $c>C_n$ and $0<k\leq K(n,c)$.
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Cohomology of coinvariant differential forms
Let $M$ be a smooth manifold and $Γ$ a group acting on $M$ by diffeomorphisms; which means that there is a group morphism $ρ:Γ\rightarrow \mathrm{Diff}(M)$ from $Γ$ to the group of diffeomorphisms of $M$. For any such action we associate a cohomology $\mathrm{H}(Ω(M)_Γ)$ which we call the cohomology of $Γ$-coinvariant forms. This is the cohomology of the graded vector space generated by the differentiable forms $ω-ρ(γ)^*ω$ where $ω$ is a differential form with compact support and $γ\in Γ$. The present paper is an introduction to the study of this cohomology. More precisely, we study the relations between this cohomology, the de Rham cohomology and the cohomology of invariant forms $\mathrm{H}(Ω(M)^Γ)$ in the case of isometric actions on compact Riemannian oriented manifolds and in the case of properly discontinuous actions on manifolds.
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On Flat Pseudo-Euclidean Nilpotent Lie Algebras
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 $(2,n-2)$ must be degenerate and all flat pseudo-Euclidean nilpotent Lie algebras of signature $(2,n-2)$ 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 $(2,n-2)$. The paper contains also some examples in low dimension.
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On para-Kähler Lie algebroids and generalized pseudo-Hessian structures
In this paper, we generalize all the results obtained on para-Kähler Lie algebras in Journal of Algebra {\bf 436} (2015) 61-101 to para-Kähler Lie algebroids. In particular, we study exact para-Kähler Lie algebroids as a generalization of exact para-Kähler Lie algebras. This study leads to a natural generalization of pseudo-Hessian manifolds. Generalized pseudo-Hessian manifolds have many similarities with Poisson manifolds. We explore these similarities which, among others, leads to a powerful machinery to build examples of non trivial pseudo-Hessian structures. Namely, we will show that given a finite dimensional commutative and associative algebra $(\mathcal{A},.)$, the orbits of the action $Φ$ of $(\mathcal{A},+)$ on $\mathcal{A}^*$ given by $Φ(a,μ)=\exp(L_a^*)(μ)$ are pseudo-Hessian manifolds, where $L_a(b)=a.b$. We illustrate this result by considering many examples of associative commutative algebras an show that the pseudo-Hessian manifolds obtained are very interesting.
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Four-dimensional homogeneous semi-symmetric Lorentzian manifolds
We give a complete description of semi-symmetric algebraic curvature tensors on a four-dimensional Lorentzian vector space and we use this description to determine all four-dimensional homogeneous semi-symmetric Lorentzian manifolds.
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Nonunimodular Lorentzian flat Lie algebras
A Lorentzian flat Lie group is a Lie group $G$ with a flat left invariant metric $μ$ with signature $(1,n-1)=(-,+,\ldots,+)$. The Lie algebra $\mathfrak{g}=T_eG$ of $G$ endowed with $\langle\;,\;\rangle=μ(e)$ 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 characterise nonunimodular Lorentzian flat Lie algebras as double extensions (in the sense of Aubert-Medina \cite{Aub-Med}) of Riemannian flat Lie algebras. As application of this result, we give all nonunimodular Lorentzian flat Lie algebras up to dimension 4.
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Biharmonic homomorphisms between Riemannian Lie groups
A Lie group $G$ endowed with a left invariant Riemannian metric $g$ is called Riemannian Lie group. Harmonic and biharmonic maps between Riemannian manifolds is an important area of investigation. In this paper, we study different aspects of harmonic and biharmonic homomorphisms between Riemannian Lie groups. We show that this class of biharmonic maps can be used at the first level to build examples but, as we will see through this paper, its study will lead to some interesting mathematical problems in the theory of Riemannian Lie groups.
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On the local structure of noncommutative deformations
Let $(M,π,\mathcal{D})$ be a Poisson manifold endowed with a flat, torsion-free contravariant connection. We show that if $\mathcal{D}$ is an $\mathcal{F}$-connection then there exists a tensor $\mathbf{T}$ such that $\mathcal{D}\mathbf{T}$ is the metacurvature tensor introduced by E. Hawkins in his work on noncommutative deformations. We compute $\mathbf{T}$ and the metacurvature tensor in this case, and show that if $\mathbf{T}=0$ then, near any regular point, $π$ and $\mathcal{D}$ are defined in a natural way by a Lie algebra action and a solution of the classical Yang-Baxter equation. Moreover, when $\mathcal{D}$ is the contravariant Levi-Civita connection associated to $π$ and a Riemannian metric, the Lie algebra action preserves the metric.
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Special bi-invariant linear connections on Lie groups and finite dimensional Poisson structures
Let $G$ be a connected Lie group and $\mathfrak{g}$ its Lie algebra. We denote by $\nabla^0$ the torsion free bi-invariant linear connection on $G$ given by $\nabla^0_XY=\frac12[X,Y],$ for any left invariant vector fields $X,Y$. A Poisson structure on $\mathfrak{g}$ is a commutative and associative product on $\mathfrak{g}$ for which $\mathrm{ad}_u$ is a derivation, for any $u\in\mathfrak{g}$. A torsion free bi-invariant linear connections on $G$ which have the same curvature as $\nabla^0$ is called special. We show that there is a bijection between the space of special connections on $G$ and the space of Poisson structures on $\mathfrak{g}$. We compute the holonomy Lie algebra of a special connection and we show that the Poisson structures associated to special connections which have the same holonomy Lie algebra as $\nabla^0$ possess interesting properties. Finally, we study Poisson structures on a Lie algebra and we give a large class of examples which gives, of course, a large class of special connections.
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