Search

k-symplectic manifolds are a convenient framework to study classical field theories and they are a generalization of polarized symplectic manifolds. This paper focus on the existence and the properties of left invariant k-symplectic structures on Lie groups having a bi-invariant pseudo-Riemannian metric. We show that compact semi-simple Lie groups and a large class of Lie groups having a bi-invariant pseudo-Riemannian metric does not carry any left invariant k-symplectic structure. This class contains the oscillator Lie groups which are the only solvable non abelian Lie groups having a bi-invariant Lorentzian metric. However, we built a natural left invariant n-symplectic structure on SL (n , R) . Moreover, up to dimension 6, only three connected and simply connected Lie groups have a bi-invariant indecomposable pseudo-Riemannian metric and a left invariant k-symplectic structure, namely, the universal covering of SL (2 , R) with a 2-symplectic structure, the universal covering of the Lorentz group SO (3 , 1) with a 2-symplectic structure, and a 2-step nilpotent 6-dimensional connected and simply connected Lie group with both a 1-symplectic structure and a 2-symplectic structure. [ABSTRACT FROM AUTHOR]

A pseudo-Euclidean non-associative algebra (g , •) is a finite dimensional algebra over a field K that has a metric, i.e., a bilinear, symmetric, and non-degenerate form 〈 , 〉. The metric is considered L-invariant (resp. R-invariant) if all left multiplications (resp. right multiplications) are skew-symmetric. The metric is called associative if 〈 u • v , w 〉 = 〈 u , v • w 〉 for all u , v , w ∈ g. These three notions coincide when g is a Lie algebra and in this case g endowed with the metric is known as a quadratic Lie algebra. This paper provides a complete description of L-invariant, R-invariant, or associative pseudo-Euclidean metrics on left Leibniz algebras over a commutative field of characteristic zero. It shows that a left Leibniz algebra with an associative metric is also right Leibniz and can be obtained easily from its underlying Lie algebra, which is a quadratic Lie algebra. Additionally, it shows that at the core of a left Leibniz algebra endowed with a L-invariant or R-invariant metric, there are two Lie algebras with one quadratic and the left Leibniz algebra can be built from these Lie algebras. We derive many important results from this complete description. Finally, the paper provides a list of left Leibniz algebras with an associative metric up to dimension 6, as well as a list of left Leibniz algebras with an L-invariant metric, up to dimension 4, and R-invariant metric up to dimension 5. [ABSTRACT FROM AUTHOR]

Let (G , Ω) be a symplectic Lie group, i.e., a Lie group endowed with a left invariant symplectic form. If g is the Lie algebra of G then we call (g , ω = Ω (e)) a symplectic Lie algebra. The product • on g defined by 3 ω (x • y , z) = ω ([ x , y ] , z) + ω ([ x , z ] , y) extends to a left invariant connection ∇ on G which is torsion free and symplectic ( ∇ Ω = 0) . When ∇ has vanishing curvature, we call (G , Ω) a flat symplectic Lie group and (g , ω) a flat symplectic Lie algebra. In this paper, we study flat symplectic Lie groups. We start by showing that the derived ideal of a flat symplectic Lie algebra is degenerate with respect to ω. We show that a flat symplectic Lie group must be nilpotent with degenerate center. This implies that the connection ∇ of a flat symplectic Lie group is always complete. We prove that the flat symplectic double extension process can be applied to characterize all flat symplectic Lie algebras. More precisely, we show that every flat symplectic Lie algebra is obtained by a sequence of flat symplectic double extension of flat symplectic Lie algebras starting from {0}. As examples in low dimensions, we classify all flat symplectic Lie algebras of dimension ≤ 6 . [ABSTRACT FROM AUTHOR]

We classify symmetric Leibniz algebras in dimensions 3 and 4 and we determine all associated Lie racks. Some of such Lie racks give rise to nontrivial topological quandles. We study some algebraic properties of these quandles and we give a necessary and sufficient condition for them to be quasi-trivial. [ABSTRACT FROM AUTHOR]

k-Para-Kähler Lie algebras are a generalization of para-Kähler Lie algebras (k = 1) and constitute a subclass of k-symplectic Lie algebras. In this paper, we show that the characterization of para-Kähler Lie algebras as left symmetric bialgebras can be generalized to k-para-Kähler Lie algebras leading to the introduction of two new structures which are different but both generalize the notion of left symmetric algebra. This permits also the introduction of generalized S-matrices. We determine then all the k-symplectic Lie algebras of dimension (k + 1) and all the six dimensional 2-para-Kähler Lie algebras. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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 [4]. 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. [ABSTRACT FROM AUTHOR]

A linear Lie rack structure on a finite dimensional vector space V is a Lie rack operation (x , y) ↦ x ⊳ y pointed at the origin and such that for any x, the left translation L x : y ↦ L x (y) = x ⊳ y is linear. A linear Lie rack operation ⊳ is called analytic if for any x , y ∈ V , x ⊳ y = y + ∑ n = 1 ∞ A n , 1 (x , ... , x , y) , where A n , 1 : V × ⋯ × V → 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 ⊳. Any left Leibniz algebra (h , [ , ]) has a canonical analytic linear Lie rack structure given by x ⊳ c y = exp ( ad x) (y) , where ad x (y) = [ x , y ]. In this paper, we show that a sequence (A n , 1) n ≥ 1 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 (h , [ , ]) , we show that there is a large class of (analytic) linear Lie rack structures on (h , [ , ]) which can be built from the canonical one and invariant multilinear symmetric maps on 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 s l 2 (R) and s o (3) are rigid. We conjecture that any simple Lie algebra is rigid as a left Leibniz algebra. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Let M be a manifold endowed with a flat torsionless connection ∇. A symmetric bivector field h on (M , ∇) is said to be a Koszul-Vinberg (K-V for short) bivector field if it satisfies: (∇ α # h) (β , γ) = (∇ β # h) (α , γ) , where α # : = h # (α) and h # : T ⁎ M → T M is the contraction of h. The pair (∇ , h) is called a K-V structure and the triple (M , ∇ , h) is called a K-V manifold. When h is non-degenerate, (M , ∇ , h − 1) is a pseudo-Hessian manifold, otherwise, we have a singular foliation defined by I m h # called the associated affine foliation whose leaves are pseudo-Hessian manifolds. In this paper, we study invariant K-V structures on homogeneous spaces. More precisely, we give an algebraic characterization of these structures in the same spirit of Nomizu's theorem on invariant connections on homogeneous spaces, and we give many classes of examples mainly for reductive or symmetric pairs (G , H). We establish many properties of pseudo-Hessian homogeneous manifolds, in particular, we show that when G is a semi-simple Lie group then G / H does not admit any non trivial G -invariant pseudo-Hessian structure. Finally, we show that the leaves of the affine foliation associated to an invariant K-V structure are homogeneous pseudo-Hessian manifolds. [ABSTRACT FROM AUTHOR]

In this paper, we generalize all the results obtained on para‐Kähler Lie algebras in [3] 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, we call them contravariant pseudo‐Hessian manifolds. Contravariant 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 (A,.), the orbits of the action Φ of (A,+) on A∗ given by Φ(a,μ)=exp(La∗)(μ) are pseudo‐Hessian manifolds, where La(b)=a.b. We illustrate this result by considering many examples of associative commutative algebras and show that the resulting pseudo‐Hessian manifolds are very interesting. [ABSTRACT FROM AUTHOR]

We study a class of Lie algebras which contains the class of quadratic Lie algebras and the class of Milnor Lie algebras, namely, Lie algebras endowed a pseudo-Euclidean metric such its Levi-Civita product is left Leibniz. We call them Levi-Civita left Leibniz Lie algebras LCLL for short. We show that a Lie group (G , h) endowed with a left invariant pseudo-Riemannian such that the corresponding Lie algebra is LCLL is complete and locally symmetric. Moreover, there is a pointed Lie rack structure on G whose corresponding left Leibniz algebra is given by the Levi-Civita product. Euclidean LCLL Lie algebras are the product of a quadratic Lie algebra and a flat Euclidean Lie algebra. We develop an adapted version of the process of double extension to construct LCLL Lie algebras. We show that Lorentzian or flat LCLL Lie algebras can be obtained by this process. [ABSTRACT FROM AUTHOR]

A flat Lorentzian Lie algebra is a left symmetric algebra endowed with a symmetric bilinear form of signature (-,+, . . . ,+) such that left multiplications are skewsymmetric. In geometrical terms, a flat Lorentzian Lie algebra is the Lie algebra of a Lie group with a left-invariant Lorentzian metric with vanishing curvature. In this article, we show that any flat nonunimodular Lorentzian Lie algebras can be obtained as a double extension of flat Riemannian Lie algebras. As an application, we give all flat nonunimodular Lorentzian Lie algebras up to dimension 4. [ABSTRACT FROM AUTHOR]

In this study, we consider Lie algebras that admit para-Kähler and hyper-para-Kähler structures. We provide new characterizations of these Lie algebras and develop many methods for building large classes of examples. Previously, Bai considered para-Kähler Lie algebras as left symmetric bialgebras. We reconsider this viewpoint and make improvements in order to obtain some new results. The study of para-Kähler and hyper-para-Kähler is intimately linked to the study of left symmetric algebras, particularly those that admit invariant symplectic forms. In this study, we provide many new classes of left symmetric algebras and complete descriptions of all the associative algebras that admit an invariant symplectic form. We also describe all four-dimensional hyper-para-Kähler Lie algebras. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]