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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]

Let (M, ∇, 〈, 〉) be a manifold endowed with a flat torsionless connection r and a Riemannian metric 〈, 〉 and (TkM)k≥1 the sequence of tangent bundles given by TkM = T(Tk−1M) and T1M = TM. We show that, for any k ≥ 1, TkM carries a Hermitian structure (Jk, gk) and a flat torsionless connection ∇k and when M is a Lie group and (∇, 〈, 〉) are left invariant there is a Lie group structure on each TkM such that (Jk, gk, ∇k) are left invariant. It is well-known that (TM, J1, g1) is Kähler if and only if 〈, 〉 is Hessian, i.e, in each system of affine coordinates (x1, . . ., xn), 〈∂xi,∂xj〉=∂2φ∂xi∂xj\left\langle {{\partial _x}_{_i},{\partial _{{x_j}}}} \right\rangle = {{{\partial ^2}\phi } \over {{\partial _x}_{_i}{\partial _x}_j}}. Having in mind many generalizations of the Kähler condition introduced recently, we give the conditions on (∇, 〈, 〉) so that (TM, J1, g1) is balanced, locally conformally balanced, locally conformally Kähler, pluriclosed, Gauduchon, Vaisman or Calabi-Yau with torsion. Moreover, we can control at the level of (∇, 〈, 〉) the conditions insuring that some (TkM, Jk, gk) or all of them satisfy a generalized Kähler condition. For instance, we show that there are some classes of (M, ∇, 〈, 〉) such that, for any k ≥ 1, (TkM, Jk, gk) is balanced non-Kähler and Calabi-Yau with torsion. By carefully studying the geometry of (M, ∇, 〈, 〉), we develop a powerful machinery to build a large classes of generalized Kähler manifolds.

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]

Kundt spacetimes are of great importance to general relativity. We show that a Kundt spacetime is a Lorentz manifold with a non-singular isotropic geodesic vector field having its orthogonal distribution integrable and determining a totally geodesic foliation. We give the local structure of Kundt spacetimes and some properties of left invariant Kundt structures on Lie groups. Finally, we classify all left invariant Kundt structures on three-dimensional simply connected unimodular Lie groups. [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]

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]

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]