Left‐symmetric algebroids

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 (, .), the orbits of the action Φ of (, +) on  * given by Φ( , ) = exp. We illustrate this result by considering many examples of associative commutative algebras and show that the resulting pseudo-Hessian manifolds are very interesting. K E Y W O R D S associative commutative algebras, left symmetric algebroids, para-Kähler Lie algebroids, pseudo-Hessian manifolds, symplectic Lie algebroids M S C ( 2 0 1 0 ) 13P25, 53A15, 53C15, 53D17 INTRODUCTIONRecall that a Lie algebroid is a vector bundle ←→ together with an anchor map ∶ ←→ and a Lie bracket [ , ] on Γ( ) such that, for any , ∈ Γ( ), ∈ ∞ ( ),[ , ] = [ , ] + ( )( ) .Lie algebroids are now a central notion in differential geometry and constitute an active domain of research. They have many applications in various part of mathematics and physics (see for instance [6][7][8]18]). It is a well-established fact that many classical geometrical structures involving the tangent bundle of a manifold (which has a natural structure of Lie algebroid) can be generalized to the context of Lie algebroids. Thus the notions of connections on Lie algebroids, symplectic Lie algebroids, pseudo-Riemannian Lie algebroids and so on are now usual notions in differential geometry with many applications in physics (see for instance [4,10]). On the other hand, it is important to point out that Lie algebroids generalize also Lie algebras and, for instance, if one obtains a result on the curvature of pseudo-Riemannian Lie algebroids this result holds for the curvature of pseudo-Euclidean Lie algebras and hence for the curvature of left invariant pseudo-Riemannian metrics on Lie groups. 1418

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“…Left symmetric algebroids Left symmetric algebroids appeared first in as Koszul–Vinberg algebroids were studied in more details in .…”

Section: Lie Algebroids Connections Levi–civita Connections Left Smentioning

confidence: 99%

“…In Section 2, we recall some basic facts about Lie algebroids and connections on Lie algebroids. A Lie algebroid with a torsionless and flat connection was called Koszul–Vinberg algebroid in and left symmetric algebroid in . These algebroids play a central role in the study of para‐kähler Lie algebroids.…”

Section: Introductionmentioning

confidence: 99%

On para‐Kähler Lie algebroids and contravariant pseudo‐Hessian structures

Benayadi

Boucetta

2019

Mathematische Nachrichten

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“…Definition 2.3. [12,22] A left-symmetric algebroid structure on a vector bundle A −→ M is a pair that consists of a left-symmetric algebra structure · A on the section space Γ(A) and a vector bundle morphism a A : A −→ T M , called the anchor, such that for all f ∈ C ∞ (M ) and x, y ∈ Γ(A), the following conditions are satisfied:…”

Section: Left-symmetric Algebroidsmentioning

confidence: 99%

“…ρ = a A and µ = 0. See [5,12] for general theory of cohomologies of right-symmetric algebras and left-symmetric algebroids respectively. The set of (n + 1)-cochains is given by…”

Section: Example 24mentioning

confidence: 99%

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Left-symmetric bialgebroids and their corresponding Manin triples

Liu¹,

Sheng²,

Bai³

2018

Differential Geometry and its Applications

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In this paper, we introduce the notion of a left-symmetric bialgebroid as a geometric generalization of a left-symmetric bialgebra and construct a left-symmetric bialgebroid from a pseudo-Hessian manifold. We also introduce the notion of a Manin triple for left-symmetric algebroids, which is equivalent to a left-symmetric bialgebroid. The corresponding double structure is a pre-symplectic algebroid rather than a left-symmetric algebroid. In particular, we establish a relation between Maurer-Cartan type equations and Dirac structures of the pre-symplectic algebroid which is the corresponding double structure for a left-symmetric bialgebroid.

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