Pre-symplectic algebroids and their applications

TL;DR

This paper introduces pre-symplectic algebroids, establishes their correspondence with symplectic Lie algebroids, and explores their classifications and geometric structures, extending the understanding of symplectic geometry in the context of Lie algebroids.

Contribution

It defines pre-symplectic algebroids, links them to symplectic Lie algebroids, and studies their classifications and geometric properties, including exact and para-complex cases.

Findings

Pre-symplectic algebroids correspond one-to-one with symplectic Lie algebroids.

Exact pre-symplectic algebroids are classified by the third cohomology group.

Para-complex pre-symplectic algebroids relate to pseudo-Riemannian Lie algebroids.

Abstract

In this paper, we introduce the notion of a pre-symplectic algebroid, and show that there is a one-to-one correspondence between pre-symplectic algebroids and symplectic Lie algebroids. This result is the geometric generalization of the relation between left-symmetric algebras and symplectic (Frobenius) Lie algebras. Although pre-symplectic algebroids are not left-symmetric algebroids, they still can be viewed as the underlying structures of symplectic Lie algebroids. %We study three classes of pre-symplectic algebroids in detail. Then we study exact pre-symplectic algebroids and show that they are classified by the third cohomology group of a left-symmetric algebroid. Finally, we study para-complex pre-symplectic algebroids. Associated to a para-complex pre-symplectic algebroid, there is a pseudo-Riemannian Lie algebroid. The multiplication in a para-complex pre-symplectic algebroid characterizes the restriction to the Lagrangian subalgebroids of the Levi-Civita connection in the corresponding pseudo-Riemannian Lie algebroid.