On contact and symplectic Lie algebroids

TL;DR

This paper explores the structure of contact and symplectic Lie algebroids, demonstrating how compatible triples induce symplectic forms on submanifolds and analyzing the properties of associated Poisson structures.

Contribution

It introduces a decomposition approach for Lie algebroids to construct integrable distributions and studies the induced symplectic and Poisson structures, extending understanding of compatible triples.

Findings

Constructed integrable distributions on Lie algebroids.

Induced symplectic forms on integral submanifolds.

Poisson structures on the base manifold derived from submanifold structures.

Abstract

In this paper, we will study compatible triples on Lie algebroids. Using a suitable decomposition for a Lie algebroid, we construct an integrable generalized distribution on the base manifold. As a result, the symplectic form on the Lie algebroid induces a symplectic form on each integral submanifold of the distribution. The induced Poisson structure on the base manifold can be represented by means of the induced Poisson structures on the integral submanifolds. Moreover, for any compatible triple with invariant metric and admissible almost complex structure, we show that the bracket annihilates on the kernel of the anchor map.