Symplectic Hodge Theory on Lie Algebroids

TL;DR

This paper extends key symplectic geometric conditions to Lie algebroids, establishing their equivalence algebraically and demonstrating their validity in this generalized setting with examples.

Contribution

It introduces algebraic analogues of symplectic conditions for Lie algebroids and proves their equivalence, expanding the theoretical framework of symplectic geometry.

Findings

Analogues of Brylinksi, Strong Lefschetz, and $d extdelta$-lemma established for Lie algebroids

Proved the equivalence of these conditions algebraically in the Lie algebroid context

Provided examples demonstrating the applicability of the theory

Abstract

We explore the natural analogues of the Brylinksi condition, Strong Lefschetz condition, and $d\delta$-lemma in Symplectic Geometry originally explored by Brylinksi, Mathieu, Yan, and Guillemin in the Symplectic Lie Algebroid case. The equivalence of the three conditions is re-established as a purely algebraic statement along with a primitive notion of the $d\delta$-lemma established by Tseng, Yau, and Ho. We then apply this algebraic theory to the desired geometric setting to show that these analogues hold, and finally provide a few classes of examples.