Symplectic Hodge Theory on Lie Algebroids

Abstract

We explore the natural analogues of the Brylinksi condition, Strong Lefschetz condition, and dδ-lemma in Symplectic Geometry originally explored by Brylinksi, Mathieu, Yan, and Guillemin to 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 dδ-lemma shown established by Tseng, Yau, and Ho. We then show that the natural analogues of these in the Lie Algebroid setting holds as well with examples given

Keywords: Symplectic Lie Algebroid, Hodge Theory, Strong Lefschetz, dδ-lemma, nilmanifold