Obstructions for Symplectic Lie Algebroids

Abstract

Several types of generically nondegenerate Poisson structures can be effectively studied as symplectic structures on naturally associated Lie algebroids. Relevant examples of this phenomenon include log-, elliptic, 𝑏ᵏ-, scattering, and elliptic-log Poisson structures. In this paper, we discuss topological obstructions to the existence of such Poisson structures, obtained through the characteristic classes of their associated symplectic Lie algebroids. In particular, we obtain the full obstructions for surfaces to carry such Poisson structures.