Fr\'echet Lie algebroids and their cohomology

Abstract

We define Lie and Courant algebroids on Fr\'{e}chet manifolds. Moreover, we construct a Dirac structure on the generalized tangent bundle of a Fr\'{e}chet manifold and show that it inherits a Fr\'{e}chet Lie algebroid structure. We show that the Lie algebroid cohomology of the $\bb$-cotangent bundle Lie algebroid of a weakly symplectic Fr\'{e}chet manifold $M$ is the Lichnerowicz-Poisson cohomology of $M$.