Fr\'echet Lie algebroids and their cohomology

TL;DR

This paper extends the concepts of Lie and Courant algebroids to Fréchet manifolds, constructs Dirac structures, and relates their cohomology to known Poisson cohomology, broadening geometric analysis in infinite-dimensional settings.

Contribution

It introduces Fréchet Lie and Courant algebroids, constructs Dirac structures on them, and links their cohomology to Lichnerowicz-Poisson cohomology, advancing the theory in infinite-dimensional geometry.

Findings

Defined Lie and Courant algebroids on Fréchet manifolds

Constructed Dirac structures on generalized tangent bundles

Connected Lie algebroid cohomology to Lichnerowicz-Poisson 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 $M$.