Courant–Dorfman Algebras and their Cohomology

Abstract

We introduce a new type of algebra, the Courant–Dorfman algebra. These are to Courant algebroids what Lie–Rinehart algebras are to Lie algebroids, or Poisson algebras to Poisson manifolds. We work with arbitrary rings and modules, without any regularity, finiteness or non-degeneracy assumptions. To each Courant–Dorfman algebra \({(\mathcal{R}, \mathcal{E})}\) we associate a differential graded algebra \({\mathcal{C}(\mathcal{E}, \mathcal{R})}\) in a functorial way by means of explicit formulas. We describe two canonical filtrations on \({\mathcal{C}(\mathcal{E}, \mathcal{R})}\) , and derive an analogue of the Cartan relations for derivations of \({\mathcal{C}(\mathcal{E}, \mathcal{R})}\) ; we classify central extensions of \({\mathcal{E}}\) in terms of \({H^2(\mathcal{E}, \mathcal{R})}\) and study the canonical cocycle \({\Theta \in \mathcal{C}^3(\mathcal{E}, \mathcal{R})}\) whose class \({[\Theta]}\) obstructs re-scalings of the Courant–Dorfman structure. In the nondegenerate case, we also explicitly describe the Poisson bracket on \({\mathcal{C}(\mathcal{E}, \mathcal{R})}\) ; for Courant–Dorfman algebras associated to Courant algebroids over finite-dimensional smooth manifolds, we prove that the Poisson dg algebra \({\mathcal{C}(\mathcal{E}, \mathcal{R})}\) is isomorphic to the one constructed in Roytenberg (On the structure of graded symplectic supermanifolds and Courant algebroids. American Mathematical Society, Providence, 2002) using graded manifolds.