Courant-Dorfman algebras and their cohomology

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 $(\mathbb{R},\text{E})$ we associate a differential graded algebra $\text{C}(\text{E},\mathbb{R})$ in a functorial way by means of explicit formulas. We describe two canonical filtrations on $\text{C}(\text{E},\mathbb{R})$, and derive an analogue of the Cartan relations for derivations of $\text{C}(\text{E},\mathbb{R})$; we classify central extensions of $\text{E}$ in terms of $H^2(\text{E},\mathbb{R})$ and study the canonical cocycle $\mathrm{\Theta }\in {\text{C}}^3(\text{E},\mathbb{R})$ whose class $[\mathrm{\Theta }]$ obstructs re-scalings of the Courant-Dorfman structure. In the nondegenerate case, we also explicitly describe the Poisson bracket on $\text{C}(\text{E},\mathbb{R})$; for Courant-Dorfman algebras associated to Courant algebroids over finite-dimensional smooth manifolds, we prove that the Poisson dg algebra $\text{C}(\text{E},\mathbb{R})$ is isomorphic to the one constructed in \cite{Roy4-GrSymp} using graded manifolds.