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 (R, e{open}) we associate a differential graded algebra C(e{open}, R) in a functorial way by means of explicit formulas. We describe two canonical filtrations on C(e{open}, R), and derive an analogue of the Cartan relations for derivations of C(e{open}, R); we classify central extensions of e{open} in terms of H2 and study the canonical cocycle Θεc3(e{open},R) whose class [Θ] obstructs re-scalings of the Courant-Dorfman structure. In the nondegenerate case, we also explicitly describe the Poisson bracket on C(e{open}, R); for Courant-Dorfman algebras associated to Courant algebroids over finite-dimensional smooth manifolds, we prove that the Poisson dg algebra C(e{open}, 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. © The Author(s) 2009.