Courant-Dorfman algebras and their cohomology

TL;DR

This paper introduces Courant-Dorfman algebras, explores their cohomology, and establishes their relation to Courant algebroids, providing explicit formulas, classifications, and connections to Poisson structures.

Contribution

It defines Courant-Dorfman algebras over arbitrary rings, constructs their cohomology, and links them to known geometric structures like Courant algebroids and Poisson brackets.

Findings

Defined Courant-Dorfman algebras and their cohomology.

Classified central extensions via second cohomology.

Connected algebraic structures to geometric Poisson brackets.

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