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 $(\R,\E)$ 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 $Θ\in\C^3(\E,\R)$ 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.

Corrected formulas for the brackets in Examples 2.27, 2.28 and 2.29. The corrections do not affect the exposition in any way

References in corpus (5)

• Higher derived brackets and homotopy algebras
• AKSZ-BV Formalism and Courant Algebroid-induced Topological Field Theories
• Current Algebras and Differential Geometry
• On weak Lie 2-algebras
• Modular Classes of Loday Algebroids

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• Superconformal structures on generalized Calabi-Yau metric manifolds
• Recent advances and open questions on the susy structure of the chiral de Rham Complex
• Lie Algebroids in the Loday-Pirashvili Category
• Cohomology of hemistrict Lie 2-algebras
• Courant-Dorfman algebras of differential operators and Dorfman connections of Courant algebroids
• Jacobi Geometry and Hamiltonian Mechanics: the Unit-Free Approach
• Higher Courant-Dorfman algebras and associated higher Poisson vertex algebras
• Higher Multi-Courant Algebroids
• The standard cohomology of regular Courant algebroids