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.
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Alekseev, A., Strobl, T.: Current algebras and differential geometry, J. High Energy Phys. (electronic) 035(3), 14 (2005). hep-th/0410183
Arias Abad, C., Crainic, M.: Representations up to homotopy of Lie algebroids. (2009). arXiv:0901.0319
Bressler P.: The first Pontryagin class, Compos. Math. 143(5), 1127–1163 (2007) arXiv:math/0509563
Courant T.: Dirac manifolds. Trans. Am. Math. Soc. 319, 631–661 (1990)
Courant, T., Weinstein, A.: Beyond Poisson structures. In: Hermann (ed.) Séminaire sud-rhodanien de géométrie VIII (Paris), Travaux en Cours, vol. 27, pp. 39–49 (1988) (Appeared first as a University of California preprint c. 1986; available at http://math.berkeley.edu/~alanw/Beyond.pdf)
Crainic M., Moerdijk I.: Deformations of Lie brackets: cohomological aspects. J. Eur. Math. Soc. 10(4), 1037–1059 (2008) arXiv:math/0403434
Dirac P.A.M.: Lectures on Quantum Mechanics. Yeshiva University, New York (1964)
Dorfman I.Y.: Dirac structures of integrable evolution equations. Phys. Lett. A 125(5), 240–246 (1987)
Dorfman I.Y.: Dirac Structures and Integrability of Nonlinear Evolution Equations, Nonlinear Science—Theory and Applications. Wiley, Chichester (1993)
Ginot, G., Grutzmann, M.: Cohomology of Courant algebroids with split base. J. Symp. Geom. (2009, to appear). arXiv:0805.3405
Hitchin N.: Generalized Calabi-Yau manifolds. Q. J. Math. Oxford Ser. 54, 281–308 (2003)
Huebschmann J.: Poisson cohomology and quantization. J. Reine Angew. Math. 408, 57–113 (1990)
Keller, F., Waldmann, S.: Deformation theory of Courant algebroids via the Rothstein algebra. (2008) arXiv:0807.0584
Liu Z.-J., Weinstein A., Xu P.: Manin triples for Lie bialgebroids. J. Differ. Geom. 45, 547–574 (1997)
Loday J.-L., Pirashvili T.: Universal enveloping algebras of Leibniz algebras and (co)homology. Math. Ann. 296, 139–158 (1993)
Rinehart G.S.: Differential forms on general commutative algebras. Trans. Am. Math. Soc. 108, 195–222 (1963)
Rothstein, M.: The structure of supersymplectic supermanifolds. In: Bartocci C., Bruzzo U., Cianci R. (eds.) Differential geometric methods in theoretical physics. Proceedings of the Nineteenth International Conference held in Rapallo, 19–24 June 1990. Lecture Notes in Physics, vol. 375, pp. 331–343. Springer, Berlin (1991)
Roytenberg, D.: Courant algebroids, derived brackets and even symplectic supermanifolds. Ph.D. thesis, UC Berkeley (1999). math.DG/9910078
Roytenberg, D.: On the Structure of graded symplectic supermanifolds and Courant algebroids. In: Voronov, T. (ed.) Quantization, Poisson brackets and beyond. Contemporary Mathematics, vol. 315. American Mathematical Society, Providence (2002). math.SG/0203110
Roytenberg D.: AKSZ-BV formalism and Courant algebroid-induced topological field theories. Lett. Math. Phys. 79(2), 143–159 (2007) hep-th/0608150
Roytenberg, D.: On weak Lie 2-algebras. XXVI Workshop on Geometrical Methods in Physics. In: Kielanowski, P., et al. (ed.) AIP Conference Proceedings, vol. 956. American Institute of Physics, Melville (2007). arXiv:0712.3461; also appeared in the MPI and ESI preprint series
Ševera P.: Letters to A. Weinstein 1998–2000. Available at http://sophia.dtp.fmph.uniba.sk/~severa/letters/
Ševera P.: Some title containing the words “homotopy” and “symplectic”, e.g. this one. Travaux Mathématiques, vol. XVI, pp. 121–137. University of Luxembourg, Luxembourg (2005). Also available at as math.SG/0105080
Stiénon M., Xu P.: Modular classes of Loday algebroids. C. R. Math. Acad. Sci. Paris 346(3–4), 193–198 (2008) arXiv:0803.2047
Voronov T.: Higher derived brackets and homotopy algebras. J. Pure Appl. Algebra 202(1–3), 133–153 (2005)
Weinstein, A.: Omni–Lie algebras. Microlocal analysis of the Schrödinger equation and related topics (Japanese) (Kyoto, 1999), vol. 1176, pp. 95–102. Srikaisekikenkysho Kkyroku (2000). arXiv:math/9912190
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To the memory of I.Ya. Dorfman (1948–1994).
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Roytenberg, D. Courant–Dorfman Algebras and their Cohomology. Lett Math Phys 90, 311–351 (2009). https://doi.org/10.1007/s11005-009-0342-3
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DOI: https://doi.org/10.1007/s11005-009-0342-3