Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 9 (2013), 014, 8 pages      arXiv:1212.0559      https://doi.org/10.3842/SIGMA.2013.014
Contribution to the Special Issue “Symmetries of Differential Equations: Frames, Invariants and Applications”

Courant Algebroids. A Short History

Yvette Kosmann-Schwarzbach
Centre de Mathématiques Laurent Schwartz, École Polytechnique, F-91128 Palaiseau, France

Received December 03, 2012, in final form February 14, 2013; Published online February 19, 2013

Abstract
The search for a geometric interpretation of the constrained brackets of Dirac led to the definition of the Courant bracket. The search for the right notion of a ''double'' for Lie bialgebroids led to the definition of Courant algebroids. We recount the emergence of these concepts.

Key words: Courant algebroid; Dorfman bracket; Lie algebroid; Lie bialgebroid; generalized geometry; Dirac structure; Loday algebra; Leibniz algebra; derived bracket.

pdf (336 kb)   tex (22 kb)

References

  1. Bressler P., The first Pontryagin class, Compos. Math. 143 (2007), 1127-1163, math.AT/0509563.
  2. Coste A., Dazord P., Weinstein A., Groupoïdes symplectiques, in Publ. Dép. Math. Nouvelle Sér. A, Vol. 2, Univ. Claude Bernard-Lyon 1, 1987, 1-62.
  3. Courant T., Dirac manifolds, Trans. Amer. Math. Soc. 319 (1990), 631-661.
  4. Courant T., Weinstein A., Beyond Poisson structures, in Actions hamiltoniennes de groupes. Troisième théorème de Lie (Lyon, 1986), Travaux en Cours, Vol. 27, Hermann, Paris, 1988, 39-49.
  5. Dirac P.A.M., Lectures on quantum mechanics, Belfer Graduate School of Science Monographs Series, Vol. 2, Belfer Graduate School of Science, New York, 1964.
  6. Dorfman I.Ya., Dirac structures and integrability of nonlinear evolution equations, Nonlinear Science: Theory and Applications, John Wiley & Sons Ltd., Chichester, 1993.
  7. Dorfman I.Ya., Dirac structures of integrable evolution equations, Phys. Lett. A 125 (1987), 240-246.
  8. Drinfel'd V.G., Hamiltonian structures on Lie groups, Lie bialgebras and the geometric meaning of classical Yang-Baxter equations, Soviet Math. Doklady 27 (1983), 68-71.
  9. Grabowski J., Marmo G., The graded Jacobi algebras and (co)homology, J. Phys. A: Math. Gen. 36 (2003), 161-181, math.DG/0207017.
  10. Graña M., Minasian R., Petrini M., Waldram D., T-duality, generalized geometry and non-geometric backgrounds, J. High Energy Phys. 2009 (2009), no. 4, 075, 39 pages, arXiv:0807.4527.
  11. Gualtieri M., Generalized complex geometry, Ann. of Math. (2) 174 (2011), 75-123, math.DG/0703298.
  12. Hitchin N., Generalized Calabi-Yau manifolds, Q. J. Math. 54 (2003), 281-308, math.DG/0209099.
  13. Ikeda N., Chern-Simons gauge theory coupled with BF theory, Internat. J. Modern Phys. A 18 (2003), 2689-2701, hep-th/0203043.
  14. Kinyon M.K., Weinstein A., Leibniz algebras, Courant algebroids, and multiplications on reductive homogeneous spaces, Amer. J. Math. 123 (2001), 525-550, math.DG/0006022.
  15. Kosmann-Schwarzbach Y., Derived brackets, Lett. Math. Phys. 69 (2004), 61-87, math.DG/0312524.
  16. Kosmann-Schwarzbach Y., Exact Gerstenhaber algebras and Lie bialgebroids, Acta Appl. Math. 41 (1995), 153-165.
  17. Kosmann-Schwarzbach Y., From Poisson algebras to Gerstenhaber algebras, Ann. Inst. Fourier (Grenoble) 46 (1996), 1243-1274.
  18. Kosmann-Schwarzbach Y., Jacobian quasi-bialgebras and quasi-Poisson Lie groups, in Mathematical Aspects of Classical Field Theory (Seattle, WA, 1991), Contemp. Math., Vol. 132, Editors M. Gotay, J.E. Marsden, V. Moncrief, Amer. Math. Soc., Providence, RI, 1992, 459-489.
  19. Kosmann-Schwarzbach Y., Poisson manifolds, Lie algebroids, modular classes: a survey, SIGMA 4 (2008), 005, 30 pages, arXiv:0710.3098.
  20. Kosmann-Schwarzbach Y., Quasi, twisted, and all that ... in Poisson geometry and Lie algebroid theory, in The Breadth of Symplectic and Poisson Geometry, Festschrift in Honor of Alan Weinstein, Progr. Math., Vol. 232, Editors J.E. Marsden, T.S. Ratiu, Birkhäuser Boston, Boston, MA, 2005, 363-389, math.DG/0310359.
  21. Kosmann-Schwarzbach Y. (Editor), Siméon-Denis Poisson. Les Mathématiques au service de la science, Éditions de l'École Polytechnique, to appear.
  22. Kosmann-Schwarzbach Y., Magri F., Poisson-Nijenhuis structures, Ann. Inst. H. Poincaré Phys. Théor. 53 (1990), 35-81.
  23. Kostant B., Sternberg S., Symplectic reduction, BRS cohomology, and infinite-dimensional Clifford algebras, Ann. Physics 176 (1987), 49-113.
  24. Li-Bland D., LA-Courant algebroids and their applications, Ph.D. thesis, University of Toronto, 2012, arXiv:1204.2796.
  25. Lichnerowicz A., Les variétés de Poisson et leurs algèbres de Lie associées, J. Differential Geometry 12 (1977), 253-300.
  26. Littlejohn R.G., A guiding center Hamiltonian: a new approach, J. Math. Phys. 20 (1979), 2445-2458.
  27. Littlejohn R.G., Geometry and guiding center motion, in Fluids and Plasmas: Geometry and Dynamics (Boulder, Colo., 1983), Contemp. Math., Vol. 28, Editor J.E. Marsden, Amer. Math. Soc., Providence, RI, 1984, 151-167.
  28. Littlejohn R.G., Hamiltonian formulation of guiding center motion, Phys. Fluids 24 (1981), 1730-1749.
  29. Liu Z.-J., Weinstein A., Xu P., Manin triples for Lie bialgebroids, J. Differential Geom. 45 (1997), 547-574, dg-ga/9508013.
  30. Loday J.-L., Une version non commutative des algèbres de Lie: les algèbres de Leibniz, Enseign. Math. (2) 39 (1993), 269-293.
  31. Mackenzie K.C.H., Double Lie algebroids and second-order geometry. I, Adv. Math. 94 (1992), 180-239.
  32. Mackenzie K.C.H., Double Lie algebroids and second-order geometry. II, Adv. Math. 154 (2000), 46-75, dg-ga/9712013.
  33. Mackenzie K.C.H., Drinfel'd doubles and Ehresmann doubles for Lie algebroids and Lie bialgebroids, Electron. Res. Announc. Amer. Math. Soc. 4 (1998), 74-87.
  34. Mackenzie K.C.H., Ehresmann doubles and Drinfel'd doubles for Lie algebroids and Lie bialgebroids, J. Reine Angew. Math. 658 (2011), 193-245, math.DG/0611799.
  35. Mackenzie K.C.H., General theory of Lie groupoids and Lie algebroids, London Mathematical Society Lecture Note Series, Vol. 213, Cambridge University Press, Cambridge, 2005.
  36. Mackenzie K.C.H., Xu P., Lie bialgebroids and Poisson groupoids, Duke Math. J. 73 (1994), 415-452.
  37. Mokhov O.I., Novikov S.P., Pogrebkov A.K., Irina Yakovlevna Dorfman, Russ. Math. Surv. 50 (1995), 1241-1246.
  38. Olver P.J., Applications of Lie groups to differential equations, Graduate Texts in Mathematics, Vol. 107, Springer-Verlag, New York, 1986.
  39. Pradines J., Théorie de Lie pour les groupoïdes différentiables. Calcul différenetiel dans la catégorie des groupoïdes infinitésimaux, C. R. Acad. Sci. Paris Sér. A-B 264 (1967), A245-A248.
  40. Roytenberg D., AKSZ-BV formalism and Courant algebroid-induced topological field theories, Lett. Math. Phys. 79 (2007), 143-159, hep-th/0608150.
  41. Roytenberg D., Courant algebroids, derived brackets and even symplectic supermanifolds, Ph.D. thesis, University of California, Berkeley, 1999, math.DG/9910078.
  42. Roytenberg D., Courant-Dorfman algebras and their cohomology, Lett. Math. Phys. 90 (2009), 311-351, arXiv:0902.4862.
  43. Roytenberg D., On the structure of graded symplectic supermanifolds and Courant algebroids, in Quantization, Poisson Brackets and Beyond (Manchester, 2001), Contemp. Math., Vol. 315, Editor T. Voronov, Amer. Math. Soc., Providence, RI, 2002, 169-185, math.SG/0203110.
  44. Roytenberg D., Quasi-Lie bialgebroids and twisted Poisson manifolds, Lett. Math. Phys. 61 (2002), 123-137, math.QA/0112152.
  45. Ševera P., Letters to Alan Weinstein written between 1998 and 2000 (no. 1, 1998, no. 7, 2000), available at http://sophia.dtp.fmph.uniba.sk/~severa/letters/.
  46. Ševera P., Weinstein A., Poisson geometry with a 3-form background, Progr. Theoret. Phys. Suppl. 144 (2001), 145-154, math.SG/0107133.
  47. Skinner R., Rusk R., Generalized Hamiltonian dynamics. I. Formulation on T*Q⊕TQ, J. Math. Phys. 24 (1983), 2589-2594.
  48. Uchino K., Remarks on the definition of a Courant algebroid, Lett. Math. Phys. 60 (2002), 171-175, math.DG/0204010.
  49. Vaintrob A.Yu., Lie algebroids and homological vector fields, Russ. Math. Surv. 52 (1997), 428-429.
  50. Voronov T., Graded manifolds and Drinfeld doubles for Lie bialgebroids, in Quantization, Poisson Brackets and Beyond (Manchester, 2001), Contemp. Math., Vol. 315, Editor T. Voronov, Amer. Math. Soc., Providence, RI, 2002, 131-168, math.DG/0105237.
  51. Voronov T., Q-manifolds and Mackenzie theory, Comm. Math. Phys. 315 (2012), 279-310, arXiv:1206.3622.
  52. Weinstein A., Coisotropic calculus and Poisson groupoids, J. Math. Soc. Japan 40 (1988), 705-727.
  53. Weinstein A., Poisson geometry, Differential Geom. Appl. 9 (1998), 213-238.
  54. Weinstein A., Sophus Lie and symplectic geometry, Exposition. Math. 1 (1983), 95-96.
  55. Yoshimura H., Marsden J.E., Dirac structures in Lagrangian mechanics. I. Implicit Lagrangian systems, J. Geom. Phys. 57 (2006), 133-156.

Previous article  Next article   Contents of Volume 9 (2013)