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

SIGMA 14 (2018), 135, 46 pages      arXiv:1708.03174

Higher-Order Analogs of Lie Algebroids via Vector Bundle Comorphisms

Michał Jóźwikowski a and Mikołaj Rotkiewicz b
a) Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, 00-656 Warszawa, Poland
b) Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Banacha 2, 02-097 Warszawa, Poland

Received January 10, 2018, in final form December 12, 2018; Published online December 29, 2018

We introduce the concept of a higher algebroid, generalizing the notions of an algebroid and a higher tangent bundle. Our ideas are based on a description of (Lie) algebroids as vector bundle comorphisms - differential relations of a special kind. In our approach higher algebroids are vector bundle comorphism between graded-linear bundles satisfying natural axioms. We provide natural examples and discuss applications in geometric mechanics.

Key words: higher algebroid; vector bundle comorphism; almost-Lie algebroid; graded manifold; graded bundle; algebroid lift; variational principle.

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  1. Abrunheiro L., Colombo L., Lagrangian Lie subalgebroids generating dynamics for second-order mechanical systems on Lie algebroids, Mediterr. J. Math. 15 (2018), 57, 19 pages, arXiv:1803.00059.
  2. Bourbaki N., Eléments de mathématique: variété différentielles et analytiques, Vol. XXIV, Hermann, Paris, 1971.
  3. Bruce A.J., Grabowska K., Grabowski J., Graded bundles in the category of Lie groupoids, SIGMA 11 (2015), 090, 25 pages, arXiv:1502.06092.
  4. Bruce A.J., Grabowska K., Grabowski J., Higher order mechanics on graded bundles, J. Phys. A: Math. Theor. 48 (2015), 205203, 32 pages, arXiv:1412.2719.
  5. Bruce A.J., Grabowska K., Grabowski J., Linear duals of graded bundles and higher analogues of (Lie) algebroids, J. Geom. Phys. 101 (2016), 71-99, arXiv:1409.0439.
  6. Bruce A.J., Grabowski J., Rotkiewicz M., Polarisation of graded bundles, SIGMA 12 (2016), 106, 30 pages, arXiv:1512.02345.
  7. Cattaneo A.S., Dherin B., Weinstein A., Integration of Lie algebroid comorphisms, Port. Math. 70 (2013), 113-144, arXiv:1210.4443.
  8. Colombo L., Second-order constrained variational problems on Lie algebroids: applications to optimal control, J. Geom. Mech. 9 (2017), 1-45, arXiv:1701.04772.
  9. Colombo L., Martín de Diego D., A variational and geometric approach for the second order Euler-Poincaré equations, Lecture Notes, Zaragoza, 2011.
  10. Colombo L., Martín de Diego D., Higher-order variational problems on Lie groups and optimal control applications, J. Geom. Mech. 6 (2014), 451-478, arXiv:1104.3221.
  11. Cortés J., de León M., Marrero J.C., Martín de Diego D., Martínez E., A survey of Lagrangian mechanics and control on Lie algebroids and groupoids, Int. J. Geom. Methods Mod. Phys. 3 (2006), 509-558, math-ph/0511009.
  12. de León M., Marrero J.C., Martínez E., Lagrangian submanifolds and dynamics on Lie algebroids, J. Phys. A: Math. Gen. 38 (2005), R241-R308, math.DG/0407528.
  13. Grabowska K., Grabowski J., Variational calculus with constraints on general algebroids, J. Phys. A: Math. Theor. 41 (2008), 175204, 25 pages, arXiv:0712.2766.
  14. Grabowska K., Urbański P., Grabowski J., Geometrical mechanics on algebroids, Int. J. Geom. Methods Mod. Phys. 3 (2006), 559-575, math-ph/0509063.
  15. Grabowski J., Modular classes of skew algebroid relations, Transform. Groups 17 (2012), 989-1010, arXiv:1108.2366.
  16. Grabowski J., Rotkiewicz M., Higher vector bundles and multi-graded symplectic manifolds, J. Geom. Phys. 59 (2009), 1285-1305, math.DG/0702772.
  17. Grabowski J., Rotkiewicz M., Graded bundles and homogeneity structures, J. Geom. Phys. 62 (2012), 21-36, arXiv:1102.0180.
  18. Grabowski J., Urbański P., Lie algebroids and Poisson-Nijenhuis structures, Rep. Math. Phys. 40 (1997), 195-208, dg-ga/9710007.
  19. Grabowski J., Urbański P., Algebroids - general differential calculi on vector bundles, J. Geom. Phys. 31 (1999), 111-141, math.DG/9909174.
  20. Guillemin V., Sternberg S., Geometric asymptotics, Mathematical Surveys, Vol. 14, Amer. Math. Soc., Providence, R.I., 1977.
  21. Higgins P.J., Mackenzie K., Algebraic constructions in the category of Lie algebroids, J. Algebra 129 (1990), 194-230.
  22. Higgins P.J., Mackenzie K.C.H., Duality for base-changing morphisms of vector bundles, modules, Lie algebroids and Poisson structures, Math. Proc. Cambridge Philos. Soc. 114 (1993), 471-488.
  23. Huebschmann J., Poisson cohomology and quantization, J. Reine Angew. Math. 408 (1990), 57-113.
  24. Jóźwikowski M., Prolongations vs. Tulczyjew triples in geometric mechanics, arXiv:1712.09858.
  25. Jóźwikowski M., Rotkiewicz M., Prototypes of higher algebroids with applications to variational calculus, arXiv:1306.3379.
  26. Jóźwikowski M., Rotkiewicz M., Bundle-theoretic methods for higher-order variational calculus, J. Geom. Mech. 6 (2014), 99-120, arXiv:1306.3097.
  27. Jóźwikowski M., Rotkiewicz M., Models for higher algebroids, J. Geom. Mech. 7 (2015), 317-359.
  28. Kolář I., Michor P.W., Slovák J., Natural operations in differential geometry, Springer-Verlag, Berlin, 1993.
  29. Konieczna K., Urbański P., Double vector bundles and duality, Arch. Math. (Brno) 35 (1999), 59-95, dg-ga/9710014.
  30. Kouotchop Wamba P.M., Ntyam A., Tangent lifts of higher order of multiplicative Dirac structures, Arch. Math. (Brno) 49 (2013), 87-104.
  31. Kruglikov B., Lychagin V., Global Lie-Tresse theorem, Selecta Math. (N.S.) 22 (2016), 1357-1411, arXiv:1111.5480.
  32. Mackenzie K.C.H., Double Lie algebroids and second-order geometry. I, Adv. Math. 94 (1992), 180-239.
  33. Mackenzie K.C.H., Duality and triple structures, in The Breadth of Symplectic and Poisson Geometry, Progr. Math., Vol. 232, Birkhäuser Boston, Boston, MA, 2005, 455-481, math.SG/0406267.
  34. 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.
  35. 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.
  36. Martínez E., Geometric formulation of mechanics on Lie algebroids, in Proceedings of the VIII Fall Workshop on Geometry and Physics (Spanish) (Medina del Campo, 1999), Publ. R. Soc. Mat. Esp., Vol. 2, R. Soc. Mat. Esp., Madrid, 2001, 209-222.
  37. Martínez E., Lagrangian mechanics on Lie algebroids, Acta Appl. Math. 67 (2001), 295-320.
  38. Martínez E., Variational calculus on Lie algebroids, ESAIM Control Optim. Calc. Var. 14 (2008), 356-380, math-ph/0603028.
  39. Martínez E., Higher-order variational calculus on Lie algebroids, J. Geom. Mech. 7 (2015), 81-108, arXiv:1501.06520.
  40. Morimoto A., Liftings of tensor fields and connections to tangent bundles of higher order, Nagoya Math. J. 40 (1970), 99-120.
  41. Pradines J., Fibres vectoriels doubles et calcul des jets non holonomes, Esquisses Mathématiques, Vol. 29, Université d'Amiens, U.E.R. de Mathématiques, Amiens, 1977.
  42. Rotkiewicz M., On structure of higher algebroids, in preparation.
  43. Saunders D.J., Prolongations of Lie groupoids and Lie algebroids, Houston J. Math. 30 (2004), 637-655.
  44. Tulczyjew W.M., Les sous-variétés lagrangiennes et la dynamique hamiltonienne, C. R. Acad. Sci. Paris Sér. A-B 283 (1976), 15-18.
  45. Tulczyjew W.M., Les sous-variétés lagrangiennes et la dynamique lagrangienne, C. R. Acad. Sci. Paris Sér. A-B 283 (1976), 675-678.
  46. Vaǐntrob A.Y., Lie algebroids and homological vector fields, Russian Math. Surveys 52 (1997), 428-429.
  47. Voronov T.T., Graded manifolds and Drinfeld doubles for Lie bialgebroids, in Quantization, Poisson Brackets and beyond (Manchester, 2001), Contemp. Math., Vol. 315, Amer. Math. Soc., Providence, RI, 2002, 131-168, math.DG/0105237.
  48. Voronov T.T., $Q$-manifolds and higher analogs of Lie algebroids, in XXIX Workshop on Geometric Methods in Physics, AIP Conf. Proc., Vol. 1307, Amer. Inst. Phys., Melville, NY, 2010, 191-202, arXiv:1010.2503.
  49. Voronov T.T., On a non-abelian Poincaré lemma, Proc. Amer. Math. Soc. 140 (2012), 2855-2872, arXiv:0905.0287.
  50. Voronov T.T., $Q$-manifolds and Mackenzie theory, Comm. Math. Phys. 315 (2012), 279-310, arXiv:1206.3622.
  51. Ševera P., Some title containing the words ''homotopy'' and ''symplectic'', e.g. this one, University of Luxembourg, Luxembourg, 2005, 121-137, math.SG/0105080.
  52. Weinstein A., Lagrangian mechanics and groupoids, in Mechanics Day (Waterloo, ON, 1992), Fields Inst. Commun., Vol. 7, Amer. Math. Soc., Providence, RI, 1996, 207-231.
  53. Zakrzewski S., Quantum and classical pseudogroups. I. Union pseudogroups and their quantization, Comm. Math. Phys. 134 (1990), 347-370.
  54. Zakrzewski S., Quantum and classical pseudogroups. II. Differential and symplectic pseudogroups, Comm. Math. Phys. 134 (1990), 371-395.

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