September  2015, 7(3): 317-359. doi: 10.3934/jgm.2015.7.317

Models for higher algebroids

1. 

Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, 00-656 Warszawa, Poland

2. 

Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Banacha 2, 02-097 Warszawa, Poland

Received  June 2013 Revised  June 2015 Published  July 2015

Reductions of higher tangent bundles of Lie groupoids provide natural examples of geometric structures which we would like to call higher algebroids. Such objects can be also constructed abstractly starting from an arbitrary almost Lie algebroid. A higher algebroid is, in principle, a graded bundle equipped with a differential relation of special kind (a Zakrzewski morphism). In the paper we investigate basic properties of higher algebroids and present some examples.
Citation: Michał Jóźwikowski, Mikołaj Rotkiewicz. Models for higher algebroids. Journal of Geometric Mechanics, 2015, 7 (3) : 317-359. doi: 10.3934/jgm.2015.7.317
References:
[1]

A. J. Bruce, K. Grabowska and J. Grabowski, Linear duals of graded bundles and higher analogues of (Lie) algebroids, preprint, arXiv:1409.0439 (2014).

[2]

A. J. Bruce, K. Grabowska and J. Grabowski, Higher order mechanics on graded bundles, J. Phys. A: Math. Theor., 48 (2015), 205203. doi: 10.1088/1751-8113/48/20/205203.

[3]

F. Cantrijn, M. Crampin, W. Sarlet and D. Saunders, The canonical isomorphism between $\T^k \T^* M$ and $\T^*\T^kM$, C. R. Acad. Sci. Paris, 309 (1989), 1509-1514.

[4]

L. Colombo and D. M. de Diego, A Variational and Geometric Approach for the Second Order Euler-Poinaré Equations, lecture notes, Zaragoza, 2011.

[5]

J. Cortes, M. de Leon, J. C. Marrero, D. Martin de Diego and E. Martínez, A survey of Lagrangian mechanics and control on Lie algebroids and groupoids, Int. J. Geom. Meth. Mod. Phys., 3 (2006), 509-558. doi: 10.1142/S0219887806001211.

[6]

M. Crainic and R. L. Fernendes, Integrability of Lie brackets, Ann. of Math., 157 (2003), 575-620. doi: 10.4007/annals.2003.157.575.

[7]

J. P. Dufour, Introduction aux tissus, in Séminaire Gaston Darboux de Géométrie et Topologie Différentielle, 1990/1991 (Montpellier, 1990/1991), Univ. Montpellier II, (1992), 55-76.

[8]

F. Gay-Balmaz, D. D. Holm, D. M. Meier, T. S. Ratiu and F. Vialard, Invariant Higher-Order Variational Problems, Comm. Math. Phys., 309 (2012), 413-458. doi: 10.1007/s00220-011-1313-y.

[9]

A. Gracia-Saz and R. A. Mehta, Lie algebroid structures on double vector bundles and representation theory of Lie algebroids, Adv. Math., 223 (2010), 1236-1275. doi: 10.1016/j.aim.2009.09.010.

[10]

K. Grabowska and J. Grabowski, Variational calculus with constraints on general algebroids, J. Phys. A: Math. Theor., 41 (2008), 175204, 25pp. doi: 10.1088/1751-8113/41/17/175204.

[11]

K. Grabowska, J. Grabowski and P. Urbański, Geometric mechanics on algebroids, Int. J. Geom. Meth. Mod. Phys., 3 (2006), 559-575. doi: 10.1142/S0219887806001259.

[12]

K. Grabowska and L. Vitagliano, Tulczyjew triples in higher derivative field theory, J. Geom. Mech., 7 (2015), 1-33. doi: 10.3934/jgm.2015.7.1.

[13]

J. Grabowski and M. Jóźwikowski, Pontryagin maximum principle on almost lie algebroids, SIAM J. Control Optim., 49 (2011), 1306-1357. doi: 10.1137/090760246.

[14]

J. Grabowski, M. de Leon, J. C. Marrero and D. Martin de Diego, Nonholonomic Constraints: A New Viewpoint, J. Math. Phys., 50 (2009), 013520, 17pp. doi: 10.1063/1.3049752.

[15]

J. Grabowski and M. Rotkiewicz, Graded bundles and homogeneity structures, J. Geom. Phys., 62 (2011), 21-36. doi: 10.1016/j.geomphys.2011.09.004.

[16]

J. Grabowski and M. Rotkiewicz, Higher vector bundles and multi-graded symplectic manifolds, J. Geom. Phys., 59 (2009), 1285-1305. doi: 10.1016/j.geomphys.2009.06.009.

[17]

J. Grabowski and P. Urbański, Lie algebroids and Poisson-Nijenhuis structures, Rep. Math. Phys., 40 (1997), 195-208. doi: 10.1016/S0034-4877(97)85916-2.

[18]

J. Grabowski and P. Urbański, Algebroids - general differential calculi on vector bundles, J. Geom. Phys., 31 (1999), 111-141. doi: 10.1016/S0393-0440(99)00007-8.

[19]

X. Gracia, J. Martin-Solano and M. Munoz-Lecenda, Some geometric aspects of variational calculus in constrained systems, Rep. Math. Phys., 51 (2003), 127-148. doi: 10.1016/S0034-4877(03)80006-X.

[20]

M. Jóźwikowski, Optimal Control Theory on Almost-Lie Algebroids, PhD thesis, arXiv:1111.1549 (2011).

[21]

M. Jóźwikowski, Jacobi vector fields for Lagrangian systems on algebroids, Int. J. Geom. Methods Mod. Phys., 10 (2013).

[22]

M. Jóźwikowski and M. Rotkiewicz, Bundle-theoretic methods in higher-order variational calculus, J. Geom. Mech., 6 (2014), 99-120. doi: 10.3934/jgm.2014.6.99.

[23]

M. Jóźwikowski and M. Rotkiewicz, Prototypes of higher algebroids with applications to variational calculus, preprint, arXiv:1306.3379 (2013).

[24]

M. Jóźwikowski and M. Rotkiewicz, Abstract higher algebroids, in preparation, 2014.

[25]

M. Jóźwikowski and M. Rotkiewicz, Variational calculus on higher algebroids, in preparation, 2014.

[26]

I. Kolar, Weil bundles as generalized jet spaces, in Handbook of Global Analysis, Elsevier, Amsterdam, 1214 (2008), 625-664. doi: 10.1016/B978-044452833-9.50013-9.

[27]

I. Kolar, P. W. Michor and J. Slovak, Natural Operations in Differential Geometry, Springer, Berlin, 1993. doi: 10.1007/978-3-662-02950-3.

[28]

M. de Leon, JC. Marrero and E. Martínez, Lagrangian submanifolds and dynamics on Lie algebroids, J. Phys. A: Math. Gen., 38 (2005), 241-308. doi: 10.1088/0305-4470/38/24/R01.

[29]

K. C. H. Mackenzie and P. Xu, Lie bialgebroids and Poisson groupoids, Duke Math. J., 73 (1994), 415-452, doi: 10.1215/S0012-7094-94-07318-3.

[30]

K. Mackenzie, General Theory of Lie groupoids and Lie Algebroids, Cambridge University Press, Cambridge, 2005. doi: 10.1017/CBO9781107325883.

[31]

E. Martínez, Lagrangian mechanics on Lie algebroids, Acta Appl. Math., 67 (2001), 295-320. doi: 10.1023/A:1011965919259.

[32]

E. Martínez, Geometric formulation of mechanics on Lie algebroids, in Proceedings of the VIII Fall Workshop on Geometry and Physics, Medina del Campo, 1999, Publicaciones de la RSME, 2 (2001), 209-222.

[33]

E. Martínez, Variational calculus on Lie algebroids, ESAIM: Control, Optimization and Calculus of Variations, 14 (2008), 356-380. doi: 10.1051/cocv:2007056.

[34]

E. Martínez, Lie algebroids in classical mechanics and optimal control, SIGMA, 3 (2007), Paper 050, 17 pp. doi: 10.3842/SIGMA.2007.050.

[35]

E. Martínez, Higher-order variational calculus on Lie algebroids, J. Geom. Mech., 7 (2015), 81-108. doi: 10.3934/jgm.2015.7.81.

[36]

A. Morimoto, Liftings of tensor fields and connections to tangent bundles of higher order, Nagoya Math. J., 40 (1970), 99-120.

[37]

D. J. Saunders, Prolongations of Lie groupoids and Lie algebroids, Houston J. Math., 30 (2004), 637-655.

[38]

W. Tulczyjew, The Lagrange differential, Bull. Acad. Polon. Sci., 24 (1976), 1089-1096.

[39]

Th.Th. Voronov, Q-manifolds and higher analogs of Lie algebroids, AIP Conf. Proc., 1307 (2010), 191-202.

[40]

Th. Th. Voronov, Microformal geometry, preprint, (2014), arXiv:1411.6720.

[41]

A. Weil, Théorie des points proches sur les varietes différentiables, in: Colloque de géométrie différentielle, CNRS, Strasbourg, 1953 (1953), 111-117.

[42]

A. Weinstein, Lagrangian mechanics and groupoids, Fields Inst. Comm., 7 (1996), 207-231.

[43]

S. Zakrzewski, Quantum and classical pseudogroups. Part I. Union pseudogroups and their quantization, Comm. Math. Phys., 134 (1990), 347-370. doi: 10.1007/BF02097707.

[44]

S. Zakrzewski, Quantum and classical pseudogroups. Part II. Differential and symplectic pseudogroups, Comm. Math. Phys., 134 (1990), 371-395. doi: 10.1007/BF02097706.

show all references

References:
[1]

A. J. Bruce, K. Grabowska and J. Grabowski, Linear duals of graded bundles and higher analogues of (Lie) algebroids, preprint, arXiv:1409.0439 (2014).

[2]

A. J. Bruce, K. Grabowska and J. Grabowski, Higher order mechanics on graded bundles, J. Phys. A: Math. Theor., 48 (2015), 205203. doi: 10.1088/1751-8113/48/20/205203.

[3]

F. Cantrijn, M. Crampin, W. Sarlet and D. Saunders, The canonical isomorphism between $\T^k \T^* M$ and $\T^*\T^kM$, C. R. Acad. Sci. Paris, 309 (1989), 1509-1514.

[4]

L. Colombo and D. M. de Diego, A Variational and Geometric Approach for the Second Order Euler-Poinaré Equations, lecture notes, Zaragoza, 2011.

[5]

J. Cortes, M. de Leon, J. C. Marrero, D. Martin de Diego and E. Martínez, A survey of Lagrangian mechanics and control on Lie algebroids and groupoids, Int. J. Geom. Meth. Mod. Phys., 3 (2006), 509-558. doi: 10.1142/S0219887806001211.

[6]

M. Crainic and R. L. Fernendes, Integrability of Lie brackets, Ann. of Math., 157 (2003), 575-620. doi: 10.4007/annals.2003.157.575.

[7]

J. P. Dufour, Introduction aux tissus, in Séminaire Gaston Darboux de Géométrie et Topologie Différentielle, 1990/1991 (Montpellier, 1990/1991), Univ. Montpellier II, (1992), 55-76.

[8]

F. Gay-Balmaz, D. D. Holm, D. M. Meier, T. S. Ratiu and F. Vialard, Invariant Higher-Order Variational Problems, Comm. Math. Phys., 309 (2012), 413-458. doi: 10.1007/s00220-011-1313-y.

[9]

A. Gracia-Saz and R. A. Mehta, Lie algebroid structures on double vector bundles and representation theory of Lie algebroids, Adv. Math., 223 (2010), 1236-1275. doi: 10.1016/j.aim.2009.09.010.

[10]

K. Grabowska and J. Grabowski, Variational calculus with constraints on general algebroids, J. Phys. A: Math. Theor., 41 (2008), 175204, 25pp. doi: 10.1088/1751-8113/41/17/175204.

[11]

K. Grabowska, J. Grabowski and P. Urbański, Geometric mechanics on algebroids, Int. J. Geom. Meth. Mod. Phys., 3 (2006), 559-575. doi: 10.1142/S0219887806001259.

[12]

K. Grabowska and L. Vitagliano, Tulczyjew triples in higher derivative field theory, J. Geom. Mech., 7 (2015), 1-33. doi: 10.3934/jgm.2015.7.1.

[13]

J. Grabowski and M. Jóźwikowski, Pontryagin maximum principle on almost lie algebroids, SIAM J. Control Optim., 49 (2011), 1306-1357. doi: 10.1137/090760246.

[14]

J. Grabowski, M. de Leon, J. C. Marrero and D. Martin de Diego, Nonholonomic Constraints: A New Viewpoint, J. Math. Phys., 50 (2009), 013520, 17pp. doi: 10.1063/1.3049752.

[15]

J. Grabowski and M. Rotkiewicz, Graded bundles and homogeneity structures, J. Geom. Phys., 62 (2011), 21-36. doi: 10.1016/j.geomphys.2011.09.004.

[16]

J. Grabowski and M. Rotkiewicz, Higher vector bundles and multi-graded symplectic manifolds, J. Geom. Phys., 59 (2009), 1285-1305. doi: 10.1016/j.geomphys.2009.06.009.

[17]

J. Grabowski and P. Urbański, Lie algebroids and Poisson-Nijenhuis structures, Rep. Math. Phys., 40 (1997), 195-208. doi: 10.1016/S0034-4877(97)85916-2.

[18]

J. Grabowski and P. Urbański, Algebroids - general differential calculi on vector bundles, J. Geom. Phys., 31 (1999), 111-141. doi: 10.1016/S0393-0440(99)00007-8.

[19]

X. Gracia, J. Martin-Solano and M. Munoz-Lecenda, Some geometric aspects of variational calculus in constrained systems, Rep. Math. Phys., 51 (2003), 127-148. doi: 10.1016/S0034-4877(03)80006-X.

[20]

M. Jóźwikowski, Optimal Control Theory on Almost-Lie Algebroids, PhD thesis, arXiv:1111.1549 (2011).

[21]

M. Jóźwikowski, Jacobi vector fields for Lagrangian systems on algebroids, Int. J. Geom. Methods Mod. Phys., 10 (2013).

[22]

M. Jóźwikowski and M. Rotkiewicz, Bundle-theoretic methods in higher-order variational calculus, J. Geom. Mech., 6 (2014), 99-120. doi: 10.3934/jgm.2014.6.99.

[23]

M. Jóźwikowski and M. Rotkiewicz, Prototypes of higher algebroids with applications to variational calculus, preprint, arXiv:1306.3379 (2013).

[24]

M. Jóźwikowski and M. Rotkiewicz, Abstract higher algebroids, in preparation, 2014.

[25]

M. Jóźwikowski and M. Rotkiewicz, Variational calculus on higher algebroids, in preparation, 2014.

[26]

I. Kolar, Weil bundles as generalized jet spaces, in Handbook of Global Analysis, Elsevier, Amsterdam, 1214 (2008), 625-664. doi: 10.1016/B978-044452833-9.50013-9.

[27]

I. Kolar, P. W. Michor and J. Slovak, Natural Operations in Differential Geometry, Springer, Berlin, 1993. doi: 10.1007/978-3-662-02950-3.

[28]

M. de Leon, JC. Marrero and E. Martínez, Lagrangian submanifolds and dynamics on Lie algebroids, J. Phys. A: Math. Gen., 38 (2005), 241-308. doi: 10.1088/0305-4470/38/24/R01.

[29]

K. C. H. Mackenzie and P. Xu, Lie bialgebroids and Poisson groupoids, Duke Math. J., 73 (1994), 415-452, doi: 10.1215/S0012-7094-94-07318-3.

[30]

K. Mackenzie, General Theory of Lie groupoids and Lie Algebroids, Cambridge University Press, Cambridge, 2005. doi: 10.1017/CBO9781107325883.

[31]

E. Martínez, Lagrangian mechanics on Lie algebroids, Acta Appl. Math., 67 (2001), 295-320. doi: 10.1023/A:1011965919259.

[32]

E. Martínez, Geometric formulation of mechanics on Lie algebroids, in Proceedings of the VIII Fall Workshop on Geometry and Physics, Medina del Campo, 1999, Publicaciones de la RSME, 2 (2001), 209-222.

[33]

E. Martínez, Variational calculus on Lie algebroids, ESAIM: Control, Optimization and Calculus of Variations, 14 (2008), 356-380. doi: 10.1051/cocv:2007056.

[34]

E. Martínez, Lie algebroids in classical mechanics and optimal control, SIGMA, 3 (2007), Paper 050, 17 pp. doi: 10.3842/SIGMA.2007.050.

[35]

E. Martínez, Higher-order variational calculus on Lie algebroids, J. Geom. Mech., 7 (2015), 81-108. doi: 10.3934/jgm.2015.7.81.

[36]

A. Morimoto, Liftings of tensor fields and connections to tangent bundles of higher order, Nagoya Math. J., 40 (1970), 99-120.

[37]

D. J. Saunders, Prolongations of Lie groupoids and Lie algebroids, Houston J. Math., 30 (2004), 637-655.

[38]

W. Tulczyjew, The Lagrange differential, Bull. Acad. Polon. Sci., 24 (1976), 1089-1096.

[39]

Th.Th. Voronov, Q-manifolds and higher analogs of Lie algebroids, AIP Conf. Proc., 1307 (2010), 191-202.

[40]

Th. Th. Voronov, Microformal geometry, preprint, (2014), arXiv:1411.6720.

[41]

A. Weil, Théorie des points proches sur les varietes différentiables, in: Colloque de géométrie différentielle, CNRS, Strasbourg, 1953 (1953), 111-117.

[42]

A. Weinstein, Lagrangian mechanics and groupoids, Fields Inst. Comm., 7 (1996), 207-231.

[43]

S. Zakrzewski, Quantum and classical pseudogroups. Part I. Union pseudogroups and their quantization, Comm. Math. Phys., 134 (1990), 347-370. doi: 10.1007/BF02097707.

[44]

S. Zakrzewski, Quantum and classical pseudogroups. Part II. Differential and symplectic pseudogroups, Comm. Math. Phys., 134 (1990), 371-395. doi: 10.1007/BF02097706.

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