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, (2014). Google Scholar

[2]

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

[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. Google Scholar

[4]

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

[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. doi: 10.1142/S0219887806001211. Google Scholar

[6]

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

[7]

J. P. Dufour, Introduction aux tissus,, in Séminaire Gaston Darboux de Géométrie et Topologie Différentielle, (1992), 55. Google Scholar

[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. doi: 10.1007/s00220-011-1313-y. Google Scholar

[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. doi: 10.1016/j.aim.2009.09.010. Google Scholar

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

[11]

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

[12]

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

[13]

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

[14]

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

[15]

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

[16]

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

[17]

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

[18]

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

[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. doi: 10.1016/S0034-4877(03)80006-X. Google Scholar

[20]

M. Jóźwikowski, Optimal Control Theory on Almost-Lie Algebroids,, PhD thesis, (2011). Google Scholar

[21]

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

[22]

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

[23]

M. Jóźwikowski and M. Rotkiewicz, Prototypes of higher algebroids with applications to variational calculus,, preprint, (2013). Google Scholar

[24]

M. Jóźwikowski and M. Rotkiewicz, Abstract higher algebroids,, in preparation, (2014). Google Scholar

[25]

M. Jóźwikowski and M. Rotkiewicz, Variational calculus on higher algebroids,, in preparation, (2014). Google Scholar

[26]

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

[27]

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

[28]

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

[29]

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

[30]

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

[31]

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

[32]

E. Martínez, Geometric formulation of mechanics on Lie algebroids, in Proceedings of the VIII Fall Workshop on Geometry and Physics,, Medina del Campo, 2 (2001), 209. Google Scholar

[33]

E. Martínez, Variational calculus on Lie algebroids,, ESAIM: Control, 14 (2008), 356. doi: 10.1051/cocv:2007056. Google Scholar

[34]

E. Martínez, Lie algebroids in classical mechanics and optimal control,, SIGMA, 3 (2007). doi: 10.3842/SIGMA.2007.050. Google Scholar

[35]

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

[36]

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

[37]

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

[38]

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

[39]

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

[40]

Th. Th. Voronov, Microformal geometry,, preprint, (2014). Google Scholar

[41]

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

[42]

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

[43]

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

[44]

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

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, (2014). Google Scholar

[2]

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

[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. Google Scholar

[4]

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

[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. doi: 10.1142/S0219887806001211. Google Scholar

[6]

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

[7]

J. P. Dufour, Introduction aux tissus,, in Séminaire Gaston Darboux de Géométrie et Topologie Différentielle, (1992), 55. Google Scholar

[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. doi: 10.1007/s00220-011-1313-y. Google Scholar

[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. doi: 10.1016/j.aim.2009.09.010. Google Scholar

[10]

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

[11]

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

[12]

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

[13]

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

[14]

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

[15]

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

[16]

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

[17]

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

[18]

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

[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. doi: 10.1016/S0034-4877(03)80006-X. Google Scholar

[20]

M. Jóźwikowski, Optimal Control Theory on Almost-Lie Algebroids,, PhD thesis, (2011). Google Scholar

[21]

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

[22]

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

[23]

M. Jóźwikowski and M. Rotkiewicz, Prototypes of higher algebroids with applications to variational calculus,, preprint, (2013). Google Scholar

[24]

M. Jóźwikowski and M. Rotkiewicz, Abstract higher algebroids,, in preparation, (2014). Google Scholar

[25]

M. Jóźwikowski and M. Rotkiewicz, Variational calculus on higher algebroids,, in preparation, (2014). Google Scholar

[26]

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

[27]

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

[28]

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

[29]

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

[30]

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

[31]

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

[32]

E. Martínez, Geometric formulation of mechanics on Lie algebroids, in Proceedings of the VIII Fall Workshop on Geometry and Physics,, Medina del Campo, 2 (2001), 209. Google Scholar

[33]

E. Martínez, Variational calculus on Lie algebroids,, ESAIM: Control, 14 (2008), 356. doi: 10.1051/cocv:2007056. Google Scholar

[34]

E. Martínez, Lie algebroids in classical mechanics and optimal control,, SIGMA, 3 (2007). doi: 10.3842/SIGMA.2007.050. Google Scholar

[35]

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

[36]

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

[37]

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

[38]

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

[39]

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

[40]

Th. Th. Voronov, Microformal geometry,, preprint, (2014). Google Scholar

[41]

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

[42]

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

[43]

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

[44]

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

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