Article
MSC:
20L05,
20N05,
22A22,
22E15,
22E60,
58H05 | MR 3583303 | Zbl 06674893 | DOI: 10.14712/1213-7243.2015.184
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Keywords:
group; Brandt groupoid; Lie group; loop; transversals; discrete mechanics
group; Brandt groupoid; Lie group; loop; transversals; discrete mechanics
Summary:
We discuss a concept of loopoid as a non-associative generalization of Brandt groupoid. We introduce and study also an interesting class of more general objects which we call semiloopoids. A differential version of loopoids is intended as a framework for Lagrangian discrete mechanics.
References:
[1] Baer R.: Nets and groups. Trans. Amer. Math. Soc. 46 (1939), 110–141. DOI 10.1090/S0002-9947-1939-0000035-5 | MR 0000035 | Zbl 0023.21502
[2] Belousov V.D.: Foundations of the Theory of Quasigroups and Loops. Nauka, Moscow, 1967 (in Russian). MR 0218483
[3] Bruck R.H.: A Survey of Binary Systems. Ergebnisse der Mathematik und ihrer Grenzgebiete. Neue Folge, Heft 20. Reihe: Gruppentheorie Springer, Berlin-Göttingen-Heidelberg, 1958. MR 0093552 | Zbl 0141.01401
[4] Bruck R.H.: What is a loop. in Studies in modern algebra, Studies in Mathematics, 2, Prentice-Hall, New Jersey, 1963, pp. 59–99. Zbl 0199.05202
[5] Fedorov Y.N., Zenkov D.V.: Discrete nonholonomic LL systems on Lie groups. Nonlinearity 18 (2005), 2211–2241. DOI 10.1088/0951-7715/18/5/017 | MR 2164739 | Zbl 1084.37049
[6] Foguel T.: Groups, transversals, and loops. Loops'99 (Prague), Comment. Math. Univ. Carolin. 41 (2000), 261–269. MR 1780870 | Zbl 1038.20047
[7] Grabowska K., Grabowski J.: Variational calculus with constraints on general algebroids. J. Phys. A 41 (2008), 175204, 25pp. DOI 10.1088/1751-8113/41/17/175204 | MR 2451669 | Zbl 1137.70011
[8] Grabowska K., Grabowski J.: Dirac algebroids in Lagrangian and Hamiltonian mechanics. J. Geom. Phys. 61 (2011), 2233–2253. DOI 10.1016/j.geomphys.2011.06.018 | MR 2827121 | Zbl 1223.37064
[9] Grabowski J., Jó'zwikowski M.: Pontryagin maximum principle on almost Lie algebroids. SIAM J. Control Optim. 49 (2011), 1306–1357. DOI 10.1137/090760246 | MR 2818883
[10] Grabowski J., de Leon M., Marrero J.C., Martin de Diego D.: Nonholonomic constraints: A new viewpoint. J. Math. Phys. 50 (2009), 013520, 17pp. DOI 10.1063/1.3049752 | MR 2492630
[11] Grabowski J., Urbański P.: Lie algebroids and Poisson-Nijenhis structures. Rep. Math. Phys. 40 (1997), 195–208. DOI 10.1016/S0034-4877(97)85916-2 | MR 1614690
[12] Grabowski J., Urbański P.: Algebroids – general differential calculi on vector bundles. J. Geom. Phys. 31 (1999), 111–141. DOI 10.1016/S0393-0440(99)00007-8 | MR 1706624 | Zbl 0954.17014
[13] Grigorian S.: $G_2$-structures and octonion bundles. arXiv:1510.04226.
[14] Iglesias D., Marrero J.C., Martín de Diego D., Martínez E.: Discrete nonholonomic Lagrangian systems on Lie groupoids. J. Nonlinear Sci. 18 (2008), 221–276. DOI 10.1007/s00332-007-9012-8 | MR 2411379
[15] Iglesias D., Marrero J.C., Martín de Diego D., Padrón E.: Discrete nonholonomic in implicit form. Discrete Contin. Dyn. Syst. 33 (2013), 1117–1135. DOI 10.3934/dcds.2013.33.1117 | MR 2988938
[16] Kinyon M.: The coquecigrue of a Leibniz algebra. preprint, 2003.
[17] Kinyon M., Weinstein A.: Leibniz algebras, Courant algebroids, and multiplications on reductive homogeneous spaces. Amer. J. Math. 123 (2001), 525–550. DOI 10.1353/ajm.2001.0017 | MR 1833152 | Zbl 0986.17001
[18] Mackenzie K.: General Theory of Lie Groupoids and Lie Algebroids. London Mathematical Society Lecture Note Series, 213, Cambridge University Press, Cambridge, 2005. MR 2157566 | Zbl 1078.58011
[19] Marrero J.C., Martín de Diego D., Martínez E.: Discrete Lagrangian and Hamiltonian mechanics on Lie groupoids. Nonlinearity 19 (2006), 1313–1348. Corrigendum: Nonlinearity 19 (2006), 3003–3004. DOI 10.1088/0951-7715/19/6/006 | MR 2230001
[20] Marrero J.C., Martín de Diego, Stern A.: Symplectic groupoids and discrete constrained Lagrangian mechanics. Discrete Contin. Dyn. Syst. 35 (2015), 367–397. MR 3286963
[21] Martínez E.: Lagrangian mechanics on Lie algebroids. Acta Appl. Math. 67 (2001), 295–320. DOI 10.1023/A:1011965919259 | MR 1861135 | Zbl 1002.70013
[22] Pflugfelder H.O.: Quasigroups and Loops: Introduction. Heldermann, Berlin, 1990. MR 1125767 | Zbl 0715.20043
[23] Sabinin L.V.: Smooth Quasigroups and Loops. Kluwer Academic Press, Dordrecht, 1999. MR 1727714 | Zbl 1038.20051
[24] Sabinin L.V.: Smooth quasigroups and loops: forty-five years of incredible growth. Comment. Math. Univ. Carolin. 41 (2000), 377–400. MR 1780879 | Zbl 1038.20051
[25] Smith J.D.H.: Loops and quasigroups: Aspects of current work and prospects for the future. Comment. Math. Univ. Carolin. 41 (2000), 415–427. MR 1780883 | Zbl 1038.20055
[26] Stern A.: Discrete Hamilton–Pontryagin mechanics and generating functions on Lie groupoids. J. Symplectic Geom. 8 (2010), 225–238. DOI 10.4310/JSG.2010.v8.n2.a5 | MR 2670166
[27] Weinstein A.: Lagrangian mechanics and groupoids. {Fields Inst. Comm.} 7 (1996), 207–231. MR 1365779 | Zbl 0844.22007
[28] Zakrzewski S.: Quantum and classical pseudogroups I. Union pseudogroups and their quantization. Comm. Math. Phys. 134 (1990), 347–370. DOI 10.1007/BF02097706 | MR 1081010 | Zbl 0708.58030
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