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Leibniz-Dirac structures and nonconservative systems with constraints
The supergeometry of Loday algebroids
1. | Polish Academy of Sciences, Institute of Mathematics, Śniadeckich 8, P.O. Box 21, 00-956 Warsaw, Poland |
2. | University of Luxembourg, Campus Kirchberg, Mathematics Research Unit, 6, rue Richard Coudenhove-Kalergi, L-1359 Luxembourg City, Luxembourg, Luxembourg |
References:
[1] |
M. Ammar and N. Poncin, Coalgebraic approach to the loday infinity category, stem differential for 2n-ary graded and homotopy algebras,, Ann. Inst. Fourier, 60 (2010), 355.
doi: 10.5802/aif.2525. |
[2] |
D. Baraglia, Leibniz algebroids, twistings and exceptional generalized geometry,, J. Geom. Phys., 62 (2012), 903.
doi: 10.1016/j.geomphys.2012.01.007. |
[3] |
Y. Bi and Y. Sheng, On higher analogues of Courant algebroid,, Sci. China Math., 54 (2011), 437.
doi: 10.1007/s11425-010-4142-0. |
[4] |
G. Bonavolontà and N. Poncin, On the category of Lie $n$-algebroids,, , (). Google Scholar |
[5] |
T. J. Courant, Dirac manifolds,, Trans. Amer. Math. Soc., 319 (1990), 631.
doi: 10.2307/2001258. |
[6] |
I. Y. Dorfman, Dirac structures of integrable evolution equations,, Phys. Lett. A, 125 (1987), 240.
doi: 10.1016/0375-9601(87)90201-5. |
[7] |
V. Drinfel'd, Quantum groups,, in, (1986).
|
[8] |
S. Eilenberg and S. Mac Lane, On the groups of $H(\Pi,n)$. I,, Ann. of Math., 58 (1953), 55. Google Scholar |
[9] |
V. T. Filippov, $n$-Lie algebras,, Sibirsk. Math. Zh., 26 (1985), 126.
|
[10] |
D. García-Beltrán and J. A. Vallejo, An approach to omni-Lie algebroids using quasi-derivations,, J. Gen. Lie Theory Appl., 5 (2011).
doi: 10.4303/jglta/G100801. |
[11] |
V. Ginzburg and M. Kapranov, Koszul duality for operads,, Duke Math. J., 76 (1994), 203.
doi: 10.1215/S0012-7094-94-07608-4. |
[12] |
K. Grabowska and J. Grabowski, Variational calculus with constraints on general algebroids,, J. Phys. A, 41 (2008).
doi: 10.1088/1751-8113/41/17/175204. |
[13] |
K. Grabowska and J. Grabowski, Dirac algebroids in Lagrangian and Hamiltonian mechanics,, J. Geom. Phys., 61 (2011), 2233.
doi: 10.1016/j.geomphys.2011.06.018. |
[14] |
K. Grabowska, J. Grabowski and P. Urbański, Geometrical mechanics on algebroids,, Int. J. Geom. Meth. Mod. Phys., 3 (2006), 559.
doi: 10.1142/S0219887806001259. |
[15] |
J. Grabowski, Abstract Jacobi and Poisson structures. Quantization and star-products,, J. Geom. Phys., 9 (1992), 45.
doi: 10.1016/0393-0440(92)90025-V. |
[16] |
J. Grabowski, Quasi-derivations and QD-algebroids,, Rep. Math. Phys., 32 (2003), 445.
doi: 10.1016/S0034-4877(03)80041-1. |
[17] |
J. Grabowski, Graded contact manifolds and contact Courant algebroids,, J. Geom. Phys., 68 (2013), 27.
doi: 10.1016/j.geomphys.2013.02.001. |
[18] |
J. Grabowski, Brackets,, to appear in Int. J. Geom. Methods Mod. Phys., (). Google Scholar |
[19] |
J. Grabowski, M. de León, D. Martín de Diego and J. C. Marrero, Nonholonomic constraints: A new viewpoint,, J. Math. Phys., 50 (2009).
doi: 10.1063/1.3049752. |
[20] |
J. Grabowski and G. Marmo, Non-antisymmetric versions of Nambu-Poisson and Lie algebroid brackets,, J. Phys. A, 34 (2001), 3803.
doi: 10.1088/0305-4470/34/18/308. |
[21] |
J. Grabowski and G. Marmo, Binary operations in classical and quantum mechanics,, in, 59 (2003), 163.
doi: 10.4064/bc59-0-8. |
[22] |
J. Grabowski and G. Marmo, The graded Jacobi algebras and (co)homology,, J. Phys. A, 36 (2003), 161.
doi: 10.1088/0305-4470/36/1/311. |
[23] |
J. Grabowski and N. Poncin, Automorphisms of quantum and classical Poisson algebras,, Compositio Math., 140 (2004), 511.
doi: 10.1112/S0010437X0300006X. |
[24] |
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. |
[25] |
J. C. Herz, Pseudo-algèbres de Lie,, C. R. Acad. Sci. Paris, 236 (1953), 1935. Google Scholar |
[26] |
Y. Hagiwara, Nambu-Dirac manifolds,, J. Phys. A, 35 (2002), 1263.
doi: 10.1088/0305-4470/35/5/310. |
[27] |
Y. Hagiwara and T. Mizutani, Leibniz algebras associated with foliations,, Kodai Math. J., 25 (2002), 151.
doi: 10.2996/kmj/1071674438. |
[28] |
R. Ibáñez, M. de León, J. C. Marrero and E. Padrón, Leibniz algebroid associated with a Nambu-Poisson structure,, J. Phys. A, 32 (1999), 8129.
doi: 10.1088/0305-4470/32/46/310. |
[29] |
N. Jacobson, On pseudo-linear transformations,, Proc. Nat. Acad. Sci., 21 (1935), 667. Google Scholar |
[30] |
N. Jacobson, Pseudo-linear transformations,, Ann. Math., 38 (1937), 485. Google Scholar |
[31] |
D. Khudaverdian, A. Mandal and N. Poncin, Higher categorified algebras versus bounded homotopy algebras,, Theo. Appl. Cat., 25 (2011), 251.
|
[32] |
A. A. Kirillov, Local Lie algebras,, (Russian) Uspekhi Mat. Nauk, 31 (1976), 57.
|
[33] |
Y. Kosmann-Schwarzbach, From Poisson algebras to Gerstenhaber algebras,, Ann. Inst. Fourier (Grenoble), 46 (1996), 1243.
doi: 10.5802/aif.1547. |
[34] |
Y. Kosmann-Schwarzbach, Derived brackets,, Lett. Math. Phys., 69 (2004), 61. Google Scholar |
[35] |
Y. Kosmann-Schwarzbach, Quasi, twisted, and all that$\ldots$in Poisson geometry and Lie algebroid theory,, in, 232 (2005), 363.
doi: 10.1007/0-8176-4419-9_12. |
[36] |
Y. Kosmann-Schwarzbach and K. Mackenzie, Differential operators and actions of Lie algebroids,, in, 315 (2002), 213.
doi: 10.1090/conm/315/05482. |
[37] |
A. Kotov and T. Strobl, Generalizing geometry-algebroids and sigma models,, in, 16 (2010), 209.
doi: 10.4171/079-1/7. |
[38] |
A. Lichnerowicz, Algèbre de Lie des automorphismes infinitésimaux d'une structure unimodulaire,, Ann. Inst. Fourier, 24 (1974), 219.
|
[39] |
Zhang-Ju Liu, A. Weinstein and Ping Xu, Manin triples for Lie bialgebroids,, J. Diff. Geom., 45 (1997), 547.
|
[40] |
J.-L. Loday, "Cyclic Homology,", Appendix E by María O. Ronco, 301 (1992).
|
[41] |
J.-L. Loday, Une version non commutative des algèbres de Lie, les algèbres de Leibniz,, Ann. Inst. Fourier, 37 (1993), 269. Google Scholar |
[42] |
J.-L. Loday and T. Pirashvili, Universal enveloping algebras of Leibniz algebras and (co)homology,, Math. Annalen, 296 (1993), 139.
doi: 10.1007/BF01445099. |
[43] |
J. M. Lodder, Leibniz cohomology for differentiable manifolds,, Ann. Inst. Fourier (Grenoble), 48 (1998), 73. Google Scholar |
[44] |
J. M. Lodder, Leibniz cohomology and the calculus of variations,, Differential Geom. Appl., 21 (2004), 113.
doi: 10.1016/j.difgeo.2004.03.010. |
[45] |
K. C. H. Mackenzie, Lie algebroids and Lie pseudoalgebras,, Bull. London Math. Soc., 27 (1995), 97.
doi: 10.1112/blms/27.2.97. |
[46] |
K. C. H. Mackenzie, "General Theory of Lie Groupoids and Lie Algebroids,", London Mathematical Society Lecture Note Series, 213 (2005).
|
[47] |
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. |
[48] |
K. Mikami and T. Mizutani, Algebroids associated with pre-Poisson structures,, in, (2007), 71.
doi: 10.1142/9789812779649_0004. |
[49] |
E. Nelson, "Tensor Analysis,", Princeton University Press, (1967). Google Scholar |
[50] |
J. P. Ortega and V. Planas-Bielsa, Dynamics on Leibniz manifolds,, J. Geom. Phys., 52 (2004), 1.
doi: 10.1016/j.geomphys.2004.01.002. |
[51] |
J. Peetre, Une caractérisation abstraite des opérateurs différentiels,, Math. Scand., 7 (1959), 211.
|
[52] |
J. Peetre, Réctifications á l'article "Une caractérisation abstraite des opératuers diffŕentiels,", Math. Scand., 8 (1960), 116.
|
[53] |
J. Pradines, Théorie de Lie pour les groupoïdes différentiables. Calcul différentiel dans la catégorie des groupoïdes infinitésimaux,, C. R. Acad. Sci. Paris, 264 (1967).
|
[54] |
D. E. Radford, A natural ring basis for the shuffle algebra and an application to group schemes,, J. Algebra, 58 (1979), 432.
doi: 10.1016/0021-8693(79)90171-6. |
[55] |
R. Ree, Lie elements and an algebra associated with shuffles,, Ann. of Math. (2), 68 (1958), 210.
doi: 10.2307/1970243. |
[56] |
R. Ree, Generalized Lie elements,, Canad. J. Math., 12 (1960), 493.
doi: 10.4153/CJM-1960-044-x. |
[57] |
D. Roytenberg, "Courant Algebroids, Derived Brackets and Even Symplectic Supermanifolds,", Ph.D. thesis, (1999).
|
[58] |
D. Roytenberg, On the structure of graded symplectic supermanifolds and Courant algebroids,, in, 315 (2002), 169.
doi: 10.1090/conm/315/05479. |
[59] |
P. Ševera and A. Weinstein, Poisson geometry with a 3-form background,, Prog. Theor. Phys. Suppl., 144 (2001), 145.
doi: 10.1143/PTPS.144.145. |
[60] |
M. Stiénon and P. Xu, Modular classes of Loday algebroids,, C. R. Acad. Sci. Paris, 346 (2008), 193.
doi: 10.1016/j.crma.2007.12.012. |
[61] |
K. Uchino, Remarks on the definition of a Courant algebroid,, Lett. Math. Phys., 60 (2002), 171.
doi: 10.1023/A,1016179410273. |
[62] |
A. M. Vinogradov, The logic algebra for the theory of linear differential operators,, Soviet. Mat. Dokl., 13 (1972), 1058. Google Scholar |
[63] |
A. Wade, Conformal Dirac structures,, Lett. Math. Phys., 53 (2000), 331.
doi: 10.1023/A,1007634407701. |
[64] |
A. Wade, On some properties of Leibniz algebroids,, in, (2002), 65.
doi: 10.1142/9789812777089_0005. |
[65] |
M. Zambon, $L_\infty$-algebras and higher analogues of Dirac structures and Courant algebroids,, J. Symplectic Geom., 10 (2012), 563.
|
[66] |
A. A. Zolotykh and A. A. Mikhalëv, The base of free shuffle superalgebras, (Russian), Uspekhi Mat. Nauk, 50 (1995), 199.
doi: 10.1070/RM1995v050n01ABEH001681. |
show all references
References:
[1] |
M. Ammar and N. Poncin, Coalgebraic approach to the loday infinity category, stem differential for 2n-ary graded and homotopy algebras,, Ann. Inst. Fourier, 60 (2010), 355.
doi: 10.5802/aif.2525. |
[2] |
D. Baraglia, Leibniz algebroids, twistings and exceptional generalized geometry,, J. Geom. Phys., 62 (2012), 903.
doi: 10.1016/j.geomphys.2012.01.007. |
[3] |
Y. Bi and Y. Sheng, On higher analogues of Courant algebroid,, Sci. China Math., 54 (2011), 437.
doi: 10.1007/s11425-010-4142-0. |
[4] |
G. Bonavolontà and N. Poncin, On the category of Lie $n$-algebroids,, , (). Google Scholar |
[5] |
T. J. Courant, Dirac manifolds,, Trans. Amer. Math. Soc., 319 (1990), 631.
doi: 10.2307/2001258. |
[6] |
I. Y. Dorfman, Dirac structures of integrable evolution equations,, Phys. Lett. A, 125 (1987), 240.
doi: 10.1016/0375-9601(87)90201-5. |
[7] |
V. Drinfel'd, Quantum groups,, in, (1986).
|
[8] |
S. Eilenberg and S. Mac Lane, On the groups of $H(\Pi,n)$. I,, Ann. of Math., 58 (1953), 55. Google Scholar |
[9] |
V. T. Filippov, $n$-Lie algebras,, Sibirsk. Math. Zh., 26 (1985), 126.
|
[10] |
D. García-Beltrán and J. A. Vallejo, An approach to omni-Lie algebroids using quasi-derivations,, J. Gen. Lie Theory Appl., 5 (2011).
doi: 10.4303/jglta/G100801. |
[11] |
V. Ginzburg and M. Kapranov, Koszul duality for operads,, Duke Math. J., 76 (1994), 203.
doi: 10.1215/S0012-7094-94-07608-4. |
[12] |
K. Grabowska and J. Grabowski, Variational calculus with constraints on general algebroids,, J. Phys. A, 41 (2008).
doi: 10.1088/1751-8113/41/17/175204. |
[13] |
K. Grabowska and J. Grabowski, Dirac algebroids in Lagrangian and Hamiltonian mechanics,, J. Geom. Phys., 61 (2011), 2233.
doi: 10.1016/j.geomphys.2011.06.018. |
[14] |
K. Grabowska, J. Grabowski and P. Urbański, Geometrical mechanics on algebroids,, Int. J. Geom. Meth. Mod. Phys., 3 (2006), 559.
doi: 10.1142/S0219887806001259. |
[15] |
J. Grabowski, Abstract Jacobi and Poisson structures. Quantization and star-products,, J. Geom. Phys., 9 (1992), 45.
doi: 10.1016/0393-0440(92)90025-V. |
[16] |
J. Grabowski, Quasi-derivations and QD-algebroids,, Rep. Math. Phys., 32 (2003), 445.
doi: 10.1016/S0034-4877(03)80041-1. |
[17] |
J. Grabowski, Graded contact manifolds and contact Courant algebroids,, J. Geom. Phys., 68 (2013), 27.
doi: 10.1016/j.geomphys.2013.02.001. |
[18] |
J. Grabowski, Brackets,, to appear in Int. J. Geom. Methods Mod. Phys., (). Google Scholar |
[19] |
J. Grabowski, M. de León, D. Martín de Diego and J. C. Marrero, Nonholonomic constraints: A new viewpoint,, J. Math. Phys., 50 (2009).
doi: 10.1063/1.3049752. |
[20] |
J. Grabowski and G. Marmo, Non-antisymmetric versions of Nambu-Poisson and Lie algebroid brackets,, J. Phys. A, 34 (2001), 3803.
doi: 10.1088/0305-4470/34/18/308. |
[21] |
J. Grabowski and G. Marmo, Binary operations in classical and quantum mechanics,, in, 59 (2003), 163.
doi: 10.4064/bc59-0-8. |
[22] |
J. Grabowski and G. Marmo, The graded Jacobi algebras and (co)homology,, J. Phys. A, 36 (2003), 161.
doi: 10.1088/0305-4470/36/1/311. |
[23] |
J. Grabowski and N. Poncin, Automorphisms of quantum and classical Poisson algebras,, Compositio Math., 140 (2004), 511.
doi: 10.1112/S0010437X0300006X. |
[24] |
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. |
[25] |
J. C. Herz, Pseudo-algèbres de Lie,, C. R. Acad. Sci. Paris, 236 (1953), 1935. Google Scholar |
[26] |
Y. Hagiwara, Nambu-Dirac manifolds,, J. Phys. A, 35 (2002), 1263.
doi: 10.1088/0305-4470/35/5/310. |
[27] |
Y. Hagiwara and T. Mizutani, Leibniz algebras associated with foliations,, Kodai Math. J., 25 (2002), 151.
doi: 10.2996/kmj/1071674438. |
[28] |
R. Ibáñez, M. de León, J. C. Marrero and E. Padrón, Leibniz algebroid associated with a Nambu-Poisson structure,, J. Phys. A, 32 (1999), 8129.
doi: 10.1088/0305-4470/32/46/310. |
[29] |
N. Jacobson, On pseudo-linear transformations,, Proc. Nat. Acad. Sci., 21 (1935), 667. Google Scholar |
[30] |
N. Jacobson, Pseudo-linear transformations,, Ann. Math., 38 (1937), 485. Google Scholar |
[31] |
D. Khudaverdian, A. Mandal and N. Poncin, Higher categorified algebras versus bounded homotopy algebras,, Theo. Appl. Cat., 25 (2011), 251.
|
[32] |
A. A. Kirillov, Local Lie algebras,, (Russian) Uspekhi Mat. Nauk, 31 (1976), 57.
|
[33] |
Y. Kosmann-Schwarzbach, From Poisson algebras to Gerstenhaber algebras,, Ann. Inst. Fourier (Grenoble), 46 (1996), 1243.
doi: 10.5802/aif.1547. |
[34] |
Y. Kosmann-Schwarzbach, Derived brackets,, Lett. Math. Phys., 69 (2004), 61. Google Scholar |
[35] |
Y. Kosmann-Schwarzbach, Quasi, twisted, and all that$\ldots$in Poisson geometry and Lie algebroid theory,, in, 232 (2005), 363.
doi: 10.1007/0-8176-4419-9_12. |
[36] |
Y. Kosmann-Schwarzbach and K. Mackenzie, Differential operators and actions of Lie algebroids,, in, 315 (2002), 213.
doi: 10.1090/conm/315/05482. |
[37] |
A. Kotov and T. Strobl, Generalizing geometry-algebroids and sigma models,, in, 16 (2010), 209.
doi: 10.4171/079-1/7. |
[38] |
A. Lichnerowicz, Algèbre de Lie des automorphismes infinitésimaux d'une structure unimodulaire,, Ann. Inst. Fourier, 24 (1974), 219.
|
[39] |
Zhang-Ju Liu, A. Weinstein and Ping Xu, Manin triples for Lie bialgebroids,, J. Diff. Geom., 45 (1997), 547.
|
[40] |
J.-L. Loday, "Cyclic Homology,", Appendix E by María O. Ronco, 301 (1992).
|
[41] |
J.-L. Loday, Une version non commutative des algèbres de Lie, les algèbres de Leibniz,, Ann. Inst. Fourier, 37 (1993), 269. Google Scholar |
[42] |
J.-L. Loday and T. Pirashvili, Universal enveloping algebras of Leibniz algebras and (co)homology,, Math. Annalen, 296 (1993), 139.
doi: 10.1007/BF01445099. |
[43] |
J. M. Lodder, Leibniz cohomology for differentiable manifolds,, Ann. Inst. Fourier (Grenoble), 48 (1998), 73. Google Scholar |
[44] |
J. M. Lodder, Leibniz cohomology and the calculus of variations,, Differential Geom. Appl., 21 (2004), 113.
doi: 10.1016/j.difgeo.2004.03.010. |
[45] |
K. C. H. Mackenzie, Lie algebroids and Lie pseudoalgebras,, Bull. London Math. Soc., 27 (1995), 97.
doi: 10.1112/blms/27.2.97. |
[46] |
K. C. H. Mackenzie, "General Theory of Lie Groupoids and Lie Algebroids,", London Mathematical Society Lecture Note Series, 213 (2005).
|
[47] |
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. |
[48] |
K. Mikami and T. Mizutani, Algebroids associated with pre-Poisson structures,, in, (2007), 71.
doi: 10.1142/9789812779649_0004. |
[49] |
E. Nelson, "Tensor Analysis,", Princeton University Press, (1967). Google Scholar |
[50] |
J. P. Ortega and V. Planas-Bielsa, Dynamics on Leibniz manifolds,, J. Geom. Phys., 52 (2004), 1.
doi: 10.1016/j.geomphys.2004.01.002. |
[51] |
J. Peetre, Une caractérisation abstraite des opérateurs différentiels,, Math. Scand., 7 (1959), 211.
|
[52] |
J. Peetre, Réctifications á l'article "Une caractérisation abstraite des opératuers diffŕentiels,", Math. Scand., 8 (1960), 116.
|
[53] |
J. Pradines, Théorie de Lie pour les groupoïdes différentiables. Calcul différentiel dans la catégorie des groupoïdes infinitésimaux,, C. R. Acad. Sci. Paris, 264 (1967).
|
[54] |
D. E. Radford, A natural ring basis for the shuffle algebra and an application to group schemes,, J. Algebra, 58 (1979), 432.
doi: 10.1016/0021-8693(79)90171-6. |
[55] |
R. Ree, Lie elements and an algebra associated with shuffles,, Ann. of Math. (2), 68 (1958), 210.
doi: 10.2307/1970243. |
[56] |
R. Ree, Generalized Lie elements,, Canad. J. Math., 12 (1960), 493.
doi: 10.4153/CJM-1960-044-x. |
[57] |
D. Roytenberg, "Courant Algebroids, Derived Brackets and Even Symplectic Supermanifolds,", Ph.D. thesis, (1999).
|
[58] |
D. Roytenberg, On the structure of graded symplectic supermanifolds and Courant algebroids,, in, 315 (2002), 169.
doi: 10.1090/conm/315/05479. |
[59] |
P. Ševera and A. Weinstein, Poisson geometry with a 3-form background,, Prog. Theor. Phys. Suppl., 144 (2001), 145.
doi: 10.1143/PTPS.144.145. |
[60] |
M. Stiénon and P. Xu, Modular classes of Loday algebroids,, C. R. Acad. Sci. Paris, 346 (2008), 193.
doi: 10.1016/j.crma.2007.12.012. |
[61] |
K. Uchino, Remarks on the definition of a Courant algebroid,, Lett. Math. Phys., 60 (2002), 171.
doi: 10.1023/A,1016179410273. |
[62] |
A. M. Vinogradov, The logic algebra for the theory of linear differential operators,, Soviet. Mat. Dokl., 13 (1972), 1058. Google Scholar |
[63] |
A. Wade, Conformal Dirac structures,, Lett. Math. Phys., 53 (2000), 331.
doi: 10.1023/A,1007634407701. |
[64] |
A. Wade, On some properties of Leibniz algebroids,, in, (2002), 65.
doi: 10.1142/9789812777089_0005. |
[65] |
M. Zambon, $L_\infty$-algebras and higher analogues of Dirac structures and Courant algebroids,, J. Symplectic Geom., 10 (2012), 563.
|
[66] |
A. A. Zolotykh and A. A. Mikhalëv, The base of free shuffle superalgebras, (Russian), Uspekhi Mat. Nauk, 50 (1995), 199.
doi: 10.1070/RM1995v050n01ABEH001681. |
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