The supergeometry of Loday algebroids

Previous Article
Semi-global symplectic invariants of the Euler top
JGM Home
This Issue
Next Article
Leibniz-Dirac structures and nonconservative systems with constraints

June 2013, 5(2): 185-213. doi: 10.3934/jgm.2013.5.185

The supergeometry of Loday algebroids

Janusz Grabowski ^1, , David Khudaverdyan ^2, and Norbert Poncin ^2,

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

Received June 2012 Revised April 2013 Published July 2013

Abstract
Related Papers

A new concept of Loday algebroid (and its pure algebraic version -- Loday pseudoalgebra) is proposed and discussed in comparison with other similar structures present in the literature. The structure of a Loday pseudoalgebra and its natural reduction to a Lie pseudoalgebra is studied. Further, Loday algebroids are interpreted as homological vector fields on a `supercommutative manifold' associated with a shuffle product and the corresponding Cartan calculus is introduced. Several examples, including Courant algebroids, Grassmann-Dorfman and twisted Courant-Dorfman brackets, as well as algebroids induced by Nambu-Poisson structures, are given.

Keywords: Algebroid, pseudoalgebra, Loday algebra, homological vector field, Cartan calculus., supercommutative manifold, Courant bracket.

Mathematics Subject Classification: 53D17, 58A50, 17A32, 17B6.

Citation: Janusz Grabowski, David Khudaverdyan, Norbert Poncin. The supergeometry of Loday algebroids. Journal of Geometric Mechanics, 2013, 5 (2) : 185-213. doi: 10.3934/jgm.2013.5.185

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), 355-387. doi: 10.5802/aif.2525.
[2]	D. Baraglia, Leibniz algebroids, twistings and exceptional generalized geometry, J. Geom. Phys., 62 (2012), 903-934. 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), 437-447. doi: 10.1007/s11425-010-4142-0.
[4]	G. Bonavolontà and N. Poncin, On the category of Lie $n$-algebroids, arXiv:1207.3590.
[5]	T. J. Courant, Dirac manifolds, Trans. Amer. Math. Soc., 319 (1990), 631-661. doi: 10.2307/2001258.
[6]	I. Y. Dorfman, Dirac structures of integrable evolution equations, Phys. Lett. A, 125 (1987), 240-246. doi: 10.1016/0375-9601(87)90201-5.
[7]	V. Drinfel'd, Quantum groups, in "Proceedings of the International Congress of Mathematicians, Vol. 1, 2" (Berkeley, Calif., 1986), American Mathematical Society, Providence, RI, 1987.
[8]	S. Eilenberg and S. Mac Lane, On the groups of $H(\Pi,n)$. I, Ann. of Math., 58 (1953), 55-106.
[9]	V. T. Filippov, $n$-Lie algebras, Sibirsk. Math. Zh., 26 (1985), 126-140, 191.
[10]	D. García-Beltrán and J. A. Vallejo, An approach to omni-Lie algebroids using quasi-derivations, J. Gen. Lie Theory Appl., 5 (2011), Art. ID G100801, 10 pp. doi: 10.4303/jglta/G100801.
[11]	V. Ginzburg and M. Kapranov, Koszul duality for operads, Duke Math. J., 76 (1994), 203-272. 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), 175204, 25 pp. 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), 2233-2253. 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), 559-575. doi: 10.1142/S0219887806001259.
[15]	J. Grabowski, Abstract Jacobi and Poisson structures. Quantization and star-products, J. Geom. Phys., 9 (1992), 45-73. doi: 10.1016/0393-0440(92)90025-V.
[16]	J. Grabowski, Quasi-derivations and QD-algebroids, Rep. Math. Phys., 32 (2003), 445-451. doi: 10.1016/S0034-4877(03)80041-1.
[17]	J. Grabowski, Graded contact manifolds and contact Courant algebroids, J. Geom. Phys., 68 (2013), 27-28. doi: 10.1016/j.geomphys.2013.02.001.
[18]	J. Grabowski, Brackets, to appear in Int. J. Geom. Methods Mod. Phys., arXiv:1301.0227.
[19]	J. Grabowski, M. de León, D. Martín de Diego and J. C. Marrero, Nonholonomic constraints: A new viewpoint, J. Math. Phys., 50 (2009), 013520, 17 pp. 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), 3803-3809. doi: 10.1088/0305-4470/34/18/308.
[21]	J. Grabowski and G. Marmo, Binary operations in classical and quantum mechanics, in "Classical and Quantum Integrability" (eds. J. Grabowski and P. Urbański), Banach Center Publ., 59, Polish Acad. Sci., Warsaw, (2003), 163-172. doi: 10.4064/bc59-0-8.
[22]	J. Grabowski and G. Marmo, The graded Jacobi algebras and (co)homology, J. Phys. A, Math. Gen., 36 (2003), 161-181. 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), 511-527. doi: 10.1112/S0010437X0300006X.
[24]	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.
[25]	J. C. Herz, Pseudo-algèbres de Lie, C. R. Acad. Sci. Paris, 236 (1953), I, 1935-1937; II, 2289-2291.
[26]	Y. Hagiwara, Nambu-Dirac manifolds, J. Phys. A, 35 (2002), 1263-1281. doi: 10.1088/0305-4470/35/5/310.
[27]	Y. Hagiwara and T. Mizutani, Leibniz algebras associated with foliations, Kodai Math. J., 25 (2002), 151-165. 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, Math. Gen., 32 (1999), 8129-8144. doi: 10.1088/0305-4470/32/46/310.
[29]	N. Jacobson, On pseudo-linear transformations, Proc. Nat. Acad. Sci., 21 (1935), 667-670.
[30]	N. Jacobson, Pseudo-linear transformations, Ann. Math., 38 (1937), 485-506.
[31]	D. Khudaverdian, A. Mandal and N. Poncin, Higher categorified algebras versus bounded homotopy algebras, Theo. Appl. Cat., 25 (2011), 251-275.
[32]	A. A. Kirillov, Local Lie algebras, (Russian) Uspekhi Mat. Nauk, 31 (1976), 57-76.
[33]	Y. Kosmann-Schwarzbach, From Poisson algebras to Gerstenhaber algebras, Ann. Inst. Fourier (Grenoble), 46 (1996), 1243-1274. doi: 10.5802/aif.1547.
[34]	Y. Kosmann-Schwarzbach, Derived brackets, Lett. Math. Phys., 69 (2004), 61-87.
[35]	Y. Kosmann-Schwarzbach, Quasi, twisted, and all that$\ldots$in Poisson geometry and Lie algebroid theory, in "The Breadth of Symplectic and Poisson Geometry," {Progr. Math.,} 232, Birkhäuser Boston, Boston, MA, (2005), 363-389. doi: 10.1007/0-8176-4419-9_12.
[36]	Y. Kosmann-Schwarzbach and K. Mackenzie, Differential operators and actions of Lie algebroids, in "Quantization, Poisson Brackets and Beyond" (ed. T. Voronov), Contemporary Math., 315, Amer. Math. Soc., Providence, RI, (2002), 213-233. doi: 10.1090/conm/315/05482.
[37]	A. Kotov and T. Strobl, Generalizing geometry-algebroids and sigma models, in "Handbook of Pseudo-Riemannian Geometry and Supersymmetry," IRMA Lect. Math. Theor. Phys., 16, Eur. Math. Soc., Zürich, (2010), 209-262. doi: 10.4171/079-1/7.
[38]	A. Lichnerowicz, Algèbre de Lie des automorphismes infinitésimaux d'une structure unimodulaire, Ann. Inst. Fourier, 24 (1974), 219-266.
[39]	Zhang-Ju Liu, A. Weinstein and Ping Xu, Manin triples for Lie bialgebroids, J. Diff. Geom., 45 (1997), 547-574.
[40]	J.-L. Loday, "Cyclic Homology," Appendix E by María O. Ronco, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 301, Springer Verlag, Berlin, 1992.
[41]	J.-L. Loday, Une version non commutative des algèbres de Lie, les algèbres de Leibniz, Ann. Inst. Fourier, 37 (1993), 269-293.
[42]	J.-L. Loday and T. Pirashvili, Universal enveloping algebras of Leibniz algebras and (co)homology, Math. Annalen, 296 (1993), 139-158. doi: 10.1007/BF01445099.
[43]	J. M. Lodder, Leibniz cohomology for differentiable manifolds, Ann. Inst. Fourier (Grenoble), 48 (1998), 73-75.
[44]	J. M. Lodder, Leibniz cohomology and the calculus of variations, Differential Geom. Appl., 21 (2004), 113-126. doi: 10.1016/j.difgeo.2004.03.010.
[45]	K. C. H. Mackenzie, Lie algebroids and Lie pseudoalgebras, Bull. London Math. Soc., 27 (1995), 97-147. 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, Cambridge University Press, Cambridge, 2005.
[47]	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.
[48]	K. Mikami and T. Mizutani, Algebroids associated with pre-Poisson structures, in "Noncommutative Geometry and Physics 2005," World Sci. Publ., Hackensack, NJ, (2007), 71-96. doi: 10.1142/9789812779649_0004.
[49]	E. Nelson, "Tensor Analysis," Princeton University Press, The University of Tokyo Press, Princeton, 1967.
[50]	J. P. Ortega and V. Planas-Bielsa, Dynamics on Leibniz manifolds, J. Geom. Phys., 52 (2004), 1-27. doi: 10.1016/j.geomphys.2004.01.002.
[51]	J. Peetre, Une caractérisation abstraite des opérateurs différentiels, Math. Scand., 7 (1959), 211-218.
[52]	J. Peetre, Réctifications á l'article "Une caractérisation abstraite des opératuers diffŕentiels," Math. Scand., 8 (1960), 116-120.
[53]	J. Pradines, Théorie de Lie pour les groupoïdes différentiables. Calcul différentiel dans la catégorie des groupoïdes infinitésimaux, C. R. Acad. Sci. Paris, Sér. A-B, 264 (1967), A245-A248.
[54]	D. E. Radford, A natural ring basis for the shuffle algebra and an application to group schemes, J. Algebra, 58 (1979), 432-454. 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), 210-220. doi: 10.2307/1970243.
[56]	R. Ree, Generalized Lie elements, Canad. J. Math., 12 (1960), 493-502. doi: 10.4153/CJM-1960-044-x.
[57]	D. Roytenberg, "Courant Algebroids, Derived Brackets and Even Symplectic Supermanifolds," Ph.D. thesis, University of California, Berkeley, 1999.
[58]	D. Roytenberg, On the structure of graded symplectic supermanifolds and Courant algebroids, in "Quantization, Poisson Brackets and Beyond" (Manchester, 2001), Contemp. Math., 315, Amer. Math. Soc., Providence, RI, (2002), 169-185. 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), 145-154. doi: 10.1143/PTPS.144.145.
[60]	M. Stiénon and P. Xu, Modular classes of Loday algebroids, C. R. Acad. Sci. Paris, 346 (2008), 193-198. doi: 10.1016/j.crma.2007.12.012.
[61]	K. Uchino, Remarks on the definition of a Courant algebroid, Lett. Math. Phys., 60 (2002), 171-175. doi: 10.1023/A,1016179410273.
[62]	A. M. Vinogradov, The logic algebra for the theory of linear differential operators, Soviet. Mat. Dokl., 13 (1972), 1058-1062.
[63]	A. Wade, Conformal Dirac structures, Lett. Math. Phys., 53 (2000), 331-348. doi: 10.1023/A,1007634407701.
[64]	A. Wade, On some properties of Leibniz algebroids, in "Infinite Dimensional Lie Groups in Geometry and Representation Theory" (Washington, DC, 2000), World Sci. Publ., River Edge, NJ, (2002), 65-78. 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), 563-599.
[66]	A. A. Zolotykh and A. A. Mikhalëv, The base of free shuffle superalgebras, (Russian) Uspekhi Mat. Nauk, 50 (1995), 199-200; translation in Russian Math. Surveys, 50 (1995), 225-226. 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), 355-387. doi: 10.5802/aif.2525.
[2]	D. Baraglia, Leibniz algebroids, twistings and exceptional generalized geometry, J. Geom. Phys., 62 (2012), 903-934. 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), 437-447. doi: 10.1007/s11425-010-4142-0.
[4]	G. Bonavolontà and N. Poncin, On the category of Lie $n$-algebroids, arXiv:1207.3590.
[5]	T. J. Courant, Dirac manifolds, Trans. Amer. Math. Soc., 319 (1990), 631-661. doi: 10.2307/2001258.
[6]	I. Y. Dorfman, Dirac structures of integrable evolution equations, Phys. Lett. A, 125 (1987), 240-246. doi: 10.1016/0375-9601(87)90201-5.
[7]	V. Drinfel'd, Quantum groups, in "Proceedings of the International Congress of Mathematicians, Vol. 1, 2" (Berkeley, Calif., 1986), American Mathematical Society, Providence, RI, 1987.
[8]	S. Eilenberg and S. Mac Lane, On the groups of $H(\Pi,n)$. I, Ann. of Math., 58 (1953), 55-106.
[9]	V. T. Filippov, $n$-Lie algebras, Sibirsk. Math. Zh., 26 (1985), 126-140, 191.
[10]	D. García-Beltrán and J. A. Vallejo, An approach to omni-Lie algebroids using quasi-derivations, J. Gen. Lie Theory Appl., 5 (2011), Art. ID G100801, 10 pp. doi: 10.4303/jglta/G100801.
[11]	V. Ginzburg and M. Kapranov, Koszul duality for operads, Duke Math. J., 76 (1994), 203-272. 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), 175204, 25 pp. 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), 2233-2253. 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), 559-575. doi: 10.1142/S0219887806001259.
[15]	J. Grabowski, Abstract Jacobi and Poisson structures. Quantization and star-products, J. Geom. Phys., 9 (1992), 45-73. doi: 10.1016/0393-0440(92)90025-V.
[16]	J. Grabowski, Quasi-derivations and QD-algebroids, Rep. Math. Phys., 32 (2003), 445-451. doi: 10.1016/S0034-4877(03)80041-1.
[17]	J. Grabowski, Graded contact manifolds and contact Courant algebroids, J. Geom. Phys., 68 (2013), 27-28. doi: 10.1016/j.geomphys.2013.02.001.
[18]	J. Grabowski, Brackets, to appear in Int. J. Geom. Methods Mod. Phys., arXiv:1301.0227.
[19]	J. Grabowski, M. de León, D. Martín de Diego and J. C. Marrero, Nonholonomic constraints: A new viewpoint, J. Math. Phys., 50 (2009), 013520, 17 pp. 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), 3803-3809. doi: 10.1088/0305-4470/34/18/308.
[21]	J. Grabowski and G. Marmo, Binary operations in classical and quantum mechanics, in "Classical and Quantum Integrability" (eds. J. Grabowski and P. Urbański), Banach Center Publ., 59, Polish Acad. Sci., Warsaw, (2003), 163-172. doi: 10.4064/bc59-0-8.
[22]	J. Grabowski and G. Marmo, The graded Jacobi algebras and (co)homology, J. Phys. A, Math. Gen., 36 (2003), 161-181. 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), 511-527. doi: 10.1112/S0010437X0300006X.
[24]	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.
[25]	J. C. Herz, Pseudo-algèbres de Lie, C. R. Acad. Sci. Paris, 236 (1953), I, 1935-1937; II, 2289-2291.
[26]	Y. Hagiwara, Nambu-Dirac manifolds, J. Phys. A, 35 (2002), 1263-1281. doi: 10.1088/0305-4470/35/5/310.
[27]	Y. Hagiwara and T. Mizutani, Leibniz algebras associated with foliations, Kodai Math. J., 25 (2002), 151-165. 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, Math. Gen., 32 (1999), 8129-8144. doi: 10.1088/0305-4470/32/46/310.
[29]	N. Jacobson, On pseudo-linear transformations, Proc. Nat. Acad. Sci., 21 (1935), 667-670.
[30]	N. Jacobson, Pseudo-linear transformations, Ann. Math., 38 (1937), 485-506.
[31]	D. Khudaverdian, A. Mandal and N. Poncin, Higher categorified algebras versus bounded homotopy algebras, Theo. Appl. Cat., 25 (2011), 251-275.
[32]	A. A. Kirillov, Local Lie algebras, (Russian) Uspekhi Mat. Nauk, 31 (1976), 57-76.
[33]	Y. Kosmann-Schwarzbach, From Poisson algebras to Gerstenhaber algebras, Ann. Inst. Fourier (Grenoble), 46 (1996), 1243-1274. doi: 10.5802/aif.1547.
[34]	Y. Kosmann-Schwarzbach, Derived brackets, Lett. Math. Phys., 69 (2004), 61-87.
[35]	Y. Kosmann-Schwarzbach, Quasi, twisted, and all that$\ldots$in Poisson geometry and Lie algebroid theory, in "The Breadth of Symplectic and Poisson Geometry," {Progr. Math.,} 232, Birkhäuser Boston, Boston, MA, (2005), 363-389. doi: 10.1007/0-8176-4419-9_12.
[36]	Y. Kosmann-Schwarzbach and K. Mackenzie, Differential operators and actions of Lie algebroids, in "Quantization, Poisson Brackets and Beyond" (ed. T. Voronov), Contemporary Math., 315, Amer. Math. Soc., Providence, RI, (2002), 213-233. doi: 10.1090/conm/315/05482.
[37]	A. Kotov and T. Strobl, Generalizing geometry-algebroids and sigma models, in "Handbook of Pseudo-Riemannian Geometry and Supersymmetry," IRMA Lect. Math. Theor. Phys., 16, Eur. Math. Soc., Zürich, (2010), 209-262. doi: 10.4171/079-1/7.
[38]	A. Lichnerowicz, Algèbre de Lie des automorphismes infinitésimaux d'une structure unimodulaire, Ann. Inst. Fourier, 24 (1974), 219-266.
[39]	Zhang-Ju Liu, A. Weinstein and Ping Xu, Manin triples for Lie bialgebroids, J. Diff. Geom., 45 (1997), 547-574.
[40]	J.-L. Loday, "Cyclic Homology," Appendix E by María O. Ronco, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 301, Springer Verlag, Berlin, 1992.
[41]	J.-L. Loday, Une version non commutative des algèbres de Lie, les algèbres de Leibniz, Ann. Inst. Fourier, 37 (1993), 269-293.
[42]	J.-L. Loday and T. Pirashvili, Universal enveloping algebras of Leibniz algebras and (co)homology, Math. Annalen, 296 (1993), 139-158. doi: 10.1007/BF01445099.
[43]	J. M. Lodder, Leibniz cohomology for differentiable manifolds, Ann. Inst. Fourier (Grenoble), 48 (1998), 73-75.
[44]	J. M. Lodder, Leibniz cohomology and the calculus of variations, Differential Geom. Appl., 21 (2004), 113-126. doi: 10.1016/j.difgeo.2004.03.010.
[45]	K. C. H. Mackenzie, Lie algebroids and Lie pseudoalgebras, Bull. London Math. Soc., 27 (1995), 97-147. 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, Cambridge University Press, Cambridge, 2005.
[47]	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.
[48]	K. Mikami and T. Mizutani, Algebroids associated with pre-Poisson structures, in "Noncommutative Geometry and Physics 2005," World Sci. Publ., Hackensack, NJ, (2007), 71-96. doi: 10.1142/9789812779649_0004.
[49]	E. Nelson, "Tensor Analysis," Princeton University Press, The University of Tokyo Press, Princeton, 1967.
[50]	J. P. Ortega and V. Planas-Bielsa, Dynamics on Leibniz manifolds, J. Geom. Phys., 52 (2004), 1-27. doi: 10.1016/j.geomphys.2004.01.002.
[51]	J. Peetre, Une caractérisation abstraite des opérateurs différentiels, Math. Scand., 7 (1959), 211-218.
[52]	J. Peetre, Réctifications á l'article "Une caractérisation abstraite des opératuers diffŕentiels," Math. Scand., 8 (1960), 116-120.
[53]	J. Pradines, Théorie de Lie pour les groupoïdes différentiables. Calcul différentiel dans la catégorie des groupoïdes infinitésimaux, C. R. Acad. Sci. Paris, Sér. A-B, 264 (1967), A245-A248.
[54]	D. E. Radford, A natural ring basis for the shuffle algebra and an application to group schemes, J. Algebra, 58 (1979), 432-454. 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), 210-220. doi: 10.2307/1970243.
[56]	R. Ree, Generalized Lie elements, Canad. J. Math., 12 (1960), 493-502. doi: 10.4153/CJM-1960-044-x.
[57]	D. Roytenberg, "Courant Algebroids, Derived Brackets and Even Symplectic Supermanifolds," Ph.D. thesis, University of California, Berkeley, 1999.
[58]	D. Roytenberg, On the structure of graded symplectic supermanifolds and Courant algebroids, in "Quantization, Poisson Brackets and Beyond" (Manchester, 2001), Contemp. Math., 315, Amer. Math. Soc., Providence, RI, (2002), 169-185. 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), 145-154. doi: 10.1143/PTPS.144.145.
[60]	M. Stiénon and P. Xu, Modular classes of Loday algebroids, C. R. Acad. Sci. Paris, 346 (2008), 193-198. doi: 10.1016/j.crma.2007.12.012.
[61]	K. Uchino, Remarks on the definition of a Courant algebroid, Lett. Math. Phys., 60 (2002), 171-175. doi: 10.1023/A,1016179410273.
[62]	A. M. Vinogradov, The logic algebra for the theory of linear differential operators, Soviet. Mat. Dokl., 13 (1972), 1058-1062.
[63]	A. Wade, Conformal Dirac structures, Lett. Math. Phys., 53 (2000), 331-348. doi: 10.1023/A,1007634407701.
[64]	A. Wade, On some properties of Leibniz algebroids, in "Infinite Dimensional Lie Groups in Geometry and Representation Theory" (Washington, DC, 2000), World Sci. Publ., River Edge, NJ, (2002), 65-78. 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), 563-599.
[66]	A. A. Zolotykh and A. A. Mikhalëv, The base of free shuffle superalgebras, (Russian) Uspekhi Mat. Nauk, 50 (1995), 199-200; translation in Russian Math. Surveys, 50 (1995), 225-226. doi: 10.1070/RM1995v050n01ABEH001681.

[1]	Franz W. Kamber and Peter W. Michor. The flow completion of a manifold with vector field. Electronic Research Announcements, 2000, 6: 95-97.
[2]	Rama Ayoub, Aziz Hamdouni, Dina Razafindralandy. A new Hodge operator in discrete exterior calculus. Application to fluid mechanics. Communications on Pure and Applied Analysis, 2021, 20 (6) : 2155-2185. doi: 10.3934/cpaa.2021062
[3]	Oǧul Esen, Hasan Gümral. Geometry of plasma dynamics II: Lie algebra of Hamiltonian vector fields. Journal of Geometric Mechanics, 2012, 4 (3) : 239-269. doi: 10.3934/jgm.2012.4.239
[4]	Shengji Li, Xiaole Guo. Calculus rules of generalized $\epsilon-$subdifferential for vector valued mappings and applications. Journal of Industrial and Management Optimization, 2012, 8 (2) : 411-427. doi: 10.3934/jimo.2012.8.411
[5]	Moon Duchin, Tom Needham, Thomas Weighill. The (homological) persistence of gerrymandering. Foundations of Data Science, 2021 doi: 10.3934/fods.2021007
[6]	Robert Roussarie. A topological study of planar vector field singularities. Discrete and Continuous Dynamical Systems, 2020, 40 (9) : 5217-5245. doi: 10.3934/dcds.2020226
[7]	Paulo Antunes, Joana M. Nunes da Costa. Hypersymplectic structures on Courant algebroids. Journal of Geometric Mechanics, 2015, 7 (3) : 255-280. doi: 10.3934/jgm.2015.7.255
[8]	David Li-Bland, Pavol Ševera. Integration of exact Courant algebroids. Electronic Research Announcements, 2012, 19: 58-76. doi: 10.3934/era.2012.19.58
[9]	Mario Roldan. Hyperbolic sets and entropy at the homological level. Discrete and Continuous Dynamical Systems, 2016, 36 (6) : 3417-3433. doi: 10.3934/dcds.2016.36.3417
[10]	Nikos Katzourakis. Nonuniqueness in vector-valued calculus of variations in $L^\infty$ and some Linear elliptic systems. Communications on Pure and Applied Analysis, 2015, 14 (1) : 313-327. doi: 10.3934/cpaa.2015.14.313
[11]	Nikos Katzourakis. Corrigendum to the paper: Nonuniqueness in Vector-Valued Calculus of Variations in $ L^\infty $ and some Linear Elliptic Systems. Communications on Pure and Applied Analysis, 2019, 18 (4) : 2197-2198. doi: 10.3934/cpaa.2019098
[12]	Honglei Lang, Yunhe Sheng. Linearization of the higher analogue of Courant algebroids. Journal of Geometric Mechanics, 2020, 12 (4) : 585-606. doi: 10.3934/jgm.2020025
[13]	Benoît Jubin, Norbert Poncin, Kyosuke Uchino. Free Courant and derived Leibniz pseudoalgebras. Journal of Geometric Mechanics, 2016, 8 (1) : 71-97. doi: 10.3934/jgm.2016.8.71
[14]	Hayato Chiba, Georgi S. Medvedev. The mean field analysis of the kuramoto model on graphs Ⅱ. asymptotic stability of the incoherent state, center manifold reduction, and bifurcations. Discrete and Continuous Dynamical Systems, 2019, 39 (7) : 3897-3921. doi: 10.3934/dcds.2019157
[15]	Nikolay A. Gusev. On the one-dimensional continuity equation with a nearly incompressible vector field. Communications on Pure and Applied Analysis, 2019, 18 (2) : 559-568. doi: 10.3934/cpaa.2019028
[16]	Tomasz Kaczynski, Marian Mrozek, Thomas Wanner. Towards a formal tie between combinatorial and classical vector field dynamics. Journal of Computational Dynamics, 2016, 3 (1) : 17-50. doi: 10.3934/jcd.2016002
[17]	Angela Aguglia, Antonio Cossidente, Giuseppe Marino, Francesco Pavese, Alessandro Siciliano. Orbit codes from forms on vector spaces over a finite field. Advances in Mathematics of Communications, 2022, 16 (1) : 135-155. doi: 10.3934/amc.2020105
[18]	Sobhan Seyfaddini. Spectral killers and Poisson bracket invariants. Journal of Modern Dynamics, 2015, 9: 51-66. doi: 10.3934/jmd.2015.9.51
[19]	Juan Carlos Marrero, David Martín de Diego, Eduardo Martínez. Local convexity for second order differential equations on a Lie algebroid. Journal of Geometric Mechanics, 2021, 13 (3) : 477-499. doi: 10.3934/jgm.2021021
[20]	Frol Zapolsky. Quasi-states and the Poisson bracket on surfaces. Journal of Modern Dynamics, 2007, 1 (3) : 465-475. doi: 10.3934/jmd.2007.1.465

2021 Impact Factor: 0.737

Tools

Metrics

Other articles
by authors

on AIMS
Janusz Grabowski David Khudaverdyan Norbert Poncin
on Google Scholar
Janusz Grabowski David Khudaverdyan Norbert Poncin

[Back to Top]