American Institute of Mathematical Sciences

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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

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:
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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.  Google Scholar

[2]

D. Baraglia, Leibniz algebroids, twistings and exceptional generalized geometry,, J. Geom. Phys., 62 (2012), 903.  doi: 10.1016/j.geomphys.2012.01.007.  Google Scholar

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Y. Bi and Y. Sheng, On higher analogues of Courant algebroid,, Sci. China Math., 54 (2011), 437.  doi: 10.1007/s11425-010-4142-0.  Google Scholar

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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.  Google Scholar

[6]

I. Y. Dorfman, Dirac structures of integrable evolution equations,, Phys. Lett. A, 125 (1987), 240.  doi: 10.1016/0375-9601(87)90201-5.  Google Scholar

[7]

V. Drinfel'd, Quantum groups,, in, (1986).   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

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V. Ginzburg and M. Kapranov, Koszul duality for operads,, Duke Math. J., 76 (1994), 203.  doi: 10.1215/S0012-7094-94-07608-4.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

J. Grabowski, Quasi-derivations and QD-algebroids,, Rep. Math. Phys., 32 (2003), 445.  doi: 10.1016/S0034-4877(03)80041-1.  Google Scholar

[17]

J. Grabowski, Graded contact manifolds and contact Courant algebroids,, J. Geom. Phys., 68 (2013), 27.  doi: 10.1016/j.geomphys.2013.02.001.  Google Scholar

[18]

J. Grabowski, Brackets,, to appear in Int. J. Geom. Methods Mod. Phys., ().   Google Scholar

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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.  Google Scholar

[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.  Google Scholar

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J. Grabowski and G. Marmo, Binary operations in classical and quantum mechanics,, in, 59 (2003), 163.  doi: 10.4064/bc59-0-8.  Google Scholar

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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.  Google Scholar

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J. Grabowski and N. Poncin, Automorphisms of quantum and classical Poisson algebras,, Compositio Math., 140 (2004), 511.  doi: 10.1112/S0010437X0300006X.  Google Scholar

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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

[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.  Google Scholar

[27]

Y. Hagiwara and T. Mizutani, Leibniz algebras associated with foliations,, Kodai Math. J., 25 (2002), 151.  doi: 10.2996/kmj/1071674438.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[32]

A. A. Kirillov, Local Lie algebras,, (Russian) Uspekhi Mat. Nauk, 31 (1976), 57.   Google Scholar

[33]

Y. Kosmann-Schwarzbach, From Poisson algebras to Gerstenhaber algebras,, Ann. Inst. Fourier (Grenoble), 46 (1996), 1243.  doi: 10.5802/aif.1547.  Google Scholar

[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.  Google Scholar

[36]

Y. Kosmann-Schwarzbach and K. Mackenzie, Differential operators and actions of Lie algebroids,, in, 315 (2002), 213.  doi: 10.1090/conm/315/05482.  Google Scholar

[37]

A. Kotov and T. Strobl, Generalizing geometry-algebroids and sigma models,, in, 16 (2010), 209.  doi: 10.4171/079-1/7.  Google Scholar

[38]

A. Lichnerowicz, Algèbre de Lie des automorphismes infinitésimaux d'une structure unimodulaire,, Ann. Inst. Fourier, 24 (1974), 219.   Google Scholar

[39]

Zhang-Ju Liu, A. Weinstein and Ping Xu, Manin triples for Lie bialgebroids,, J. Diff. Geom., 45 (1997), 547.   Google Scholar

[40]

J.-L. Loday, "Cyclic Homology,", Appendix E by María O. Ronco, 301 (1992).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[45]

K. C. H. Mackenzie, Lie algebroids and Lie pseudoalgebras,, Bull. London Math. Soc., 27 (1995), 97.  doi: 10.1112/blms/27.2.97.  Google Scholar

[46]

K. C. H. Mackenzie, "General Theory of Lie Groupoids and Lie Algebroids,", London Mathematical Society Lecture Note Series, 213 (2005).   Google Scholar

[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.  Google Scholar

[48]

K. Mikami and T. Mizutani, Algebroids associated with pre-Poisson structures,, in, (2007), 71.  doi: 10.1142/9789812779649_0004.  Google Scholar

[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.  Google Scholar

[51]

J. Peetre, Une caractérisation abstraite des opérateurs différentiels,, Math. Scand., 7 (1959), 211.   Google Scholar

[52]

J. Peetre, Réctifications á l'article "Une caractérisation abstraite des opératuers diffŕentiels,", Math. Scand., 8 (1960), 116.   Google Scholar

[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).   Google Scholar

[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.  Google Scholar

[55]

R. Ree, Lie elements and an algebra associated with shuffles,, Ann. of Math. (2), 68 (1958), 210.  doi: 10.2307/1970243.  Google Scholar

[56]

R. Ree, Generalized Lie elements,, Canad. J. Math., 12 (1960), 493.  doi: 10.4153/CJM-1960-044-x.  Google Scholar

[57]

D. Roytenberg, "Courant Algebroids, Derived Brackets and Even Symplectic Supermanifolds,", Ph.D. thesis, (1999).   Google Scholar

[58]

D. Roytenberg, On the structure of graded symplectic supermanifolds and Courant algebroids,, in, 315 (2002), 169.  doi: 10.1090/conm/315/05479.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[61]

K. Uchino, Remarks on the definition of a Courant algebroid,, Lett. Math. Phys., 60 (2002), 171.  doi: 10.1023/A,1016179410273.  Google Scholar

[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.  Google Scholar

[64]

A. Wade, On some properties of Leibniz algebroids,, in, (2002), 65.  doi: 10.1142/9789812777089_0005.  Google Scholar

[65]

M. Zambon, $L_\infty$-algebras and higher analogues of Dirac structures and Courant algebroids,, J. Symplectic Geom., 10 (2012), 563.   Google Scholar

[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.  Google Scholar

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.  Google Scholar

[2]

D. Baraglia, Leibniz algebroids, twistings and exceptional generalized geometry,, J. Geom. Phys., 62 (2012), 903.  doi: 10.1016/j.geomphys.2012.01.007.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

I. Y. Dorfman, Dirac structures of integrable evolution equations,, Phys. Lett. A, 125 (1987), 240.  doi: 10.1016/0375-9601(87)90201-5.  Google Scholar

[7]

V. Drinfel'd, Quantum groups,, in, (1986).   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[11]

V. Ginzburg and M. Kapranov, Koszul duality for operads,, Duke Math. J., 76 (1994), 203.  doi: 10.1215/S0012-7094-94-07608-4.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

J. Grabowski, Quasi-derivations and QD-algebroids,, Rep. Math. Phys., 32 (2003), 445.  doi: 10.1016/S0034-4877(03)80041-1.  Google Scholar

[17]

J. Grabowski, Graded contact manifolds and contact Courant algebroids,, J. Geom. Phys., 68 (2013), 27.  doi: 10.1016/j.geomphys.2013.02.001.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

J. Grabowski and G. Marmo, Binary operations in classical and quantum mechanics,, in, 59 (2003), 163.  doi: 10.4064/bc59-0-8.  Google Scholar

[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.  Google Scholar

[23]

J. Grabowski and N. Poncin, Automorphisms of quantum and classical Poisson algebras,, Compositio Math., 140 (2004), 511.  doi: 10.1112/S0010437X0300006X.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

Y. Hagiwara and T. Mizutani, Leibniz algebras associated with foliations,, Kodai Math. J., 25 (2002), 151.  doi: 10.2996/kmj/1071674438.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[32]

A. A. Kirillov, Local Lie algebras,, (Russian) Uspekhi Mat. Nauk, 31 (1976), 57.   Google Scholar

[33]

Y. Kosmann-Schwarzbach, From Poisson algebras to Gerstenhaber algebras,, Ann. Inst. Fourier (Grenoble), 46 (1996), 1243.  doi: 10.5802/aif.1547.  Google Scholar

[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.  Google Scholar

[36]

Y. Kosmann-Schwarzbach and K. Mackenzie, Differential operators and actions of Lie algebroids,, in, 315 (2002), 213.  doi: 10.1090/conm/315/05482.  Google Scholar

[37]

A. Kotov and T. Strobl, Generalizing geometry-algebroids and sigma models,, in, 16 (2010), 209.  doi: 10.4171/079-1/7.  Google Scholar

[38]

A. Lichnerowicz, Algèbre de Lie des automorphismes infinitésimaux d'une structure unimodulaire,, Ann. Inst. Fourier, 24 (1974), 219.   Google Scholar

[39]

Zhang-Ju Liu, A. Weinstein and Ping Xu, Manin triples for Lie bialgebroids,, J. Diff. Geom., 45 (1997), 547.   Google Scholar

[40]

J.-L. Loday, "Cyclic Homology,", Appendix E by María O. Ronco, 301 (1992).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[45]

K. C. H. Mackenzie, Lie algebroids and Lie pseudoalgebras,, Bull. London Math. Soc., 27 (1995), 97.  doi: 10.1112/blms/27.2.97.  Google Scholar

[46]

K. C. H. Mackenzie, "General Theory of Lie Groupoids and Lie Algebroids,", London Mathematical Society Lecture Note Series, 213 (2005).   Google Scholar

[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.  Google Scholar

[48]

K. Mikami and T. Mizutani, Algebroids associated with pre-Poisson structures,, in, (2007), 71.  doi: 10.1142/9789812779649_0004.  Google Scholar

[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.  Google Scholar

[51]

J. Peetre, Une caractérisation abstraite des opérateurs différentiels,, Math. Scand., 7 (1959), 211.   Google Scholar

[52]

J. Peetre, Réctifications á l'article "Une caractérisation abstraite des opératuers diffŕentiels,", Math. Scand., 8 (1960), 116.   Google Scholar

[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).   Google Scholar

[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.  Google Scholar

[55]

R. Ree, Lie elements and an algebra associated with shuffles,, Ann. of Math. (2), 68 (1958), 210.  doi: 10.2307/1970243.  Google Scholar

[56]

R. Ree, Generalized Lie elements,, Canad. J. Math., 12 (1960), 493.  doi: 10.4153/CJM-1960-044-x.  Google Scholar

[57]

D. Roytenberg, "Courant Algebroids, Derived Brackets and Even Symplectic Supermanifolds,", Ph.D. thesis, (1999).   Google Scholar

[58]

D. Roytenberg, On the structure of graded symplectic supermanifolds and Courant algebroids,, in, 315 (2002), 169.  doi: 10.1090/conm/315/05479.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[61]

K. Uchino, Remarks on the definition of a Courant algebroid,, Lett. Math. Phys., 60 (2002), 171.  doi: 10.1023/A,1016179410273.  Google Scholar

[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.  Google Scholar

[64]

A. Wade, On some properties of Leibniz algebroids,, in, (2002), 65.  doi: 10.1142/9789812777089_0005.  Google Scholar

[65]

M. Zambon, $L_\infty$-algebras and higher analogues of Dirac structures and Courant algebroids,, J. Symplectic Geom., 10 (2012), 563.   Google Scholar

[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.  Google Scholar

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