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

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

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.
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 $n$-algebroidsarXiv: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 $n$-algebroidsarXiv: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]

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