The supergeometry of Loday algebroids

Janusz Grabowski; David Khudaverdyan; Norbert Poncin

doi:10.3934/jgm.2013.5.185

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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

Abstract
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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. Google Scholar
[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. Google Scholar
[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. 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), 631-661. doi: 10.2307/2001258. Google Scholar
[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. Google Scholar
[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. Google Scholar
[8]	S. Eilenberg and S. Mac Lane, On the groups of $H(\Pi,n)$. I, Ann. of Math., 58 (1953), 55-106. Google Scholar
[9]	V. T. Filippov, $n$-Lie algebras, Sibirsk. Math. Zh., 26 (1985), 126-140, 191. 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), Art. ID G100801, 10 pp. doi: 10.4303/jglta/G100801. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[16]	J. Grabowski, Quasi-derivations and QD-algebroids, Rep. Math. Phys., 32 (2003), 445-451. 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), 27-28. 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), 013520, 17 pp. 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), 3803-3809. doi: 10.1088/0305-4470/34/18/308. Google Scholar
[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. Google Scholar
[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. Google Scholar
[23]	J. Grabowski and N. Poncin, Automorphisms of quantum and classical Poisson algebras, Compositio Math., 140 (2004), 511-527. doi: 10.1112/S0010437X0300006X. Google Scholar
[24]	J. Grabowski and P. Urbański, Algebroids-general differential calculi on vector bundles, J. Geom. Phys., 31 (1999), 111-141. 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), I, 1935-1937; II, 2289-2291. Google Scholar
[26]	Y. Hagiwara, Nambu-Dirac manifolds, J. Phys. A, 35 (2002), 1263-1281. 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), 151-165. 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, Math. Gen., 32 (1999), 8129-8144. doi: 10.1088/0305-4470/32/46/310. Google Scholar
[29]	N. Jacobson, On pseudo-linear transformations, Proc. Nat. Acad. Sci., 21 (1935), 667-670. Google Scholar
[30]	N. Jacobson, Pseudo-linear transformations, Ann. Math., 38 (1937), 485-506. Google Scholar
[31]	D. Khudaverdian, A. Mandal and N. Poncin, Higher categorified algebras versus bounded homotopy algebras, Theo. Appl. Cat., 25 (2011), 251-275. Google Scholar
[32]	A. A. Kirillov, Local Lie algebras, (Russian) Uspekhi Mat. Nauk, 31 (1976), 57-76. Google Scholar
[33]	Y. Kosmann-Schwarzbach, From Poisson algebras to Gerstenhaber algebras, Ann. Inst. Fourier (Grenoble), 46 (1996), 1243-1274. doi: 10.5802/aif.1547. Google Scholar
[34]	Y. Kosmann-Schwarzbach, Derived brackets, Lett. Math. Phys., 69 (2004), 61-87. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[38]	A. Lichnerowicz, Algèbre de Lie des automorphismes infinitésimaux d'une structure unimodulaire, Ann. Inst. Fourier, 24 (1974), 219-266. Google Scholar
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[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. Google Scholar
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[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[49]	E. Nelson, "Tensor Analysis," Princeton University Press, The University of Tokyo Press, Princeton, 1967. Google Scholar
[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. Google Scholar
[51]	J. Peetre, Une caractérisation abstraite des opérateurs différentiels, Math. Scand., 7 (1959), 211-218. Google Scholar
[52]	J. Peetre, Réctifications á l'article "Une caractérisation abstraite des opératuers diffŕentiels," Math. Scand., 8 (1960), 116-120. 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, Sér. A-B, 264 (1967), A245-A248. Google Scholar
[54]	D. E. Radford, A natural ring basis for the shuffle algebra and an application to group schemes, J. Algebra, 58 (1979), 432-454. 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), 210-220. doi: 10.2307/1970243. Google Scholar
[56]	R. Ree, Generalized Lie elements, Canad. J. Math., 12 (1960), 493-502. doi: 10.4153/CJM-1960-044-x. Google Scholar
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[62]	A. M. Vinogradov, The logic algebra for the theory of linear differential operators, Soviet. Mat. Dokl., 13 (1972), 1058-1062. Google Scholar
[63]	A. Wade, Conformal Dirac structures, Lett. Math. Phys., 53 (2000), 331-348. doi: 10.1023/A,1007634407701. Google Scholar
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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. Google Scholar
[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. Google Scholar
[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. 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), 631-661. doi: 10.2307/2001258. Google Scholar
[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. Google Scholar
[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. Google Scholar
[8]	S. Eilenberg and S. Mac Lane, On the groups of $H(\Pi,n)$. I, Ann. of Math., 58 (1953), 55-106. Google Scholar
[9]	V. T. Filippov, $n$-Lie algebras, Sibirsk. Math. Zh., 26 (1985), 126-140, 191. 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), Art. ID G100801, 10 pp. doi: 10.4303/jglta/G100801. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[16]	J. Grabowski, Quasi-derivations and QD-algebroids, Rep. Math. Phys., 32 (2003), 445-451. 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), 27-28. 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), 013520, 17 pp. 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), 3803-3809. doi: 10.1088/0305-4470/34/18/308. Google Scholar
[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. Google Scholar
[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. Google Scholar
[23]	J. Grabowski and N. Poncin, Automorphisms of quantum and classical Poisson algebras, Compositio Math., 140 (2004), 511-527. doi: 10.1112/S0010437X0300006X. Google Scholar
[24]	J. Grabowski and P. Urbański, Algebroids-general differential calculi on vector bundles, J. Geom. Phys., 31 (1999), 111-141. 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), I, 1935-1937; II, 2289-2291. Google Scholar
[26]	Y. Hagiwara, Nambu-Dirac manifolds, J. Phys. A, 35 (2002), 1263-1281. 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), 151-165. 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, Math. Gen., 32 (1999), 8129-8144. doi: 10.1088/0305-4470/32/46/310. Google Scholar
[29]	N. Jacobson, On pseudo-linear transformations, Proc. Nat. Acad. Sci., 21 (1935), 667-670. Google Scholar
[30]	N. Jacobson, Pseudo-linear transformations, Ann. Math., 38 (1937), 485-506. Google Scholar
[31]	D. Khudaverdian, A. Mandal and N. Poncin, Higher categorified algebras versus bounded homotopy algebras, Theo. Appl. Cat., 25 (2011), 251-275. Google Scholar
[32]	A. A. Kirillov, Local Lie algebras, (Russian) Uspekhi Mat. Nauk, 31 (1976), 57-76. Google Scholar
[33]	Y. Kosmann-Schwarzbach, From Poisson algebras to Gerstenhaber algebras, Ann. Inst. Fourier (Grenoble), 46 (1996), 1243-1274. doi: 10.5802/aif.1547. Google Scholar
[34]	Y. Kosmann-Schwarzbach, Derived brackets, Lett. Math. Phys., 69 (2004), 61-87. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[38]	A. Lichnerowicz, Algèbre de Lie des automorphismes infinitésimaux d'une structure unimodulaire, Ann. Inst. Fourier, 24 (1974), 219-266. Google Scholar
[39]	Zhang-Ju Liu, A. Weinstein and Ping Xu, Manin triples for Lie bialgebroids, J. Diff. Geom., 45 (1997), 547-574. Google Scholar
[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. Google Scholar
[41]	J.-L. Loday, Une version non commutative des algèbres de Lie, les algèbres de Leibniz, Ann. Inst. Fourier, 37 (1993), 269-293. Google Scholar
[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. Google Scholar
[43]	J. M. Lodder, Leibniz cohomology for differentiable manifolds, Ann. Inst. Fourier (Grenoble), 48 (1998), 73-75. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[49]	E. Nelson, "Tensor Analysis," Princeton University Press, The University of Tokyo Press, Princeton, 1967. Google Scholar
[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. Google Scholar
[51]	J. Peetre, Une caractérisation abstraite des opérateurs différentiels, Math. Scand., 7 (1959), 211-218. Google Scholar
[52]	J. Peetre, Réctifications á l'article "Une caractérisation abstraite des opératuers diffŕentiels," Math. Scand., 8 (1960), 116-120. 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, Sér. A-B, 264 (1967), A245-A248. Google Scholar
[54]	D. E. Radford, A natural ring basis for the shuffle algebra and an application to group schemes, J. Algebra, 58 (1979), 432-454. 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), 210-220. doi: 10.2307/1970243. Google Scholar
[56]	R. Ree, Generalized Lie elements, Canad. J. Math., 12 (1960), 493-502. doi: 10.4153/CJM-1960-044-x. Google Scholar
[57]	D. Roytenberg, "Courant Algebroids, Derived Brackets and Even Symplectic Supermanifolds," Ph.D. thesis, University of California, Berkeley, 1999. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[61]	K. Uchino, Remarks on the definition of a Courant algebroid, Lett. Math. Phys., 60 (2002), 171-175. doi: 10.1023/A,1016179410273. Google Scholar
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The supergeometry of Loday algebroids

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The supergeometry of Loday algebroids

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