Free Courant and derived Leibniz pseudoalgebras

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March 2016, 8(1): 71-97. doi: 10.3934/jgm.2016.8.71

Free Courant and derived Leibniz pseudoalgebras

Benoît Jubin ^1, , Norbert Poncin ^2, and Kyosuke Uchino ^1,

1.	University of Luxembourg, Mathematics Research Unit, 6, rue Richard Coudenhove-Kalergi, L-1359 Luxembourg City, Luxembourg, Luxembourg
2.	University of Luxembourg, Campus Kirchberg, Mathematics Research Unit, 6, rue Richard Coudenhove-Kalergi, L-1359 Luxembourg City

Received August 2014 Revised September 2015 Published February 2016

Abstract
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We introduce the category of generalized Courant pseudoalgebras and show that it admits a free object on any anchored module over `functions'. The free generalized Courant pseudoalgebra is built from two components: the generalized Courant pseudoalgebra associated to a symmetric Leibniz pseudoalgebra and the free symmetric Leibniz pseudoalgebra on an anchored module. Our construction is thus based on the new concept of symmetric Leibniz algebroid. We compare this subclass of Leibniz algebroids with the subclass made of Loday algebroids, which were introduced in [12] as geometric replacements of standard Leibniz algebroids. Eventually, we apply our algebro-categorical machinery to associate a differential graded Lie algebra to any symmetric Leibniz pseudoalgebra, such that the Leibniz bracket of the latter coincides with the derived bracket of the former.

Keywords: free object, pseudoalgebra, Loday algebroid, Leibniz algebroid, derived bracket., Courant algebroid.

Mathematics Subject Classification: Primary: 17A32, 53D17; Secondary: 17B0.

Citation: 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

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[2]	A. Alekseev and P. Xu, Derived brackets and Courant algebroids, unpublished manuscript.
[3]	H. Bursztyn, D. Iglesias and P. Ševera, Courant morphisms and moment maps, Math. Res. Lett., 16 (2009), 215-232. doi: 10.4310/MRL.2009.v16.n2.a2.
[4]	G. Bonavolontà and N. Poncin, On the category of Lie $n$-algebroids, J. Geo. Phys., 73 (2013), 70-90. doi: 10.1016/j.geomphys.2013.05.004.
[5]	G. Di Brino, D. Pištalo and N. Poncin, Model structure on differential graded commutative algebras over the ring of differential operators, preprint, arXiv:1505.07720.
[6]	T. J. Courant, Dirac manifolds, Trans. Amer. Math. Soc., 319 (1990), 631-661. doi: 10.1090/S0002-9947-1990-0998124-1.
[7]	T. Courant and A. Weinstein, Beyond Poisson structures, Hermann, Travaux en cours, 27 (1988), 39-49.
[8]	I. Y. Dorfman, Dirac structures of integrable evolution equations, Phys. Lett. A, 125 (1987), 240-246. doi: 10.1016/0375-9601(87)90201-5.
[9]	V. Dotsenko and N. Poncin, A tale of three homotopies, Appl. Cat. Structures, (2015), 1-29, arXiv:1208.4695v2. doi: 10.1007/s10485-015-9407-x.
[10]	V. Drinfeld, Quantum groups, in Proceedings of the International Congress of Mathematicians (Berkeley), Amer. Math. Soc., 1 (1987), 798-820.
[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]	J. Grabowski, D. Khudaverdyan and N. Poncin, The supergeometry of Loday algebroids, J. Geo. Mech., 5 (2013), 185-213. doi: 10.3934/jgm.2013.5.185.
[13]	J. Grabowski and G. Marmo, Non-antisymmetric versions of Nambu-Poisson and Lie algebroid brackets, J. Phys. A: Math. Gen., 34 (2001), 3803-3809. doi: 10.1088/0305-4470/34/18/308.
[14]	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.
[15]	J. Grabowski, Abstract Jacobi and Poisson structures. Quantization and star-products, J. Geo. Phys., 9 (1992), 45-73. doi: 10.1016/0393-0440(92)90025-V.
[16]	J. Grabowski, Quasi-derivations and QD-algebroids, Rep. Math. Phys., 52 (2003), 445-451. doi: 10.1016/S0034-4877(03)80041-1.
[17]	D. García-Beltrán and J. A. Vallejo, An approach to omni-Lie algebroids using quasi-derivations, J. Gen. Lie Theo. Appl., 5 (2011), Art. ID G100801, 10 pp.
[18]	Y. Hagiwara, Nambu-Dirac manifolds, J. Phys. A: Math. Gen., 35 (2002), 1263-1281. doi: 10.1088/0305-4470/35/5/310.
[19]	Y. Hagiwara and T. Mizutani, Leibniz algebras associated with foliations, Kodai Math. J., 25 (2002), 151-165. doi: 10.2996/kmj/1071674438.
[20]	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.
[21]	M. Kapranov, Free Lie algebroids and the space of paths, Selecta Mathematica, 13 (2007), p277, arXiv:math/0702584. doi: 10.1007/s00029-007-0041-9.
[22]	D. Khudaverdian, A. Mandal and N. Poncin, Higher categorified algebras versus bounded homotopy algebras, Theo. Appl. Cat., 25 (2011), 251-275.
[23]	Y. Kosmann-Schwarzbach, From Poisson algebras to Gerstenhaber algebras, Ann. Inst. Fourier, 46 (1996), 1243-1274. doi: 10.5802/aif.1547.
[24]	Y. Kosmann-Schwarzbach, Derived brackets, Lett. Math. Phys., 69 (2004), 61-87. doi: 10.1007/s11005-004-0608-8.
[25]	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 (2005), 363-389. doi: 10.1007/0-8176-4419-9_12.
[26]	D. Khudaverdian, N. Poncin and J. Qiu, On the infinity category of homotopy Leibniz algebras, Theo. Appl. Cat., 29 (2014), 332-370.
[27]	A. Kotov and T. Strobl, Generalizing geometry - algebroids and sigma models, in Handbook of Pseudo-Riemannian Geometry and Supersymmetry, IRMA Lect. Math. Theo. Phys., {16}, 2010, 209-262. doi: 10.4171/079-1/7.
[28]	J.-L. Loday, Une version non commutative des algèbres de Lie: les algèbres de Leibniz, Ann. Inst. Fourier, 39 (1993), 269-293.
[29]	J.-L. Loday and T. Pirashvili, Universal enveloping algebras of Leibniz algebras and (co)homology, Math. Ann., 296 (1993), 139-158. doi: 10.1007/BF01445099.
[30]	Z.-J. Liu, A. Weinstein and P. Xu, Manin triples for Lie bialgebroids, J. Diff. Geo., 45 (1997), 547-574.
[31]	K. Mikami and T. Mizutani, Algebroids associated with pre-Poisson structures, Noncommutative Geometry and Physics, (2005), 71-96. doi: 10.1142/9789812779649_0004.
[32]	D. Roytenberg, Courant Algebroids, Derived Brackets and Even Symplectic Supermanifolds, Ph.D thesis, Berkeley, 1999.
[33]	D. Roytenberg, On the structure of graded symplectic supermanifolds and Courant algebroids, Contemp. Math., 315 (2002), 169-185.
[34]	D. Roytenberg, Courant-Dorfman algebras and their cohomology, Lett. Math. Phys., 90 (2009), 311-351. doi: 10.1007/s11005-009-0342-3.
[35]	M. Stiénon and P. Xu, Modular classes of Loday algebroids, C. R. Acad. Sci. Paris, Ser. I, 346 (2008), 193-198. doi: 10.1016/j.crma.2007.12.012.
[36]	K. Uchino, Remarks on the definition of a Courant algebroid, Lett. Math. Phys., 60 (2002), 171-175. doi: 10.1023/A:1016179410273.
[37]	T. Voronov, Higher derived brackets for arbitrary derivations, Trav. Math., XVI (2005), 163-186.
[38]	A. Wade, On some properties of Leibniz algebroids, in Infinite Dimensional Lie Groups in Geometry and Representation Theory, World Sci. Publ., River Edge, NJ, 2002, 65-78. doi: 10.1142/9789812777089_0005.

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]	A. Alekseev and P. Xu, Derived brackets and Courant algebroids, unpublished manuscript.
[3]	H. Bursztyn, D. Iglesias and P. Ševera, Courant morphisms and moment maps, Math. Res. Lett., 16 (2009), 215-232. doi: 10.4310/MRL.2009.v16.n2.a2.
[4]	G. Bonavolontà and N. Poncin, On the category of Lie $n$-algebroids, J. Geo. Phys., 73 (2013), 70-90. doi: 10.1016/j.geomphys.2013.05.004.
[5]	G. Di Brino, D. Pištalo and N. Poncin, Model structure on differential graded commutative algebras over the ring of differential operators, preprint, arXiv:1505.07720.
[6]	T. J. Courant, Dirac manifolds, Trans. Amer. Math. Soc., 319 (1990), 631-661. doi: 10.1090/S0002-9947-1990-0998124-1.
[7]	T. Courant and A. Weinstein, Beyond Poisson structures, Hermann, Travaux en cours, 27 (1988), 39-49.
[8]	I. Y. Dorfman, Dirac structures of integrable evolution equations, Phys. Lett. A, 125 (1987), 240-246. doi: 10.1016/0375-9601(87)90201-5.
[9]	V. Dotsenko and N. Poncin, A tale of three homotopies, Appl. Cat. Structures, (2015), 1-29, arXiv:1208.4695v2. doi: 10.1007/s10485-015-9407-x.
[10]	V. Drinfeld, Quantum groups, in Proceedings of the International Congress of Mathematicians (Berkeley), Amer. Math. Soc., 1 (1987), 798-820.
[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]	J. Grabowski, D. Khudaverdyan and N. Poncin, The supergeometry of Loday algebroids, J. Geo. Mech., 5 (2013), 185-213. doi: 10.3934/jgm.2013.5.185.
[13]	J. Grabowski and G. Marmo, Non-antisymmetric versions of Nambu-Poisson and Lie algebroid brackets, J. Phys. A: Math. Gen., 34 (2001), 3803-3809. doi: 10.1088/0305-4470/34/18/308.
[14]	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.
[15]	J. Grabowski, Abstract Jacobi and Poisson structures. Quantization and star-products, J. Geo. Phys., 9 (1992), 45-73. doi: 10.1016/0393-0440(92)90025-V.
[16]	J. Grabowski, Quasi-derivations and QD-algebroids, Rep. Math. Phys., 52 (2003), 445-451. doi: 10.1016/S0034-4877(03)80041-1.
[17]	D. García-Beltrán and J. A. Vallejo, An approach to omni-Lie algebroids using quasi-derivations, J. Gen. Lie Theo. Appl., 5 (2011), Art. ID G100801, 10 pp.
[18]	Y. Hagiwara, Nambu-Dirac manifolds, J. Phys. A: Math. Gen., 35 (2002), 1263-1281. doi: 10.1088/0305-4470/35/5/310.
[19]	Y. Hagiwara and T. Mizutani, Leibniz algebras associated with foliations, Kodai Math. J., 25 (2002), 151-165. doi: 10.2996/kmj/1071674438.
[20]	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.
[21]	M. Kapranov, Free Lie algebroids and the space of paths, Selecta Mathematica, 13 (2007), p277, arXiv:math/0702584. doi: 10.1007/s00029-007-0041-9.
[22]	D. Khudaverdian, A. Mandal and N. Poncin, Higher categorified algebras versus bounded homotopy algebras, Theo. Appl. Cat., 25 (2011), 251-275.
[23]	Y. Kosmann-Schwarzbach, From Poisson algebras to Gerstenhaber algebras, Ann. Inst. Fourier, 46 (1996), 1243-1274. doi: 10.5802/aif.1547.
[24]	Y. Kosmann-Schwarzbach, Derived brackets, Lett. Math. Phys., 69 (2004), 61-87. doi: 10.1007/s11005-004-0608-8.
[25]	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 (2005), 363-389. doi: 10.1007/0-8176-4419-9_12.
[26]	D. Khudaverdian, N. Poncin and J. Qiu, On the infinity category of homotopy Leibniz algebras, Theo. Appl. Cat., 29 (2014), 332-370.
[27]	A. Kotov and T. Strobl, Generalizing geometry - algebroids and sigma models, in Handbook of Pseudo-Riemannian Geometry and Supersymmetry, IRMA Lect. Math. Theo. Phys., {16}, 2010, 209-262. doi: 10.4171/079-1/7.
[28]	J.-L. Loday, Une version non commutative des algèbres de Lie: les algèbres de Leibniz, Ann. Inst. Fourier, 39 (1993), 269-293.
[29]	J.-L. Loday and T. Pirashvili, Universal enveloping algebras of Leibniz algebras and (co)homology, Math. Ann., 296 (1993), 139-158. doi: 10.1007/BF01445099.
[30]	Z.-J. Liu, A. Weinstein and P. Xu, Manin triples for Lie bialgebroids, J. Diff. Geo., 45 (1997), 547-574.
[31]	K. Mikami and T. Mizutani, Algebroids associated with pre-Poisson structures, Noncommutative Geometry and Physics, (2005), 71-96. doi: 10.1142/9789812779649_0004.
[32]	D. Roytenberg, Courant Algebroids, Derived Brackets and Even Symplectic Supermanifolds, Ph.D thesis, Berkeley, 1999.
[33]	D. Roytenberg, On the structure of graded symplectic supermanifolds and Courant algebroids, Contemp. Math., 315 (2002), 169-185.
[34]	D. Roytenberg, Courant-Dorfman algebras and their cohomology, Lett. Math. Phys., 90 (2009), 311-351. doi: 10.1007/s11005-009-0342-3.
[35]	M. Stiénon and P. Xu, Modular classes of Loday algebroids, C. R. Acad. Sci. Paris, Ser. I, 346 (2008), 193-198. doi: 10.1016/j.crma.2007.12.012.
[36]	K. Uchino, Remarks on the definition of a Courant algebroid, Lett. Math. Phys., 60 (2002), 171-175. doi: 10.1023/A:1016179410273.
[37]	T. Voronov, Higher derived brackets for arbitrary derivations, Trav. Math., XVI (2005), 163-186.
[38]	A. Wade, On some properties of Leibniz algebroids, in Infinite Dimensional Lie Groups in Geometry and Representation Theory, World Sci. Publ., River Edge, NJ, 2002, 65-78. doi: 10.1142/9789812777089_0005.

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