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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

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.  Google Scholar

[2]

A. Alekseev and P. Xu, Derived brackets and Courant algebroids,, unpublished manuscript., ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

G. Di Brino, D. Pištalo and N. Poncin, Model structure on differential graded commutative algebras over the ring of differential operators,, preprint, ().   Google Scholar

[6]

T. J. Courant, Dirac manifolds, Trans. Amer. Math. Soc., 319 (1990), 631-661. doi: 10.1090/S0002-9947-1990-0998124-1.  Google Scholar

[7]

T. Courant and A. Weinstein, Beyond Poisson structures, Hermann, Travaux en cours, 27 (1988), 39-49.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

V. Drinfeld, Quantum groups, in Proceedings of the International Congress of Mathematicians (Berkeley), Amer. Math. Soc., 1 (1987), 798-820.  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]

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

J. Grabowski, Quasi-derivations and QD-algebroids, Rep. Math. Phys., 52 (2003), 445-451. doi: 10.1016/S0034-4877(03)80041-1.  Google Scholar

[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.  Google Scholar

[18]

Y. Hagiwara, Nambu-Dirac manifolds, J. Phys. A: Math. Gen., 35 (2002), 1263-1281. doi: 10.1088/0305-4470/35/5/310.  Google Scholar

[19]

Y. Hagiwara and T. Mizutani, Leibniz algebras associated with foliations, Kodai Math. J., 25 (2002), 151-165. doi: 10.2996/kmj/1071674438.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

D. Khudaverdian, A. Mandal and N. Poncin, Higher categorified algebras versus bounded homotopy algebras, Theo. Appl. Cat., 25 (2011), 251-275.  Google Scholar

[23]

Y. Kosmann-Schwarzbach, From Poisson algebras to Gerstenhaber algebras, Ann. Inst. Fourier, 46 (1996), 1243-1274. doi: 10.5802/aif.1547.  Google Scholar

[24]

Y. Kosmann-Schwarzbach, Derived brackets, Lett. Math. Phys., 69 (2004), 61-87. doi: 10.1007/s11005-004-0608-8.  Google Scholar

[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.  Google Scholar

[26]

D. Khudaverdian, N. Poncin and J. Qiu, On the infinity category of homotopy Leibniz algebras, Theo. Appl. Cat., 29 (2014), 332-370.  Google Scholar

[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.  Google Scholar

[28]

J.-L. Loday, Une version non commutative des algèbres de Lie: les algèbres de Leibniz, Ann. Inst. Fourier, 39 (1993), 269-293.  Google Scholar

[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.  Google Scholar

[30]

Z.-J. Liu, A. Weinstein and P. Xu, Manin triples for Lie bialgebroids, J. Diff. Geo., 45 (1997), 547-574.  Google Scholar

[31]

K. Mikami and T. Mizutani, Algebroids associated with pre-Poisson structures, Noncommutative Geometry and Physics, (2005), 71-96. doi: 10.1142/9789812779649_0004.  Google Scholar

[32]

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

[33]

D. Roytenberg, On the structure of graded symplectic supermanifolds and Courant algebroids, Contemp. Math., 315 (2002), 169-185.  Google Scholar

[34]

D. Roytenberg, Courant-Dorfman algebras and their cohomology, Lett. Math. Phys., 90 (2009), 311-351. doi: 10.1007/s11005-009-0342-3.  Google Scholar

[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.  Google Scholar

[36]

K. Uchino, Remarks on the definition of a Courant algebroid, Lett. Math. Phys., 60 (2002), 171-175. doi: 10.1023/A:1016179410273.  Google Scholar

[37]

T. Voronov, Higher derived brackets for arbitrary derivations, Trav. Math., XVI (2005), 163-186.  Google Scholar

[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.  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), 355-387. doi: 10.5802/aif.2525.  Google Scholar

[2]

A. Alekseev and P. Xu, Derived brackets and Courant algebroids,, unpublished manuscript., ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

G. Di Brino, D. Pištalo and N. Poncin, Model structure on differential graded commutative algebras over the ring of differential operators,, preprint, ().   Google Scholar

[6]

T. J. Courant, Dirac manifolds, Trans. Amer. Math. Soc., 319 (1990), 631-661. doi: 10.1090/S0002-9947-1990-0998124-1.  Google Scholar

[7]

T. Courant and A. Weinstein, Beyond Poisson structures, Hermann, Travaux en cours, 27 (1988), 39-49.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

V. Drinfeld, Quantum groups, in Proceedings of the International Congress of Mathematicians (Berkeley), Amer. Math. Soc., 1 (1987), 798-820.  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]

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

J. Grabowski, Quasi-derivations and QD-algebroids, Rep. Math. Phys., 52 (2003), 445-451. doi: 10.1016/S0034-4877(03)80041-1.  Google Scholar

[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.  Google Scholar

[18]

Y. Hagiwara, Nambu-Dirac manifolds, J. Phys. A: Math. Gen., 35 (2002), 1263-1281. doi: 10.1088/0305-4470/35/5/310.  Google Scholar

[19]

Y. Hagiwara and T. Mizutani, Leibniz algebras associated with foliations, Kodai Math. J., 25 (2002), 151-165. doi: 10.2996/kmj/1071674438.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

D. Khudaverdian, A. Mandal and N. Poncin, Higher categorified algebras versus bounded homotopy algebras, Theo. Appl. Cat., 25 (2011), 251-275.  Google Scholar

[23]

Y. Kosmann-Schwarzbach, From Poisson algebras to Gerstenhaber algebras, Ann. Inst. Fourier, 46 (1996), 1243-1274. doi: 10.5802/aif.1547.  Google Scholar

[24]

Y. Kosmann-Schwarzbach, Derived brackets, Lett. Math. Phys., 69 (2004), 61-87. doi: 10.1007/s11005-004-0608-8.  Google Scholar

[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.  Google Scholar

[26]

D. Khudaverdian, N. Poncin and J. Qiu, On the infinity category of homotopy Leibniz algebras, Theo. Appl. Cat., 29 (2014), 332-370.  Google Scholar

[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.  Google Scholar

[28]

J.-L. Loday, Une version non commutative des algèbres de Lie: les algèbres de Leibniz, Ann. Inst. Fourier, 39 (1993), 269-293.  Google Scholar

[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.  Google Scholar

[30]

Z.-J. Liu, A. Weinstein and P. Xu, Manin triples for Lie bialgebroids, J. Diff. Geo., 45 (1997), 547-574.  Google Scholar

[31]

K. Mikami and T. Mizutani, Algebroids associated with pre-Poisson structures, Noncommutative Geometry and Physics, (2005), 71-96. doi: 10.1142/9789812779649_0004.  Google Scholar

[32]

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

[33]

D. Roytenberg, On the structure of graded symplectic supermanifolds and Courant algebroids, Contemp. Math., 315 (2002), 169-185.  Google Scholar

[34]

D. Roytenberg, Courant-Dorfman algebras and their cohomology, Lett. Math. Phys., 90 (2009), 311-351. doi: 10.1007/s11005-009-0342-3.  Google Scholar

[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.  Google Scholar

[36]

K. Uchino, Remarks on the definition of a Courant algebroid, Lett. Math. Phys., 60 (2002), 171-175. doi: 10.1023/A:1016179410273.  Google Scholar

[37]

T. Voronov, Higher derived brackets for arbitrary derivations, Trav. Math., XVI (2005), 163-186.  Google Scholar

[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.  Google Scholar

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