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Free Courant and derived Leibniz pseudoalgebras
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 |
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. |
[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A. Alekseev and P. Xu, Derived brackets and Courant algebroids,, unpublished manuscript., ().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.
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.
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, (). Google Scholar |
[6] |
T. J. Courant, Dirac manifolds,, Trans. Amer. Math. Soc., 319 (1990), 631.
doi: 10.1090/S0002-9947-1990-0998124-1. |
[7] |
T. Courant and A. Weinstein, Beyond Poisson structures,, Hermann, 27 (1988), 39.
|
[8] |
I. Y. Dorfman, Dirac structures of integrable evolution equations,, Phys. Lett. A, 125 (1987), 240.
doi: 10.1016/0375-9601(87)90201-5. |
[9] |
V. Dotsenko and N. Poncin, A tale of three homotopies,, Appl. Cat. Structures, (2015), 1.
doi: 10.1007/s10485-015-9407-x. |
[10] |
V. Drinfeld, Quantum groups,, in Proceedings of the International Congress of Mathematicians (Berkeley), 1 (1987), 798.
|
[11] |
V. Ginzburg and M. Kapranov, Koszul duality for operads,, Duke Math. J., 76 (1994), 203.
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.
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.
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.
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.
doi: 10.1016/0393-0440(92)90025-V. |
[16] |
J. Grabowski, Quasi-derivations and QD-algebroids,, Rep. Math. Phys., 52 (2003), 445.
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).
|
[18] |
Y. Hagiwara, Nambu-Dirac manifolds,, J. Phys. A: Math. Gen., 35 (2002), 1263.
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.
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.
doi: 10.1088/0305-4470/32/46/310. |
[21] |
M. Kapranov, Free Lie algebroids and the space of paths,, Selecta Mathematica, 13 (2007).
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.
|
[23] |
Y. Kosmann-Schwarzbach, From Poisson algebras to Gerstenhaber algebras,, Ann. Inst. Fourier, 46 (1996), 1243.
doi: 10.5802/aif.1547. |
[24] |
Y. Kosmann-Schwarzbach, Derived brackets,, Lett. Math. Phys., 69 (2004), 61.
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, 232 (2005), 363.
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.
|
[27] |
A. Kotov and T. Strobl, Generalizing geometry - algebroids and sigma models,, in Handbook of Pseudo-Riemannian Geometry and Supersymmetry, (2010), 209.
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.
|
[29] |
J.-L. Loday and T. Pirashvili, Universal enveloping algebras of Leibniz algebras and (co)homology,, Math. Ann., 296 (1993), 139.
doi: 10.1007/BF01445099. |
[30] |
Z.-J. Liu, A. Weinstein and P. Xu, Manin triples for Lie bialgebroids,, J. Diff. Geo., 45 (1997), 547.
|
[31] |
K. Mikami and T. Mizutani, Algebroids associated with pre-Poisson structures,, Noncommutative Geometry and Physics, (2005), 71.
doi: 10.1142/9789812779649_0004. |
[32] |
D. Roytenberg, Courant Algebroids, Derived Brackets and Even Symplectic Supermanifolds,, Ph.D thesis, (1999).
|
[33] |
D. Roytenberg, On the structure of graded symplectic supermanifolds and Courant algebroids,, Contemp. Math., 315 (2002), 169.
|
[34] |
D. Roytenberg, Courant-Dorfman algebras and their cohomology,, Lett. Math. Phys., 90 (2009), 311.
doi: 10.1007/s11005-009-0342-3. |
[35] |
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. |
[36] |
K. Uchino, Remarks on the definition of a Courant algebroid,, Lett. Math. Phys., 60 (2002), 171.
doi: 10.1023/A:1016179410273. |
[37] |
T. Voronov, Higher derived brackets for arbitrary derivations,, Trav. Math., XVI (2005), 163.
|
[38] |
A. Wade, On some properties of Leibniz algebroids,, in Infinite Dimensional Lie Groups in Geometry and Representation Theory, (2002), 65.
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.
doi: 10.5802/aif.2525. |
[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.
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.
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, (). Google Scholar |
[6] |
T. J. Courant, Dirac manifolds,, Trans. Amer. Math. Soc., 319 (1990), 631.
doi: 10.1090/S0002-9947-1990-0998124-1. |
[7] |
T. Courant and A. Weinstein, Beyond Poisson structures,, Hermann, 27 (1988), 39.
|
[8] |
I. Y. Dorfman, Dirac structures of integrable evolution equations,, Phys. Lett. A, 125 (1987), 240.
doi: 10.1016/0375-9601(87)90201-5. |
[9] |
V. Dotsenko and N. Poncin, A tale of three homotopies,, Appl. Cat. Structures, (2015), 1.
doi: 10.1007/s10485-015-9407-x. |
[10] |
V. Drinfeld, Quantum groups,, in Proceedings of the International Congress of Mathematicians (Berkeley), 1 (1987), 798.
|
[11] |
V. Ginzburg and M. Kapranov, Koszul duality for operads,, Duke Math. J., 76 (1994), 203.
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.
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.
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.
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.
doi: 10.1016/0393-0440(92)90025-V. |
[16] |
J. Grabowski, Quasi-derivations and QD-algebroids,, Rep. Math. Phys., 52 (2003), 445.
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).
|
[18] |
Y. Hagiwara, Nambu-Dirac manifolds,, J. Phys. A: Math. Gen., 35 (2002), 1263.
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.
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.
doi: 10.1088/0305-4470/32/46/310. |
[21] |
M. Kapranov, Free Lie algebroids and the space of paths,, Selecta Mathematica, 13 (2007).
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.
|
[23] |
Y. Kosmann-Schwarzbach, From Poisson algebras to Gerstenhaber algebras,, Ann. Inst. Fourier, 46 (1996), 1243.
doi: 10.5802/aif.1547. |
[24] |
Y. Kosmann-Schwarzbach, Derived brackets,, Lett. Math. Phys., 69 (2004), 61.
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, 232 (2005), 363.
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.
|
[27] |
A. Kotov and T. Strobl, Generalizing geometry - algebroids and sigma models,, in Handbook of Pseudo-Riemannian Geometry and Supersymmetry, (2010), 209.
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.
|
[29] |
J.-L. Loday and T. Pirashvili, Universal enveloping algebras of Leibniz algebras and (co)homology,, Math. Ann., 296 (1993), 139.
doi: 10.1007/BF01445099. |
[30] |
Z.-J. Liu, A. Weinstein and P. Xu, Manin triples for Lie bialgebroids,, J. Diff. Geo., 45 (1997), 547.
|
[31] |
K. Mikami and T. Mizutani, Algebroids associated with pre-Poisson structures,, Noncommutative Geometry and Physics, (2005), 71.
doi: 10.1142/9789812779649_0004. |
[32] |
D. Roytenberg, Courant Algebroids, Derived Brackets and Even Symplectic Supermanifolds,, Ph.D thesis, (1999).
|
[33] |
D. Roytenberg, On the structure of graded symplectic supermanifolds and Courant algebroids,, Contemp. Math., 315 (2002), 169.
|
[34] |
D. Roytenberg, Courant-Dorfman algebras and their cohomology,, Lett. Math. Phys., 90 (2009), 311.
doi: 10.1007/s11005-009-0342-3. |
[35] |
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. |
[36] |
K. Uchino, Remarks on the definition of a Courant algebroid,, Lett. Math. Phys., 60 (2002), 171.
doi: 10.1023/A:1016179410273. |
[37] |
T. Voronov, Higher derived brackets for arbitrary derivations,, Trav. Math., XVI (2005), 163.
|
[38] |
A. Wade, On some properties of Leibniz algebroids,, in Infinite Dimensional Lie Groups in Geometry and Representation Theory, (2002), 65.
doi: 10.1142/9789812777089_0005. |
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