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Extensions and crossed modules of \(n\)-Lie-Rinehart algebras. (English) Zbl 1501.17003

Summary: We introduce a notion of \(n\)-Lie-Rinehart algebras as a generalization of Lie-Rinehart algebras to \(n\)-ary case. This notion is also an algebraic analogue of \(n\)-Lie algebroids. We develop representation theory and describe a cohomology complex of \(n\)-Lie-Rinehart algebras. Furthermore, we investigate extension theory of \(n\)-Lie-Rinehart algebras by means of 2-cocycles. Finally, we introduce crossed modules of \(n\)-Lie-Rinehart algebras to gain a better understanding of their third cohomology groups.

MSC:

17A42 Other \(n\)-ary compositions \((n \ge 3)\)
17A30 Nonassociative algebras satisfying other identities
53D17 Poisson manifolds; Poisson groupoids and algebroids
17A32 Leibniz algebras
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[1] Alekseevsky, D.; Guha, P., On decomposability of Nambu-Poisson tensor, Acta Math. Univ. Comenian. (N.S.), 65, 1, 1-9 (1996) · Zbl 0864.70012
[2] Ammar, F., Mabrouk, S., Makhlouf, A.: Constructions of quadratic \(n\)-ary Hom-Nambu algebras. In: Algebra, geometry and mathematical physics, Springer Proc. Math. Stat., vol. 85, pp. 201-232. Springer, Heidelberg (2014). doi:10.1007/978-3-642-55361-5_12 · Zbl 1345.17005
[3] Bai, R., Bai, C., Wang, J.: Realizations of 3-Lie algebras. J. Math. Phys. 51(6),(2010). doi:10.1063/1.3436555 · Zbl 1311.17002
[4] Bai, R.; Li, Y., \(T^*_\theta \)-extensions of \(n\)-Lie algebras, ISRN Algebra, 11, 381875 (2011) · Zbl 1292.17005 · doi:10.5402/2011/381875
[5] Bai, R.; Li, Y.; Wi, W., Extensions of \(n\)-Lie algebras, Sci. Sin. Math., 7, 4, 689-698 (2012) · Zbl 1488.17024 · doi:10.1360/012011-369
[6] Bai, R.; Song, G.; Zhang, Y., On classification of \(n\)-Lie algebras, Front. Math. China, 6, 4, 581-606 (2011) · Zbl 1282.17003 · doi:10.1007/s11464-011-0107-z
[7] Bajo, I.; Benayadi, S.; Medina, A., Symplectic structures on quadratic Lie algebras, J. Algebra, 316, 1, 174-188 (2007) · Zbl 1124.17005 · doi:10.1016/j.jalgebra.2007.06.001
[8] Ben Hassine, A., Chtioui, T., Elhamdadi, M., Mabrouk, S.: Cohomology and deformations of left-symmetric Rinehart algebras (2020). arXiv:2010.00335
[9] Ben Hassine, A.; Chtioui, T.; Mabrouk, S.; Silvestrov, S., Structure and cohomology of 3-Lie-Rinehart superalgebras, Comm. Algebra, 49, 11, 4883-4904 (2021) · Zbl 1487.17008 · doi:10.1080/00927872.2021.1931266
[10] Bkouche, R., Structures \((K,\, A)\)-linéaires, C. R. Acad. Sci. Paris Sér. A B, 262, 5 (1966) · Zbl 0139.25801
[11] Bordemann, M., Nondegenerate invariant bilinear forms on nonassociative algebras, Acta Math. Univ. Comenian. (N.S.), 66, 2, 151-201 (1997) · Zbl 1014.17003
[12] Casas, JM, Obstructions to Lie-Rinehart algebra extensions, Algebra Colloq., 18, 1, 83-104 (2011) · Zbl 1237.17020 · doi:10.1142/S1005386711000046
[13] Casas, JM; García-Martínez, X., Abelian extensions and crossed modules of Hom-Lie algebras, J. Pure Appl. Algebra, 224, 3, 987-1008 (2020) · Zbl 1436.17028 · doi:10.1016/j.jpaa.2019.06.018
[14] Casas, JM; Khmaladze, E.; Ladra, M., Crossed modules for Leibniz \(n\)-algebras, Forum Math., 20, 5, 841-858 (2008) · Zbl 1236.17005 · doi:10.1515/FORUM.2008.040
[15] Casas, JM; Ladra, M.; Pirashvili, T., Crossed modules for Lie-Rinehart algebras, J. Algebra, 274, 5 (2004) · Zbl 1046.17006 · doi:10.1016/j.jalgebra.2003.10.001
[16] Casas, JM; Ladra, M.; Pirashvili, T., Triple cohomology of Lie-Rinehart algebras and the canonical class of associative algebras, J. Algebra, 291, 1, 144-163 (2005) · Zbl 1151.17308 · doi:10.1016/j.jalgebra.2005.05.018
[17] Chebotar, MA; Ke, WF, On skew-symmetric maps on Lie algebras, Proc. R. Soc. Edinb. Sect. A, 133, 6 (2003) · Zbl 1083.17012 · doi:10.1017/S0308210500002924
[18] Chemla, S., Operations for modules on Lie-Rinehart superalgebras, Manuscr. Math., 87, 2, 199-223 (1995) · Zbl 0999.17009 · doi:10.1007/BF02570471
[19] Chen, Z.; Liu, Z.; Zhong, D., Lie-Rinehart bialgebras for crossed products, J. Pure Appl. Algebra, 215, 6, 1270-1283 (2011) · Zbl 1217.17014 · doi:10.1016/j.jpaa.2010.08.011
[20] Daletskii, YL; Takhtajan, LA, Leibniz and Lie algebra structures for Nambu algebra, Lett. Math. Phys., 39, 2, 127-141 (1997) · Zbl 0869.58024 · doi:10.1023/A:1007316732705
[21] Das, A.: Crossed extensions of lie algebras (2018). arXiv:1812.10680 · Zbl 1486.17032
[22] Dokas, I., Cohomology of restricted Lie-Rinehart algebras and the Brauer group, Adv. Math., 231, 5, 2573-2592 (2012) · Zbl 1335.17009 · doi:10.1016/j.aim.2012.08.003
[23] Figueroa-O’Farrill, JM, Deformations of 3-algebras, J. Math. Phys., 50, 11, 113514 (2009) · Zbl 1304.17005 · doi:10.1063/1.3262528
[24] Filippov, VT, \(n\)-Lie algebras, Sibirsk. Mat. Zh., 26, 6, 126-140 (1985) · Zbl 0585.17002
[25] Gautheron, P., Simple facts concerning Nambu algebras, Comm. Math. Phys., 195, 2, 417-434 (1998) · Zbl 0931.37031 · doi:10.1007/s002200050396
[26] Grabowski, J.; Marmo, G., On Filippov algebroids and multiplicative Nambu-Poisson structures, Differ. Geom. Appl., 12, 1, 35-50 (2000) · Zbl 1026.17006 · doi:10.1016/S0926-2245(99)00042-X
[27] Guo, S., Zhang, X., Wang, S.: On split regular Hom-Leibniz-Rinehart algebras (2020). arXiv:2002.06017 · Zbl 1510.17030
[28] Herz, JC, Pseudo-algèbres de Lie, I. C. R. Acad. Sci. Paris, 236, 1935-1937 (1953) · Zbl 0050.03201
[29] Higgins, PJ; Mackenzie, K., Algebraic constructions in the category of Lie algebroids, J. Algebra, 129, 1, 194-230 (1990) · Zbl 0696.22007 · doi:10.1016/0021-8693(90)90246-K
[30] Huebschmann, J., Poisson cohomology and quantization, J. Reine Angew. Math., 408, 57-113 (1990) · Zbl 0699.53037 · doi:10.1515/crll.1990.408.57
[31] Huebschmann, J., Duality for Lie-Rinehart algebras and the modular class, J. Reine Angew. Math., 510, 103-159 (1999) · Zbl 1034.53083 · doi:10.1515/crll.1999.043
[32] Huebschmann, J.: Lie-Rinehart algebras, descent, and quantization. In: Galois theory, Hopf algebras, and semiabelian categories, Fields Inst. Commun., vol. 43, pp. 295-316. Amer. Math. Soc., Providence, RI (2004). doi:10.1090/fic/043 · Zbl 1096.17006
[33] Kasymov, SM, On a theory of \(n\)-Lie algebras, Algebra i Logika, 26, 3, 277-297 (1987) · Zbl 0647.17001
[34] Krähmer, U.; Rovi, A., A Lie-Rinehart algebra with no antipode, Comm. Algebra, 43, 10, 4049-4053 (2015) · Zbl 1381.17010 · doi:10.1080/00927872.2014.896375
[35] Lin, J.; Wang, Y.; Deng, S., \(T^*\)-extension of Lie triple systems, Linear Algebra Appl., 431, 11, 2071-2083 (2009) · Zbl 1181.17004 · doi:10.1016/j.laa.2009.07.001
[36] Liu, J.; Sheng, Y.; Bai, C., Left-symmetric bialgebroids and their corresponding Manin triples, Differ. Geom. Appl., 59, 91-111 (2018) · Zbl 1431.53089 · doi:10.1016/j.difgeo.2018.04.003
[37] Liu, J.; Sheng, Y.; Bai, C., Pre-symplectic algebroids and their applications, Lett. Math. Phys., 108, 3, 779-804 (2018) · Zbl 1390.17029 · doi:10.1007/s11005-017-0973-8
[38] Liu, J.; Sheng, Y.; Bai, C.; Chen, Z., Left-symmetric algebroids, Math. Nachr., 289, 14-15, 1893-1908 (2016) · Zbl 1354.18003 · doi:10.1002/mana.201300339
[39] Liu, W.; Zhang, Z., \(T^*\)-extension of a 3-Lie algebra, Linear Multilinear Algebra, 60, 5, 538-594 (2012) · Zbl 1266.17001 · doi:10.1080/03081087.2011.616202
[40] Liu, W.; Zhang, Z., \(T^*\)-extension of \(n\)-Lie algebras, Linear Multilinear Algebra, 61, 4, 527-542 (2013) · Zbl 1285.17004 · doi:10.1080/03081087.2012.693922
[41] Mackenzie, K.: Lie groupoids and Lie algebroids in differential geometry, London Mathematical Society Lecture Note Series, vol. 124. Cambridge University Press, Cambridge (1987). doi:10.1017/CBO9780511661839 · Zbl 0683.53029
[42] Mackenzie, KCH, Lie algebroids and Lie pseudoalgebras, Bull. Lond. Math. Soc., 27, 2, 97-147 (1995) · Zbl 0829.22001 · doi:10.1112/blms/27.2.97
[43] Makhlouf, A.: On deformations of \(n\)-Lie algebras. In: Non-associative and non-commutative algebra and operator theory, Springer Proc. Math. Stat., vol. 160, pp. 55-81. Springer, Cham (2016). doi:10.1007/978-3-319-32902-4_4 · Zbl 1419.17007
[44] Marmo, G.; Vilasi, G.; Vinogradov, AM, The local structure of \(n\)-Poisson and \(n\)-Jacobi manifolds, J. Geom. Phys., 25, 1-2, 141-182 (1998) · Zbl 0978.53126 · doi:10.1016/S0393-0440(97)00057-0
[45] Medina, A., Revoy, P.: Algèbres de Lie et produit scalaire invariant. Ann. Sci. École Norm. Sup. (4) 18(3), 553-561 (1985). http://www.numdam.org/item?id=ASENS_1985_4_18_3_553_0 · Zbl 0592.17006
[46] Michor, P.W., Vinogradov, A.M.: \(n\)-ary Lie and associative algebras. Rend. Sem. Mat. Univ. Politec. Torino 54(4), 373-392 (1996). Special issue dedicated to the conference on Geometrical Structures for Physical Theories, II (Vietri, 1996) · Zbl 0928.17029
[47] Mishra, S.K., Mukherjee, G., Naolekar, A.: Cohomology and deformations of Filippov algebroids (2019). arXiv:1912.13193 · Zbl 1503.17009
[48] Nakanishi, N., On Nambu-Poisson manifolds, Rev. Math. Phys., 10, 4, 499-510 (1998) · Zbl 0929.70015 · doi:10.1142/S0129055X98000161
[49] Nambu, Y., Generalized Hamiltonian dynamics, Phys. Rev. D, 3, 7, 2405-2412 (1973) · Zbl 1027.70503 · doi:10.1103/PhysRevD.7.2405
[50] Papadopoulos, G., M2-branes, 3-Lie algebras and Plücker relations, J. High Energy Phys., 5, 054 (2008) · doi:10.1088/1126-6708/2008/05/054
[51] Rotkiewicz, M., Cohomology ring of \(n\)-Lie algebras, Extracta Math., 20, 3, 219-232 (2005) · Zbl 1163.17300
[52] Sheng, Y., On deformations of Lie algebroids, Results Math., 62, 1-2, 103-120 (2012) · Zbl 1318.53094 · doi:10.1007/s00025-011-0133-x
[53] Sheng, Y.; Zhu, C., Higher extensions of Lie algebroids, Commun. Contemp. Math., 19, 3, 1650034 (2017) · Zbl 1419.58015 · doi:10.1142/S0219199716500346
[54] Takhtajan, L.: On foundation of the generalized Nambu mechanics. Comm. Math. Phys. 160(2), 295-315 (1994). http://projecteuclid.org/euclid.cmp/1104269612 · Zbl 0808.70015
[55] Takhtajan, LA, Higher order analog of Chevalley-Eilenberg complex and deformation theory of \(n\)-gebras, Algebra i Analiz, 6, 2, 262-272 (1994) · Zbl 0833.17021
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