Advanced search
Publication Cover
Communications in Algebra Volume 49, 2021 - Issue 11
Submit an article Journal homepage
945
Views
6
CrossRef citations to date
0
Altmetric
Research Article

Structure and cohomology of 3-Lie-Rinehart superalgebras

Abdelkader Ben Hassinea Department of Mathematics, Faculty of Science and Arts at Belqarn, University of Bisha, Sabt Al-Alaya, Kingdom of Saudi Arabia;b Faculty of Sciences, University of Sfax, Sfax, TunisiaORCID Iconhttps://orcid.org/0000-0001-5440-8616
ORCID Icon,
Taoufik Chtiouib Faculty of Sciences, University of Sfax, Sfax, TunisiaORCID Iconhttps://orcid.org/0000-0002-1950-217X
ORCID Icon,
Sami Mabroukc Faculty of Sciences, University of Gafsa, Gafsa, TunisiaORCID Iconhttps://orcid.org/0000-0003-2610-3262
ORCID Icon &
Sergei Silvestrovd Division of Mathematics and Physics, School of Education, Culture and Communication, Mälardalen University, Västeras, SwedenCorrespondence[email protected]
ORCID Iconhttps://orcid.org/0000-0003-4554-6528
ORCID Icon
Pages 4883-4904 | Received 24 Dec 2020, Accepted 07 May 2021, Published online: 22 Jul 2021

References

  • Abramov, V. (2017). Super 3-Lie algebras induced by super Lie algebras. Adv. Appl. Clifford Algebras 27(1):9–16. DOI: 10.1007/s00006-015-0604-3.
  • Abramov, V. (2019). 3-Lie superalgebras induced by Lie superalgebras. Axioms. 8(1):21. DOI: 10.3390/axioms8010021.
  • Abramov, V. (2020). Weil algebra, 3-Lie algebra and B.R.S. algebra. In: Silvestrov, S., Malyarenko, A., Rancic, M., eds. Algebraic Structures and Applications. Springer Proceedings in Mathematics and Statistics, Vol. 317, Ch 1. Cham: Springer, pp. 1–12.
  • Abramov, V., Lätt, P. (2016). Classification of low dimensional 3-Lie superalgebras. In: Silvestrov S., Rancic M., eds. Engineering Mathematics II. Springer Proceedings in Mathematics and Statistics, Vol. 179. Cham: Springer, pp. 1–12.
  • Abramov, V., Lätt, P. (2020). Ternary Lie superalgebras and Nambu-Hamilton equation in superspace. In: Silvestrov S., Malyarenko A., Rancić, M., eds. Algebraic Structures and Applications. Springer Proceedings in Mathematics and Statistics, Vol. 317, Ch 3. Springer, pp. 47–80.
  • Abramov, V., Silvestrov, S. (2020). 3-Hom-Lie algebras based on σ-derivation and involution. Adv. Appl. Clifford Algebras 30(3):45. DOI: 10.1007/s00006-020-01068-6.
  • Arnlind, J., Kitouni, A., Makhlouf, A., Silvestrov, S. (2014). Structure and cohomology of 3-Lie algebras induced by Lie algebras. In: Makhlouf, A., Paal, E., Silvestrov, S. D., Stolin, A., eds. Algebra, Geometry and Mathematical Physics. Springer Proceedings in Mathematics and Statistics, Vol. 85, Berlin, Heidelberg: Springer, pp. 123–144.
  • Arnlind, J., Makhlouf, A., Silvestrov, S. (2010). Ternary Hom-Nambu-Lie algebras induced by Hom-Lie algebras. J. Math. Phys. 51(4):043515. DOI: 10.1063/1.3359004.
  • Arnlind, J., Makhlouf, A., Silvestrov, S. (2011). Construction of n-Lie algebras and n-ary Hom-Nambu-Lie algebras. J. Math. Phys. 52(12):123502–123513. DOI: 10.1063/1.3653197.
  • Albuquerque, H., Barreiro, E., Calderón, A. J., Sánchez, J. M. (2020) Split Lie-Rinehart algebras. J. Algebra Appl. DOI: 10.1142/S0219498821501644.
  • Awata, H., Li, M., Minic, D., Yoneya, T. (2001). On the quantization of Nambu brackets. J. High Energy Phys. 2001(02):013–017. DOI: 10.1088/1126-6708/2001/02/013.
  • Bai, R., Li, X., Wu, Y. (2019). 3-Lie-Rinehart algebras. arXiv:1903.12283v1.
  • Bai, R., Li, X., Wu, Y. (2020). Hom 3-Lie-Rinehart algebras. arXiv:2001.07570.
  • Bai, R., Bai, C., Wang, J. (2010). Realizations of 3-Lie algebras. J. Math. Phys. 51(6):063505. DOI: 10.1063/1.3436555.
  • Bai, R., Wu, Y. (2015). Constructions of 3-Lie algebras. Lin. Multilin. Alg. 63(11):2171–2186. DOI: 10.1080/03081087.2014.986121.
  • Bai, R., Li, Z., Wang, W. (2017). Infnite-dimensional 3-Lie algebras and their connections to Harish-Chandra module. Front. Math. China 12(3):515–530. DOI: 10.1007/s11464-017-0606-7.
  • Ben Hassine, A., Mabrouk, S., Ncib, O. (2019). Some constructions of multiplicative n-ary Hom-Nambu algebras. Adv. Appl. Clifford Algebras 29(5):88. DOI: 10.1007/s00006-019-0996-6.
  • Ben Hassine, A., Chtioui, T., Elhamdadi, M., Mabrouk, S. (2021). Extensions and crossed modules of n-Lie Rinehart algebras. arXiv:2103.15006.
  • Bkouche, R. (1966). Structures (K, A)-linéaires. C. R. Acad. Sci. Paris Sér. A-B. 262:A373–A376. (in French)
  • Calderon Martin, A. J., Forero Piulestan, M. (2011). Split 3-Lie algebras. J. Math. Phys. 52(12):123503.
  • Cantarini, N., Kac, V. G. (2010). Classification of simple linearly compact n-Lie superalgebras. Commun. Math. Phys. 298(3):833–853. DOI: 10.1007/s00220-010-1049-0.
  • Casas, J. M. (2011). Obstructions to Lie-Rinehart algebra extensions. Algebra Colloq. 18(01):83–104. DOI: 10.1142/S1005386711000046.
  • Casas, J. M., Ladra, M., Pirashvili, T. (2004). Crossed modules for Lie-Rinehart algebras. J. Algebra 274(1):192–201. DOI: 10.1016/j.jalgebra.2003.10.001.
  • Casas, J. M., Ladra, M., Pirashvili, T. (2005). Triple cohomology of Lie-Rinehart algebras and the canonical class of associative algebras J. Algebra 291(1):144–163. DOI: 10.1016/j.jalgebra.2005.05.018.
  • Chemla, S. (1995). Operations for modules on Lie-Rinehart superalgebras. Manuscripta Math. 87(1):199–224. DOI: 10.1007/BF02570471.
  • Chen, Z., Liu, Z., Zhong, D. (2011). Lie-Rinehart bialgebras for crossed products. J. Pure Appl. Algebra 215(6):1270–1283. DOI: 10.1016/j.jpaa.2010.08.011.
  • Dokas, I. (2012). Cohomology of restricted Lie-Rinehart algebras. Adv. Math. 231(5):2573–2592. DOI: 10.1016/j.aim.2012.08.003.
  • Daletskii, Y. L., Kushnirevich, V. A. (1996). Inclusion of the Nambu-Takhtajan algebra in the structure of formal differential geometry. Dopov. Akad. Nauk Ukr. 4:12–18.
  • Filippov, V. (1985). n-Lie algebras. Sib. Mat. Zh. 26:126–140.
  • Guan, B., Chen, L., Sun, B. (2017). 3-ary Hom-Lie superalgebras induced by Hom-Lie superalgebras. Adv. Appl. Clifford Algebras 27(4):3063–3082. DOI: 10.1007/s00006-017-0801-3.
  • Guo, S., Zhang, X., Wang, S. (2019). On 3-Hom-Lie-Rinehart algebras. arXiv:1911.10992.
  • Herz, J. (1953). Pseudo-algebras de Lie. I, II. C. R. Acad. Sci. Paris. 236:2289–2291, 1935-1937.
  • Higgins, P. J., Mackenzie, K. (1990). Algebraic constructions in the category of Lie algebroids. J. Algebra 129(1):194–230. DOI: 10.1016/0021-8693(90)90246-K.
  • Huebschmann, J. (1990). Poisson cohomology and quantization. J. Reine Angew. Math. 408:57–113.
  • Huebschmann, J. (1998). Lie-Rinehart algebras, Gerstenhaber algebras and Batalin-Vilkovisky algebras. Ann. Inst. Fourier (Grenoble). 48(2):425–440. DOI: 10.5802/aif.1624.
  • Huebschmann, J. (1999). Duality for Lie-Rinehart algebras and the modular class. J. Reine Angew. Math. 1999(510):103–159. DOI: 10.1515/crll.1999.043.
  • Huebschmann, J. (2004). Lie-Rinehart algebras, descent, and quantization. In: Galois Theory, Hopf Algebras, and Semiabelian Categories, (Fields Institute Communications), Vol. 43. Providence, RI: American Mathematical Society, pp. 295–316.
  • Huebschmann, J., Perlmutter, M., Ratiu, T. S. (2013). Extensions of Lie-Rinehart algebras and cotangent bundle reduction. Proc. Lond. Math. Soc. 107(5):1135–1172. DOI: 10.1112/plms/pdt030.
  • Huebschmann, J. (2017). Multi derivation Maurer-Cartan algebras and sh Lie-Rinehart algebras. J. Algebra 472:437–479. DOI: 10.1016/j.jalgebra.2016.10.008.
  • Krahmer, U., Rovi, A. (2015). A Lie-Rinehart algebra with no antipode. Commun. Algebra 43(10):4049–4053. DOI: 10.1080/00927872.2014.896375.
  • Kac, V. G. (1977). A sketch of Lie superalgebra theory. Commun. Math. Phys. 52(1):31–64. DOI: 10.1007/BF01609166.
  • Kitouni, A., Makhlouf, A., Silvestrov, S. (2016). On (n + 1)-Hom-Lie algebras induced by n-Hom-Lie algebras. Georgian Math. J. 23(1):75–95.
  • Kitouni, A., Makhlouf, A., Silvestrov, S. (2020). On solvability and nilpotency for n-Hom-Lie algebras and (n + 1)-Hom-Lie algebras induced by n-Hom-Lie algebras. In: Silvestrov, S., Malyarenko, A., Rancic, M., eds. Algebraic Structures and Applications, Springer Proceedings in Mathematics and Statistics, Vol 317, Ch 6, Cham: Springer, pp. 127–157.
  • Mackenzie, K. (1987). Lie groupoids and Lie algebroids in differential geometry. London Math. Soc. Lecture Note Series 124, Cambridge: Cambridge University Press.
  • Mackenzie, K. C. H. (1995). Lie algebroids and Lie pseudoalgebras. Bull. London Math. Soc. 27(2):97–147. DOI: 10.1112/blms/27.2.97.
  • Mandal, A., Mishra, S. K. (2018). Hom-Lie-Rinehart algebras. Commun. Algebra 46(9):3722–3744. DOI: 10.1080/00927872.2018.1424865.
  • Mandal, A., Mishra, S. K. (2018). On hom-Gerstenhaber algebras, and hom-Lie algebroids. J. Geom. Phys. 133:287–302. DOI: 10.1016/j.geomphys.2018.07.018.
  • Mishra, S. K., Silvestrov, S. (2020). A review on Hom-Gerstenhaber algebras and Hom-Lie algebroids. In: Silvestrov S., Malyarenko A., Rancić, M., eds. Algebraic Structures and Applications, Springer Proceedings in Mathematics and Statistics, Vol. 317, Ch 11. Cham: Springer, pp. 285–315.
  • Palais, R. (1961). The cohomology of Lie rings. Proc. Symp. Pure Math., Amer. Math. Soc. Providence, R. I., pp. 130–137
  • Rinehart, G. (1963). Differential forms on general commutative algebras. Trans. Amer. Math. Soc. 108(2):195–222. DOI: 10.1090/S0002-9947-1963-0154906-3.
  • Rota, G. (1969). Baxter algebras and combinatorial identities I, II. Bull. Amer. Math. Soc. 75(2):330–334. DOI: 10.1090/S0002-9904-1969-12156-7.
  • Takhtajan, L. (1994). On foundation of the generalized Nambu mechanics. Commun. Math. Phys. 160(2):295–315. DOI: 10.1007/BF02103278.

Related research

People also read lists articles that other readers of this article have read.

Recommended articles lists articles that we recommend and is powered by our AI driven recommendation engine.

Cited by lists all citing articles based on Crossref citations.
Articles with the Crossref icon will open in a new tab.