Nonabelian embedding tensors on 3-Lie algebras and 3-Leibniz-Lie algebras

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Abstract

The purpose of this paper is to study nonabelian embedding tensors on 3-Lie algebras, and to explore the fundamental algebraic structures, cohomology and deformations associated with them. First, we introduce the concept of nonabelian embedding tensors on 3-Lie algebras. Then, we present the concept of a 3-Leibniz-Lie algebra, which constitutes the fundamental algebraic framework for a nonabelian embedding tensor on a 3-Lie algebra. Additionally, we examine the 3-Leibniz- Lie algebras that are derived from Leibniz-Lie algebras. Finally, we develop the cohomology of nonabelian embedding tensors on 3-Lie algebras and utilize the first cohomology group to characterize infinitesimal deformations.

Keywords

3-Lie algebra / nonabelian embedding tensor / 3-Leibniz algebra / 3-Leibniz-Lie algebra / cohomology / infinitesimal deformation

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References

[1] H. Nicolai, H. Samtleben, Maximal gauged supergravity in three dimensions, Phys. Rev. Lett., 86 (2001), 1686. https://doi.org/10.1103/PhysRevLett.86.1686

[2] E. A. Bergshoeff, M. de Roo, O. Hohm, Multiple M2-branes and the embedding tensor, Classical Quantum Gravity, 25 (2008), 142001. https://doi.org/10.1088/0264-9381/25/14/142001

[3] R. Bonezzi, O. Hohm, Leibniz gauge theories and infinity structures, Commun. Math. Phys., 377 (2020), 2027-2077. https://doi.org/10.1007/s00220-020-03785-2

[4] B. de Wit, H. Nicolai, H. Samtleben, Gauged supergravities, tensor hierarchies, and M-theory, J. High Energy Phys., 2008 (2008), 044. https://doi.org/10.1088/1126-6708/2008/02/044

[5] B. de Wit, H. Samtleben, Gauged maximal supergravities and hierarchies of nonabelian vector-tensor systems, Fortschr. Phys., 53 (2005), 442-449. https://doi.org/10.1002/prop.200510202

[6] A. Kotov, T. Strobl, The embedding tensor, Leibniz-Loday algebras and their higher gauge theories, Commun. Math. Phys., 376 (2020), 235-258. https://doi.org/10.1007/s00220-019-03569-3

[7] Y. Sheng, R. Tang, C. Zhu, The controlling L_∞-algebra, cohomology and homotopy of embedding tensors and Lie-Leibniz triples , Commun. Math. Phys., 386 (2021), 269-304. https://doi.org/10.1007/s00220-021-04032-y

[8] M. Hu, S. Hou, L. Song, Y. Zhou, Deformations and cohomologies of embedding tensors on 3-Lie algebras, Commun. Algebra, 51 (2023), 4622-4639. https://doi.org/10.1080/00927872.2023.2214215

[9] R. Tang, Y. Sheng, Nonabelian embedding tensors, Lett. Math. Physics, 113 (2023), 14. https://doi.org/10.1007/s11005-023-01637-3

[10] J. Pei, C. Bai, L. Guo, X. Ni, Replicating of binary operads, Koszul duality, Manin products and average operators, preprint, arXiv:1212.0177.

[11] A. Das, R. Mandal, Averaging algebras of any nonzero weight, preprint, arXiv:2304.12593.

[12] W. Teng, J. Jin, Y. Zhang, Cohomology of nonabelian embedding tensors on Hom-Lie algebras, AIMS Math., 8 (2023), 21176-21190. https://doi.org/10.3934/math.20231079

[13] V. T. Filippov, n-Lie algebras, Sib. Math. J., 26 (1985), 879-891. https://doi.org/10.1007/BF00969110

[14] Y. Nambu, Generalized hamiltonian dynamics, Phys. Rev. D, 7 (1973), 2405. https://doi.org/10.1103/PhysRevD.7.2405

[15] J. Bagger, N. Lambert, Gauge symmetry and supersymmetry of multiple M2-branes, Phys. Rev. D, 77 (2008), 065008. https://doi.org/10.1103/PhysRevD.77.065008

[16] A. Gustavsson, Algebraic structures on parallel M2-branes, Nuclear Phys. B, 811 (2009), 66-76. https://doi.org/10.1016/j.nuclphysb.2008.11.014

[17] R. Bai, C. Bai, J. Wang, Realizatons of 3-Lie algebras, J. Math. Phys., 51 (2010), 0633505. https://doi.org/10.1063/1.3436555

[18] S. Guo, Y. Qin, K. Wang, G. Zhou, Deformations and cohomology theory of Rota-Baxter 3-Lie algebras of arbitrary weights, J. Geom. Phys., 183 (2023), 104704. https://doi.org/10.1016/j.geomphys.2022.104704

[19] S. Hou, Y. Sheng, Y. Zhou, 3-post-Lie algebras and relative Rota-Baxter operators of nonzero weight on 3-Lie algebras, J. Algebra, 615 (2023), 103-129. https://doi.org/10.1016/j.jalgebra.2022.10.016

[20] S. M. Kasymov, Theory of n-Lie algebras, Algebra Logic, 26 (1987), 155-166. https://doi.org/10.1007/BF02009328

[21] J. M. Casas, J. Loday, T. Pirashvili, Leibniz n-algebras, Forum Math., 14 (2002), 189-207. https://doi.org/10.1515/form.2002.009

[22] J. M. Casas, Homology with trivial coefcients of Leibniz n-algebras, Commun. Algebra, 31 (2003), 1377-1386. https://doi.org/10.1081/AGB-120017771

[23] Y. L. Daletskii, L. A. Takhtajan, Leibniz and Lie algebra structures for Nambu algebra, Lett. Math. Phys., 39 (1997), 127-141. https://doi.org/10.1023/A:1007316732705

[24] L. Takhtajan, Higher order analog of Chevalley-Eilenberg complex and deformation theory of n-algebras, Algebra Anal., 6 (1994), 262-272.