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Keywords:
quasi-trace function; 3-Lie algebra; Leibniz algebra
quasi-trace function; 3-Lie algebra; Leibniz algebra
Summary:
We introduce the notion of quasi-trace functions on Lie algebras. As applications we study realizations of 3-dimensional and 4-dimensional 3-Lie algebras. Some comparison results on cohomologies of 3-Lie algebras and Leibniz algebras arising from quasi-trace functions are obtained.
References:
[1] Amayo, R. K.: Quasi-ideals of Lie algebras. I. Proc. Lond. Math. Soc., III. Ser. 33 (1976), 28-36. DOI 10.1112/plms/s3-33.1.28 | MR 0409573 | Zbl 0337.17004
[2] Amayo, R. K.: Quasi-ideals of Lie algebras. II. Proc. Lond. Math. Soc., III. Ser. 33 (1976), 37-64. DOI 10.1112/plms/s3-33.1.37 | MR 0409574 | Zbl 0337.17005
[3] Arnlind, J., Kitouni, A., Makhlouf, A., Silvestrov, S.: Structure and cohomology of 3-Lie algebras induced by Lie algebras. Algebra, Geometry and Mathematical Physics Springer Proceedings in Mathematics and Statistics 85. Springer, Berlin (2014), 123-144. DOI 10.1007/978-3-642-55361-5_9 | MR 3275936 | Zbl 1358.17004
[4] Arnlind, J., Makhlouf, A., Silvestrov, S.: Ternary Hom-Nambu-Lie algebras induced by Hom-Lie algebras. J. Math. Phys. 51 (2010), Article ID 043515, 11 pages. DOI 10.1063/1.3359004 | MR 2662502 | Zbl 1310.17001
[5] Awata, H., Li, M., Minic, D., Yoneya, T.: On the quantization of Nambu brackets. J. High Energy Phys. 2001 (2001), Article ID 13, 17 pages. DOI 10.1088/1126-6708/2001/02/013 | MR 1825236
[6] Bai, R., Bai, C., Wang, J.: Realizations of 3-Lie algebras. J. Math. Phys. 51 (2010), Article ID 063505, 12 pages. DOI 10.1063/1.3436555 | MR 2676482 | Zbl 1311.17002
[7] Burde, D., Steinhoff, C.: Classification of orbit closures of 4-dimensional complex Lie algebras. J. Algebra 214 (1999), 729-739. DOI 10.1006/jabr.1998.7714 | MR 1680532 | Zbl 0932.17005
[8] Carter, R.: Lie Algebras of Finite and Affine Type. Cambridge Studies in Advanced Mathematics 96. Cambridge Univesity Press, Cambridge (2005). DOI 10.1017/CBO9780511614910 | MR 2188930 | Zbl 1110.17001
[9] Daletskii, Y. L., Takhtajan, L. A.: Leibniz and Lie algebra structures for Nambu algebra. Lett. Math. Phys. 39 (1997), 127-141. DOI 10.1023/A:1007316732705 | MR 1437747 | Zbl 0869.58024
[10] Azcárraga, J. A. de, Izquierdo, J. M.: $n$-ary algebras: A review with applications. J. Phys. A, Math. Theor. 43 (2010), Article ID 293001, 117 pages. DOI 10.1088/1751-8113/43/29/293001 | Zbl 1202.81187
[11] Dixmier, J.: Enveloping Algebras. Graduate Studies in Mathematics 11. American Mathematical Society, Providence (1996). DOI 10.1090/gsm/011 | MR 1393197 | Zbl 0867.17001
[12] Dudek, W. A.: On some old and new problems in $n$-ary groups. Quasigroups Relat. Syst. 8 (2001), 15-36. MR 1876783 | Zbl 1052.20048
[13] Erdmann, K., Wildon, M. J.: Introduction to Lie Algebras. Springer Undergraduate Mathematics Series. Springer, London (2006). DOI 10.1007/1-84628-490-2 | MR 2218355 | Zbl 1139.17001
[14] Figueroa-O'Farrill, J. M.: Deformations of 3-algebras. J. Math. Phys. 50 (2009), Article ID 113514, 27 pages. DOI 10.1063/1.3262528 | MR 2567220 | Zbl 1304.17005
[15] Filippov, V. T.: $n$-Lie algebras. Sib. Math. J. 26 (1985), 879-891. DOI 10.1007/BF00969110 | MR 0816511 | Zbl 0594.17002
[16] García-Martínez, X., Turdibaev, R., Linden, T. Van der: Do $n$-Lie algebras have universal enveloping algebras?. J. Lie Theory 28 (2018), 43-55. MR 3673814 | Zbl 1433.17006
[17] Jacobson, N.: Lie Algebras. Interscience Tracts in Pure and Applied Mathematics 10. Interscience Publishers, New York (1962). MR 0559927 | Zbl 0121.27504
[18] Kasymov, S. M.: Theory of $n$-Lie algebras. Algebra Logic 26 (1987), 155-166. DOI 10.1007/BF02009328 | MR 0962883 | Zbl 0658.17003
[19] Liu, J., Makhlouf, A., Sheng, Y.: A new approach to representations of 3-Lie algebras and abelian extensions. Algebr. Represent. Theory 20 (2017), 1415-1431. DOI 10.1007/s10468-017-9693-0 | MR 3735913 | Zbl 1430.17011
[20] Loday, J.-L.: Une version non commutative des algèbres de Lie: Les algèbres de Leibniz. Enseign. Math., II. Sér. 39 (1993), 269-293 French. MR 1252069 | Zbl 0806.55009
[21] Loday, J.-L., Pirashvili, T.: Universal enveloping algebras of Leibniz algebras and (co)homology. Math. Ann. 296 (1993), 139-158. DOI 10.1007/BF01445099 | MR 1213376 | Zbl 0821.17022
[22] Song, L., Jiang, J.: Generalized derivations extensions of 3-Lie algebras and corresponding Nambu-Poisson structures. J. Geom. Phys. 124 (2018), 74-85. DOI 10.1016/j.geomphys.2017.10.011 | MR 3754499 | Zbl 1430.17010
[23] Takhtajan, L.: On foundation of the generalized Nambu mechanics. Commun. Math. Phys. 160 (1994), 295-315. DOI 10.1007/BF02103278 | MR 1262199 | Zbl 0808.70015
[24] Tan, Y., Xu, S.: The Wells map for abelian extensions of 3-Lie algebras. Czech. Math. J. 69 (2019), 1133-1164. DOI 10.21136/CMJ.2019.0098-18 | MR 4039627 | Zbl 07144882
[25] Zhang, T.: Cohomology and deformations of 3-Lie colour algebras. Linear Multilinear Algebra 63 (2015), 651-671. DOI 10.1080/03081087.2014.891589 | MR 3291556 | Zbl 1387.17007
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