| Title:
|
Quasi-trace functions on Lie algebras and their applications to 3-Lie algebras (English) |
| Author:
|
Tan, Youjun |
| Author:
|
Xu, Senrong |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
72 |
| Issue:
|
2 |
| Year:
|
2022 |
| Pages:
|
559-591 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| 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. (English) |
| Keyword:
|
quasi-trace function |
| Keyword:
|
3-Lie algebra |
| Keyword:
|
Leibniz algebra |
| MSC:
|
17A32 |
| MSC:
|
17A42 |
| MSC:
|
17B05 |
| MSC:
|
17B56 |
| idZBL:
|
Zbl 07547220 |
| idMR:
|
MR4412775 |
| DOI:
|
10.21136/CMJ.2022.0059-21 |
| . |
| Date available:
|
2022-04-21T19:05:35Z |
| Last updated:
|
2024-07-01 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/150417 |
| . |
| Reference:
|
[1] Amayo, R. K.: Quasi-ideals of Lie algebras. I.Proc. Lond. Math. Soc., III. Ser. 33 (1976), 28-36. Zbl 0337.17004, MR 0409573, 10.1112/plms/s3-33.1.28 |
| Reference:
|
[2] Amayo, R. K.: Quasi-ideals of Lie algebras. II.Proc. Lond. Math. Soc., III. Ser. 33 (1976), 37-64. Zbl 0337.17005, MR 0409574, 10.1112/plms/s3-33.1.37 |
| Reference:
|
[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. Zbl 1358.17004, MR 3275936, 10.1007/978-3-642-55361-5_9 |
| Reference:
|
[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. Zbl 1310.17001, MR 2662502, 10.1063/1.3359004 |
| Reference:
|
[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. MR 1825236, 10.1088/1126-6708/2001/02/013 |
| Reference:
|
[6] Bai, R., Bai, C., Wang, J.: Realizations of 3-Lie algebras.J. Math. Phys. 51 (2010), Article ID 063505, 12 pages. Zbl 1311.17002, MR 2676482, 10.1063/1.3436555 |
| Reference:
|
[7] Burde, D., Steinhoff, C.: Classification of orbit closures of 4-dimensional complex Lie algebras.J. Algebra 214 (1999), 729-739. Zbl 0932.17005, MR 1680532, 10.1006/jabr.1998.7714 |
| Reference:
|
[8] Carter, R.: Lie Algebras of Finite and Affine Type.Cambridge Studies in Advanced Mathematics 96. Cambridge Univesity Press, Cambridge (2005). Zbl 1110.17001, MR 2188930, 10.1017/CBO9780511614910 |
| Reference:
|
[9] Daletskii, Y. L., Takhtajan, L. A.: Leibniz and Lie algebra structures for Nambu algebra.Lett. Math. Phys. 39 (1997), 127-141. Zbl 0869.58024, MR 1437747, 10.1023/A:1007316732705 |
| Reference:
|
[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. Zbl 1202.81187, 10.1088/1751-8113/43/29/293001 |
| Reference:
|
[11] Dixmier, J.: Enveloping Algebras.Graduate Studies in Mathematics 11. American Mathematical Society, Providence (1996). Zbl 0867.17001, MR 1393197, 10.1090/gsm/011 |
| Reference:
|
[12] Dudek, W. A.: On some old and new problems in $n$-ary groups.Quasigroups Relat. Syst. 8 (2001), 15-36. Zbl 1052.20048, MR 1876783 |
| Reference:
|
[13] Erdmann, K., Wildon, M. J.: Introduction to Lie Algebras.Springer Undergraduate Mathematics Series. Springer, London (2006). Zbl 1139.17001, MR 2218355, 10.1007/1-84628-490-2 |
| Reference:
|
[14] Figueroa-O'Farrill, J. M.: Deformations of 3-algebras.J. Math. Phys. 50 (2009), Article ID 113514, 27 pages. Zbl 1304.17005, MR 2567220, 10.1063/1.3262528 |
| Reference:
|
[15] Filippov, V. T.: $n$-Lie algebras.Sib. Math. J. 26 (1985), 879-891. Zbl 0594.17002, MR 0816511, 10.1007/BF00969110 |
| Reference:
|
[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. Zbl 1433.17006, MR 3673814 |
| Reference:
|
[17] Jacobson, N.: Lie Algebras.Interscience Tracts in Pure and Applied Mathematics 10. Interscience Publishers, New York (1962). Zbl 0121.27504, MR 0559927 |
| Reference:
|
[18] Kasymov, S. M.: Theory of $n$-Lie algebras.Algebra Logic 26 (1987), 155-166. Zbl 0658.17003, MR 0962883, 10.1007/BF02009328 |
| Reference:
|
[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. Zbl 1430.17011, MR 3735913, 10.1007/s10468-017-9693-0 |
| Reference:
|
[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. Zbl 0806.55009, MR 1252069 |
| Reference:
|
[21] Loday, J.-L., Pirashvili, T.: Universal enveloping algebras of Leibniz algebras and (co)homology.Math. Ann. 296 (1993), 139-158. Zbl 0821.17022, MR 1213376, 10.1007/BF01445099 |
| Reference:
|
[22] Song, L., Jiang, J.: Generalized derivations extensions of 3-Lie algebras and corresponding Nambu-Poisson structures.J. Geom. Phys. 124 (2018), 74-85. Zbl 1430.17010, MR 3754499, 10.1016/j.geomphys.2017.10.011 |
| Reference:
|
[23] Takhtajan, L.: On foundation of the generalized Nambu mechanics.Commun. Math. Phys. 160 (1994), 295-315. Zbl 0808.70015, MR 1262199, 10.1007/BF02103278 |
| Reference:
|
[24] Tan, Y., Xu, S.: The Wells map for abelian extensions of 3-Lie algebras.Czech. Math. J. 69 (2019), 1133-1164. Zbl 07144882, MR 4039627, 10.21136/CMJ.2019.0098-18 |
| Reference:
|
[25] Zhang, T.: Cohomology and deformations of 3-Lie colour algebras.Linear Multilinear Algebra 63 (2015), 651-671. Zbl 1387.17007, MR 3291556, 10.1080/03081087.2014.891589 |
| . |
