On ideals and derived and central descending series of \(n\)-ary Hom-algebras. (English) Zbl 07822809
Albuquerque, Helena (ed.) et al., Non-associative algebras and related topics. NAART II. Selected papers based on the presentations at the 2nd conference, Coimbra, Portugal, July 18–22, 2022. Cham: Springer. Springer Proc. Math. Stat. 427, 261-286 (2023).
Summary: The aim of this work is to explore some properties of \(n\)-ary skew-symmetric Hom-algebras and \(n\)-Hom-Lie algebras related to their ideals, derived series and central descending series. We extend the notions of derived series and central descending series to \(n\)-ary skew-symmetric Hom-algebras and provide various general conditions for their members to be Hom-subalgebras, weak ideals or Hom-ideals in the algebra or relatively to each other. In particular we study the invariance under the twisting maps of the derived series and central descending series and their subalgebra and ideal properties for a class of 3-dimensional Hom-Lie algebras and some 4-dimensional 3-Hom-Lie algebras. We also introduce a type of generalized ideals in \(n\)-ary Hom-algebras and present a few basic properties.
For the entire collection see [Zbl 1531.17001].
For the entire collection see [Zbl 1531.17001].
MSC:
| 17B61 | Hom-Lie and related algebras |
| 17D30 | (non-Lie) Hom algebras and topics |
| 17A40 | Ternary compositions |
| 17A42 | Other \(n\)-ary compositions \((n \ge 3)\) |
| 17B30 | Solvable, nilpotent (super)algebras |
References:
| [1] | Abdaoui, K., Mabrouk, S., Makhlouf, A.: Cohomology of Hom-Leibniz and \(n\)-ary Hom-Nambu-Lie superalgebras, p. 24. arXiv:1406.3776 [math.RT] |
| [2] | Abramov, V., Super 3-Lie algebras induced by super Lie algebras, Adv. Appl. Clifford Algebr., 27, 1, 9-16 (2017) · Zbl 1419.17005 · doi:10.1007/s00006-015-0604-3 |
| [3] | Abramov, V.: Weil algebra, \(3\)-Lie algebra and B.R.S. algebra. In: Silvestrov, S., Malyarenko, A., Rančić, M. (eds.) Algebraic Structures and Applications, Springer Proceedings in Mathematics and Statistics, vol. 317, Ch. 1 (2020). arXiv:1802.05576 [math.RA] · Zbl 1467.16001 |
| [4] | Abramov, V.; Lätt, P.; Silvestrov, S.; Rancic, M., Classification of low dimensional \(3\)-Lie superalgebras, Engineering Mathematics II, Springer Proceedings in Mathematics and Statistics, 1-12 (2016), Cham: Springer, Cham · Zbl 1418.17012 |
| [5] | Abramov, V., Lätt, P.: Ternary Lie superalgebras and Nambu-Hamilton equation in superspace. In: Silvestrov, S., Malyarenko, A., Rančić, M. (eds.) Algebraic Structures and Applications, Springer Proceedings in Mathematics and Statistics, vol. 317, Ch. 3 (2020) · Zbl 1511.17052 |
| [6] | Abramov, V.; Silvestrov, S., \(3\)-Hom-Lie algebras based on \(\sigma \)-derivation and involution, Adv. Appl. Clifford Algebras, 30, 45 (2020) · Zbl 1468.17031 · doi:10.1007/s00006-020-01068-6 |
| [7] | Aizawa, N., Sato, H.: \(q\)-Deformation of the Virasoro algebra with central extension. Phys. Lett. B 256, 185-190 (1991) (Hiroshima University preprint, preprint HUPD-9012 (1990)) · Zbl 1332.17011 |
| [8] | Ammar, F.; Ejbehi, Z.; Makhlouf, A., Cohomology and deformations of Hom-algebras, J. Lie Theory, 21, 4, 813-836 (2011) · Zbl 1237.17003 |
| [9] | Ammar, F.; Mabrouk, S.; Makhlouf, A., Representation and cohomology of \(n\)-ary multiplicative Hom-Nambu-Lie algebras, J. Geom. Phys., 61, 1898-1913 (2011) · Zbl 1258.17007 · doi:10.1016/j.geomphys.2011.04.022 |
| [10] | Arnlind, J., Kitouni, A., Makhlouf, A., Silvestrov, S.: 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, vol. 85, pp. 123-144. Springer Proceedings in Mathematics and Statistics, Springer (2014) · Zbl 1358.17004 |
| [11] | Arnlind, J.; Makhlouf, A.; Silvestrov, S., Ternary Hom-Nambu-Lie algebras induced by Hom-Lie algebras, J. Math. Phys., 51, 43515, 11 (2010) · Zbl 1310.17001 |
| [12] | Arnlind, J.; Makhlouf, A.; Silvestrov, S., Construction of \(n\)-Lie algebras and \(n\)-ary Hom-Nambu-Lie algebras, J. Math. Phys., 52, 123502, 13 (2011) · Zbl 1273.17005 |
| [13] | Ataguema, H.; Makhlouf, A.; Silvestrov, S., Generalization of \(n\)-ary Nambu algebras and beyond, J. Math. Phys., 50 (2009) · Zbl 1328.17004 · doi:10.1063/1.3167801 |
| [14] | Awata, H.; Li, M.; Minic, D.; Yoneya, T., On the quantization of Nambu brackets, J. High Energy Phys., 2, 13, 17 (2001) |
| [15] | Bai, C.; Guo, L.; Sheng, Y., Bialgebras, the classical Yang-Baxter equation and Manin triples for \(3\)-Lie algebras, Adv. Theor. Math. Phys., 23, 1, 27-74 (2019) · Zbl 1479.17008 · doi:10.4310/ATMP.2019.v23.n1.a2 |
| [16] | Bai, R.; An, H.; Li, Z., Centroid structures of \(n\)-Lie algebras, Linear Algebra Appl., 430, 229-240 (2009) · Zbl 1221.17004 · doi:10.1016/j.laa.2008.07.007 |
| [17] | Bai, R.; Bai, C.; Wang, J., Realizations of \(3\)-Lie algebras, J. Math. Phys., 51 (2010) · Zbl 1311.17002 · doi:10.1063/1.3436555 |
| [18] | Bai, R.; Chen, L.; Meng, D., The Frattini subalgebra of \(n\)-Lie algebras, Acta Math. Sinica. Eng. Ser., 23, 5, 847-856 (2007) · Zbl 1152.17004 · doi:10.1007/s10114-005-0923-8 |
| [19] | Bai, R.; Meng, D., The central extension of \(n\)-Lie algebras, Chinese Ann. Math., 27, 4, 491-502 (2006) |
| [20] | Bai, R.; Meng, D., The centroid of \(n\)-Lie algebras, Algebras Groups Geom., 25, 2, 29-38 (2004) · Zbl 1093.17003 |
| [21] | Bai, R.; Song, G.; Zhang, Y., On classification of \(n\)-Lie algebras, Front. Math. China, 6, 581-606 (2011) · Zbl 1282.17003 · doi:10.1007/s11464-011-0107-z |
| [22] | Bai, R.; Wang, X.; Xiao, W.; An, H., The structure of low dimensional \(n\)-Lie algebras over the field of characteristic \(2\), Linear Algebra Appl., 428, 8-9, 1912-1920 (2008) · Zbl 1205.17007 · doi:10.1016/j.laa.2007.10.035 |
| [23] | Bai, R.; Wu, Y.; Li, J.; Zhou, H., Constructing \((n+1)\)-Lie algebras from \(n\)-Lie algebras, J. Phys. A: Math. Theor., 45, 47 (2012) · Zbl 1259.17001 · doi:10.1088/1751-8113/45/47/475206 |
| [24] | Bai, R.; Zhang, Z.; Li, H.; Shi, H., The inner derivation algebras of \((n+1)\)-dimensional \(n\)-Lie algebras, Comm. Algebra, 28, 6, 2927-2934 (2000) · Zbl 0963.17002 · doi:10.1080/00927870008827001 |
| [25] | Bakayoko, I., Silvestrov, S.: Multiplicative \(n\)-Hom-Lie color algebras. In: Silvestrov, S., Malyarenko, A., Rančić, M. (eds.) Algebraic Structures and Applications, vol. 317, Ch. 7. Springer Proceedings in Mathematics and Statistics (2020). arXiv:1912.10216 [math.QA] · Zbl 1479.17010 |
| [26] | Ben Hassine, A.; Mabrouk, S.; Ncib, O., Some constructions of multiplicative \(n\)-ary hom-Nambu algebras, Adv. Appl. Clifford Algebras, 29, 5, 88 (2019) · Zbl 1435.17030 · doi:10.1007/s00006-019-0996-6 |
| [27] | Ben Abdeljelil, A., Elhamdadi, M., Kaygorodov, I., Makhlouf, A.: Generalized derivations of \(n\)-BiHom-Lie algebras. In: Silvestrov, S., Malyarenko, A., Rančić, M. (eds.) Algebraic Structures and Applications, vol. 317, Ch. 4. Springer Proceedings in Mathematics and Statistics (2020). arXiv:1901.09750 [math.RA] · Zbl 1485.17014 |
| [28] | Benayadi, S.; Makhlouf, A., Hom-Lie algebras with symmetric invariant nondegenerate bilinear forms, J. Geom. Phys., 76, 38-60 (2014) · Zbl 1331.17028 · doi:10.1016/j.geomphys.2013.10.010 |
| [29] | Beites, P.D., Kaygorodov, I., Popov, Y.: Generalized derivations of multiplicative \(n\)-ary Hom-\( \Omega\) color algebras. Bull. Malay. Math. Sci. Soc. 41 (2018) |
| [30] | Carlsson, R., \(n\)-Ary algebras, Nagoya Math. J., 78, 45-65 (1980) · Zbl 0466.16017 · doi:10.1017/S0027763000018791 |
| [31] | Casas, JM; Loday, J-L; Pirashvili, T., Leibniz \(n\)-algebras, Forum Math., 14, 189-207 (2002) · Zbl 1037.17002 · doi:10.1515/form.2002.009 |
| [32] | Chaichian, M.; Ellinas, D.; Popowicz, Z., Quantum conformal algebra with central extension, Phys. Lett. B, 248, 95-99 (1990) · doi:10.1016/0370-2693(90)90021-W |
| [33] | Chaichian, M.; Isaev, AP; Lukierski, J.; Popowic, Z.; Prešnajder, P., \(q\)-Deformations of Virasoro algebra and conformal dimensions, Phys. Lett. B, 262, 1, 32-38 (1991) · doi:10.1016/0370-2693(91)90638-7 |
| [34] | Chaichian, M.; Kulish, P.; Lukierski, J., \(q\)-Deformed Jacobi identity, \(q\)-oscillators and \(q\)-deformed infinite-dimensional algebras, Phys. Lett. B, 237, 401-406 (1990) · doi:10.1016/0370-2693(90)91196-I |
| [35] | Chaichian, M.; Popowicz, Z.; Prešnajder, P., \(q\)-Virasoro algebra and its relation to the \(q\)-deformed KdV system, Phys. Lett. B, 249, 63-65 (1990) · doi:10.1016/0370-2693(90)90527-D |
| [36] | Curtright, TL; Zachos, CK, Deforming maps for quantum algebras, Phys. Lett. B, 243, 237-244 (1990) · doi:10.1016/0370-2693(90)90845-W |
| [37] | Damaskinsky, E.V., Kulish, P.P.: Deformed oscillators and their applications. Zap. Nauch. Semin. LOMI 189, 37-74 (1991) (in Russian) [Engl. transl. in J. Sov. Math. 62, 2963-2986 (1992)] · Zbl 0783.17004 |
| [38] | Daskaloyannis, C., Generalized deformed Virasoro algebras, Modern Phys. Lett. A, 7, 9, 809-816 (1992) · Zbl 1021.81566 · doi:10.1142/S0217732392000793 |
| [39] | Daletskii, YL; Takhtajan, LA, Leibniz and Lie algebra structures for Nambu algebra, Lett. Math. Phys., 39, 127-141 (1997) · Zbl 0869.58024 · doi:10.1023/A:1007316732705 |
| [40] | De Azcárraga, JA; Izquierdo, JM, \(n\)-Ary algebras: a review with applications, J. Phys. A: Math. Theor., 43, 29 (2010) · Zbl 1202.81187 · doi:10.1088/1751-8113/43/29/293001 |
| [41] | Elchinger, O.; Lundengård, K.; Makhlouf, A.; Silvestrov, SD, Brackets with \((\tau,\sigma )\)-derivations and \((p, q)\)-deformations of Witt and Virasoro algebras, Forum Math., 28, 4, 657-673 (2016) · Zbl 1404.17034 · doi:10.1515/forum-2014-0132 |
| [42] | Filippov, V.T.: \(n\)-Lie algebras. Siberian Math. J. 26, 879-891 (1985). Translated from Russian: Sib. Mat. Zh. 26, 126-140 (1985) · Zbl 0585.17002 |
| [43] | Hartwig, J.T., Larsson, D., Silvestrov, S.D.: Deformations of Lie algebras using \(\sigma \)-derivations. J. Algebra 295, 314-361 (2006) (Preprints in Mathematical Sciences 2003:32, LUTFMA-5036-2003, Centre for Mathematical Sciences, Lund University, pp. 52 (2003)) · Zbl 1138.17012 |
| [44] | Hu, N., \(q\)-Witt algebras, \(q\)-Lie algebras, \(q\)-holomorph structure and representations, Algebra Colloq., 6, 1, 51-70 (1999) · Zbl 0943.17007 |
| [45] | Kassel, C., Cyclic homology of differential operators, the Virasoro algebra and a \(q\)-analogue, Comm. Math. Phys., 146, 2, 343-356 (1992) · Zbl 0761.17020 · doi:10.1007/BF02102632 |
| [46] | Kasymov, ShM, Theory of \(n\)-Lie algebras, Algebra Logic, 26, 155-166 (1987) · Zbl 0658.17003 · doi:10.1007/BF02009328 |
| [47] | Kitouni, A., Makhlouf, A.: On structure and central extensions of \((n+1)\)-Lie algebras induced by \(n\)-Lie algebras. arXiv:1405.5930 (2014) |
| [48] | Kitouni, A.; Makhlouf, A.; Silvestrov, S., On \((n+1)\)-Hom-Lie algebras induced by \(n\)-Hom-Lie algebras, Georgian Math. J., 23, 1, 75-95 (2016) · Zbl 1335.17003 · doi:10.1515/gmj-2015-0063 |
| [49] | Kitouni, A., Makhlouf, A., Silvestrov, S.: 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., Rančić, M. (eds.) Algebraic Structures and Applications, vol. 317, Ch 6, pp. 127-157. Springer Proceedings in Mathematics and Statistics, Springer (2020) · Zbl 1479.17009 |
| [50] | Kitouni, A., Makhlouf, A., Silvestrov, S.: On \(n\)-ary generalization of BiHom-Lie algebras and BiHom-associative algebras. In: Silvestrov, S., Malyarenko, A., Rančić, M. (eds.) Algebraic Structures and Applications, vol. 317, Ch 5. Springer Proceedings in Mathematics and Statistics (2020) · Zbl 1485.17015 |
| [51] | Kitouni, A., Silvestrov, S.: On Classification of \((n+1)\)-dimensional \(n\)-Hom-Lie algebras with nilpotent twisting maps. In: Silvestrov, S., Malyarenko, A. (eds.) Non-commutative and Non-associative Algebra and Analysis Structures, vol. 426, Ch. 19. Springer Proceedings in Mathematics and Statistics, Springer (2023) · Zbl 1535.17022 |
| [52] | Kitouni, A., Silvestrov, S.: On properties and classification of a class of \(4\)-dimensional \(3\)-Hom-Lie algebras with a nilpotent twisting map. arXiv:2304.10674 [math.RA] (2023) |
| [53] | Larsson, D.; Sigurdsson, G.; Silvestrov, SD, Quasi-Lie deformations on the algebra \(\mathbb{F} [t]/(t^N)\), J. Gen. Lie Theory Appl., 2, 201-205 (2008) · Zbl 1220.17008 · doi:10.4303/jglta/S080318 |
| [54] | Larsson, D., Silvestrov, S.D.: Quasi-hom-Lie algebras, central extensions and \(2\)-cocycle-like identities. J. Algebra 288, 321-344 (2005) (Preprints in Mathematical Sciences 2004:3, LUTFMA-5038-2004, Centre for Mathematical Sciences, Lund University (2004)) · Zbl 1099.17015 |
| [55] | Larsson, D., Silvestrov, S.D.: Quasi-Lie algebras. In: Noncommutative Geometry and Representation Theory in Mathematical Physics. Contemp. Math., 391, Amer. Math. Soc., Providence, RI, 241-248 (2005) (Preprints in Mathematical Sciences 2004:30, LUTFMA-5049-2004, Centre for Mathematical Sciences, Lund University (2004)) · Zbl 1105.17005 |
| [56] | Larsson, D.; Silvestrov, SD, Graded quasi-Lie agebras, Czechoslovak J. Phys., 55, 1473-1478 (2005) · doi:10.1007/s10582-006-0028-3 |
| [57] | Larsson, D.; Silvestrov, SD, Quasi-deformations of \(sl_2(\mathbb{F} )\) using twisted derivations, Comm. Algebra, 35, 4303-4318 (2007) · Zbl 1131.17010 · doi:10.1080/00927870701545127 |
| [58] | Larsson, D., Silvestrov, S.D.: On generalized \(N\)-complexes comming from twisted derivations. In: Silvestrov, S., Paal, E., Abramov, V., Stolin, A. (eds.) Generalized Lie Theory in Mathematics, Ch. 7, pp. 81-88. Physics and Beyond, Springer-Verlag (2009) · Zbl 1162.81024 |
| [59] | Ling, W.X.: On the structure of \(n\)-Lie algebras, PhD Thesis. University-GHS-Siegen, Siegen (1993) · Zbl 0841.17002 |
| [60] | Liu, K.Q.: Quantum central extensions. C. R. Math. Rep. Acad. Sci. Canada 13(4), 135-140 (1991) · Zbl 0767.17012 |
| [61] | Liu, KQ, Characterizations of the quantum Witt algebra, Lett. Math. Phys., 24, 4, 257-265 (1992) · Zbl 0759.17007 · doi:10.1007/BF00420485 |
| [62] | Liu, K.Q.: The quantum Witt algebra and quantization of some modules over Witt algebra, PhD Thesis. Department of Mathematics, University of Alberta, Edmonton, Canada (1992) |
| [63] | Mabrouk, S., Ncib, O., Silvestrov, S.: Generalized derivations and Rota-Baxter operators of \(n\)-ary Hom-Nambu superalgebras. Adv. Appl. Clifford Algebras 31, 32 (2021). (arXiv:2003.01080 [math.QA]) · Zbl 1507.17041 |
| [64] | Makhlouf, A., Silvestrov, S.D.: Hom-algebra structures. J. Gen. Lie Theory Appl. 2(2), 51-64 (2008) (Preprints in Mathematical Sciences 2006:10, LUTFMA-5074-2006, Centre for Mathematical Sciences, Lund University (2006)) · Zbl 1184.17002 |
| [65] | Makhlouf, A., Silvestrov, S.: Hom-Lie admissible Hom-coalgebras and Hom-Hopf algebras. In: Silvestrov, S., Paal, E., Abramov, V., Stolin, A. (eds.) Generalized Lie Theory in Mathematics, Physics and Beyond, Ch. 17, pp. 189-206. Springer-Verlag, Berlin, Heidelberg (2009) (Preprints in Mathematical Sciences 2007:25, LUTFMA-5091-2007, Centre for Mathematical Sciences, Lund University (2007). (arXiv:0709.2413 [math.RA] (2007)) · Zbl 1173.16019 |
| [66] | Makhlouf, A., Silvestrov, S.D.: Hom-algebras and Hom-coalgebras. J. Algebra Appl. 9(4), 553-589 (2010) (Preprints in Mathematical Sciences 2008:19, LUTFMA-5103-2008, Centre for Mathematical Sciences, Lund University (2008)). arXiv:0811.0400 [math.RA] (2008) · Zbl 1259.16041 |
| [67] | Makhlouf, A., Silvestrov, S.: Notes on \(1\)-parameter formal deformations of Hom-associative and Hom-Lie algebras. Forum Math. 22(4), 715-739 (2010) (Preprints in Mathematical Sciences 2007:31, LUTFMA-5095-2007, Centre for Mathematical Sciences, Lund University (2007)). arXiv:0712.3130v1 [math.RA] (2007) · Zbl 1201.17012 |
| [68] | Nambu, Y.: Generalized Hamiltonian dynamics. Phys. Rev. D 3(7), 2405-2412 (1973) · Zbl 1027.70503 |
| [69] | Richard, L.; Silvestrov, SD, Quasi-Lie structure of \(\sigma \)-derivations of \(\mathbb{C} [t^{\pm 1}]\), J. Algebra, 319, 3, 1285-1304 (2008) · Zbl 1161.17017 · doi:10.1016/j.jalgebra.2007.09.029 |
| [70] | Rotkiewicz, M., Cohomology ring of \(n\)-Lie algebras, Extracta Math., 20, 219-232 (2005) · Zbl 1163.17300 |
| [71] | Sheng, Y., Representations of hom-Lie algebras, Algebr. Reprensent. Theory, 15, 6, 1081-1098 (2012) · Zbl 1294.17001 · doi:10.1007/s10468-011-9280-8 |
| [72] | Sigurdsson, G., Silvestrov, S.: Lie color and hom-Lie algebras of Witt type and their central extensions. In: Generalized Lie Theory in Mathematics. Physics and Beyond, pp. 247-255. Springer, Berlin (2009) · Zbl 1169.17016 |
| [73] | Sigurdsson, G.; Silvestrov, S., Graded quasi-Lie algebras of Witt type, Czech. J. Phys., 56, 1287-1291 (2006) · Zbl 1109.17014 · doi:10.1007/s10582-006-0439-1 |
| [74] | Takhtajan, LA, On foundation of the generalized Nambu mechanics, Comm. Math. Phys., 160, 2, 295-315 (1994) · Zbl 0808.70015 · doi:10.1007/BF02103278 |
| [75] | Takhtajan, LA, Higher order analog of Chevalley-Eilenberg complex and deformation theory of \(n\)-gebras, St. Petersburg Math. J., 6, 2, 429-438 (1995) |
| [76] | Vainerman, L.; Kerner, R., On special classes of \(n\)-algebras, J. Math. Phys., 37, 5, 2553-2565 (1996) · Zbl 0864.17002 · doi:10.1063/1.531526 |
| [77] | Yau, D.: A Hom-associative analogue of Hom-Nambu algebras. arXiv:1005.2373 [math.RA] (2010) |
| [78] | Yau, D., Enveloping algebras of Hom-Lie algebras, J. Gen. Lie Theory Appl., 2, 2, 95-108 (2008) · Zbl 1214.17001 · doi:10.4303/jglta/S070209 |
| [79] | Yau, D., Hom-algebras and homology, J. Lie Theory, 19, 2, 409-421 (2009) · Zbl 1252.17002 |
| [80] | Yau, D., On \(n\)-ary Hom-Nambu and Hom-Nambu-Lie algebras, J. Geom. Phys., 62, 506-522 (2012) · Zbl 1280.17006 · doi:10.1016/j.geomphys.2011.11.006 |
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