Kupershmidt operators on Hom-Malcev algebras and their deformation. (English) Zbl 1542.17053
Summary: The main feature of Hom-algebras is that the identities defining the structures are twisted by linear maps. The purpose of this paper is to introduce and study a Hom-type generalization of pre-Malcev algebras, called Hom-pre-Malcev algebras. We also introduce the notion of Kupershmidt operators of Hom-Malcev and Hom-pre-Malcev algebras and show the connections between Hom-Malcev and Hom-pre-Malcev algebras using Kupershmidt operators. Hom-pre-Malcev algebras generalize Hom-pre-Lie algebras to the Hom-alternative setting and fit into a bigger framework with a close relationship with Hom-pre-alternative algebras. Finally, we establish a deformation theory of Kupershmidt operators on a Hom-Malcev algebra in consistence with the general principles of deformation theories and introduce the notion of Nijenhuis elements.
MSC:
| 17D30 | (non-Lie) Hom algebras and topics |
| 17B61 | Hom-Lie and related algebras |
| 17D10 | Mal’tsev rings and algebras |
| 17A01 | General theory of nonassociative rings and algebras |
| 17A30 | Nonassociative algebras satisfying other identities |
| 17B10 | Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) |
References:
| [1] | Bai, C. M., A unified algebraic approach to classical Yang-Baxter equation, J. Phys. A: Math. Theor.40(36) (2007) 11073-11082. · Zbl 1118.17008 |
| [2] | Bakayoko, I., Laplacian of Hom-Lie quasi-bialgebras, Int. J. Algebra8(15) (2014) 713-727. |
| [3] | Bakayoko, I., L-modules, L-comodules and Hom-Lie quasi-bialgebras, African Diaspora J. Math.17 (2014) 49-64. · Zbl 1321.16023 |
| [4] | I. Bakayoko, Hom-post-Lie, O-operators and some functors on Hom-algebras, preprint (2016), arXiv 1610.02845. |
| [5] | Bakayoko, I. and Banagoura, M., Bimodules and Rota-Baxter relations, J. Appl. Mech. Eng.4(5) (2015) 1000178. |
| [6] | Bakayoko, I. and Silvestrov, S., Multiplicative n-hom-Lie color algebras, in Algebraic Structures and Applications, eds. Silvestrov, S. and Malyarenko, A., , Vol. 317 (Springer, 2020), pp. 159-187, arXiv:1912.10216[math.QA]. · Zbl 1479.17010 |
| [7] | Bakayoko, I. and Silvestrov, S., Hom-left-symmetric color dialgebras, Hom-tri-dendri-form color algebras and Yaus twisting generalizations, Afr. Mat.32 (2021) 941-958, arXiv:1912.01441[math.RA]. · Zbl 1488.17059 |
| [8] | Balavoine, D., Deformations of algebras over a quadratic operad, in Operads: Proceedings of Renaissance Conferences (Hartford, CT/Luminy, , 1995), , Vol. 202 (American Mathematical Society, Providence, RI, 1997), pp. 207-234. · Zbl 0883.17004 |
| [9] | Burde, D., Simple left-symmetric algebras with solvable Lie algebra, Manuscripta Math.95 (1998) 397-411. · Zbl 0907.17008 |
| [10] | Caenepeel, S. and Goyvaerts, I., Monoidal Hom-Hopf Algebras, Comm. Algebra39(6) (2011) 2216-2240. · Zbl 1255.16032 |
| [11] | Carinena, J., Grabowski, J. and Marmo, G., Quantum bi-Hamiltonian systems, Int. J. Mod. Phys. A15(30) (2000) 4797-4810. · Zbl 1002.81026 |
| [12] | Chtioui, T., Mabrouk, S. and Makhlouf, A., BiHom-alternative, BiHom-Malcev and BiHom-Jordan algebras, Rocky Mountain J. Math.50(1) (2020) 69-90. · Zbl 1484.17051 |
| [13] | Chtioui, T., Mabrouk, S. and Makhlouf, A., BiHom-pre-alternative algebras and BiHom-alternative quadri-algebras, Bull. Math. Soc. Sci. Math. Roumanie.63 (111)(1) (2020) 3-21. · Zbl 1513.17023 |
| [14] | M. L. Dassoundo and S. Silvestrov, Nearly associative and nearly Hom-associative algebras and bialgebras, preprint (2021), arXiv:2101.12377 [math.RA]. |
| [15] | Dorfman, I. Y., Dirac Structures and Integrability of Nonlinear Evolution Equations (John Wiley, 1993). |
| [16] | Fox, T. F., An introduction to algebraic deformation theory, J. Pure Appl. Algebra84 (1993) 17-41. · Zbl 0772.18006 |
| [17] | Gerstenhaber, M., On the deformation of rings and algebras, Ann. Math.79 (1964) 59-103. · Zbl 0123.03101 |
| [18] | Gerstenhaber, M., On the deformation of rings and algebras II, Ann. Math.84 (1966) 1-19. · Zbl 0147.28903 |
| [19] | Gerstenhaber, M., On the deformation of rings and algebras III, Ann. Math.88 (1968) 1-34. · Zbl 0182.05902 |
| [20] | Gerstenhaber, M., On the deformation of rings and algebras IV., Ann. Math.99 (1974) 257-276. · Zbl 0281.16016 |
| [21] | Gürsey, F. and Tze, C.-H., On the Role of Division, Jordan and Related Algebras in Particle Physics (World Scientific, 1996). · Zbl 0956.81502 |
| [22] | J. T. Hartwig, D. Larsson and S. D. Silvestrov, Deformations of Lie algebras using \(\sigma \)-derivations, J. Algebra295(2) (2006) 314-361; Preprints in Mathematical Sciences 2003:32, LUTFMA-5036-2003, Centre for Mathematical Sciences, Department of Mathematics, Lund University, 52 pp (2003). · Zbl 1138.17012 |
| [23] | Hu, Y. W., Liu, J. F. and Sheng, Y. H., Kupershmidt-(dual-)Nijenhuis structures on a Lie algebra with a representation, J. Math. Phys.59 (2018) 081702. · Zbl 1398.17010 |
| [24] | M. Kontsevich and Y. Soibelman, Deformation theory. I (2010), http://www.math.ksu.edu/soibel/Book-vol1.ps, 2010. |
| [25] | Kerdman, F. S., Analytic Moufang loops in the large, Algebra Logic18 (1980) 325-347. · Zbl 0457.22002 |
| [26] | Kupershmidt, B. A., What a Classical r-Matrix Really Is, J. Nonlin. Math. Phys.6(4) (1999) 448-488. · Zbl 1015.17015 |
| [27] | Kuzmin, E. N., The connection between Malcev algebras and analytic Moufang loops, Algebra Logic10 (1971) 1-14. · Zbl 0248.17001 |
| [28] | Kuzmin, E. N., Malcev algebras and their representations, Algebra Logic7 (1968) 233-244. · Zbl 0204.36104 |
| [29] | D. Larsson and S. D. Silvestrov, Quasi-hom-Lie algebras, central extensions and 2-cocycle-like identities, J. Algebra288 (2005) 321-344; Preprints in Mathematical Sciences, 2004:3, LUTFMA-5038-2004, Centre for Mathematical Sciences, Lund University (2004). · Zbl 1099.17015 |
| [30] | D. Larsson and S. D. Silvestrov, Quasi-Lie algebras, in Noncommutative Geometry and Representation Theory in Mathematical Physics, eds. J. Fuchs, J. Mickelsson, G. Rozemnblioum, A. Stolin and A. Westerberg, Contemporary Mathematics, Vol. 391 (American Mathematical Society, Providence, RI, 2005), pp. 241-248; Preprints in Mathematical Sciences, 2004:30, LUTFMA-5049-2004, Centre for Mathematical Sciences, Lund University (2004). · Zbl 1105.17005 |
| [31] | Larsson, D. and Silvestrov, S. D., Graded quasi-Lie agebras, Czechoslovak J. Phys.55(11) (2005) 1473-1478. |
| [32] | Larsson, D., Sigurdsson, G. and Silvestrov, S. D., Quasi-Lie deformations on the algebra \(\mathbb{F}[t]/( t^N)\), J. Gen. Lie Theory Appl.2(3) (2008) 201-205. · Zbl 1220.17008 |
| [33] | D. Larsson and S. D. Silvestrov, Quasi-deformations of \(s l_2(\mathbb{F})\) using twisted derivations, Comm. Algebra35(12) (2007) 4303-4318; Preprints in Mathematical Sciences, 2004:26, LUTFMA-5047-2004, Centre for Mathematical Sciences, Lund University (2004), arXiv:math/0506172 [math.RA] (2005). · Zbl 1131.17010 |
| [34] | Larsson, D. and Silvestrov, S. D., On generalized N-complexes comming from twisted derivations, in Generalized Lie Theory in Mathematics, Physics and Beyond, eds. Silvestrov, S., Paal, E., Abramov, V. and Stolin, A. (Springer-Verlag, 2009), pp. 81-89. · Zbl 1151.22002 |
| [35] | Loday, J.-L. and Vallette, B., Algebraic Operads (Springer, 2012). · Zbl 1260.18001 |
| [36] | T. Ma, A. Makhlouf and S. Silvestrov, Curved \(\mathcal{O} \)-operator systems, preprint (2017), arXiv:1710.05232 . |
| [37] | T. Ma, A. Makhlouf and S. Silvestrov, Rota-Baxter bisystems and covariant bialgebras, preprint (2017), arXiv:1710.05161 . |
| [38] | Ma, T., Makhlouf, A. and Silvestrov, S., Rota-Baxter cosystems and coquasitriangular mixed bialgebras, J. Algebra Appl.20(4) (2021) 2150064. · Zbl 1476.16030 |
| [39] | Madariaga, S., Splitting of operations for alternative and Malcev structures, Comm. Algebra45(1) (2014) 183-197. · Zbl 1418.17004 |
| [40] | Mabrouk, S., Deformation of Kupershmidt operators and Kupershmidt-Nijenhuis structures of a Malcev algebra, Hacet. J. Math. Stat.51(1) (2021) 199-217. · Zbl 1499.17020 |
| [41] | Mabrouk, S., Ncib, O. and Silvestrov, S., Generalized derivations and Rota-Baxter operators of n-ary Hom-Nambu superalgebras, Adv. Appl. Clifford Algebra31 (2021) 32. · Zbl 1507.17041 |
| [42] | Makhlouf, A., Hom-alternative algebras and Hom-Jordan algebras, Int. Elect. J. Algebra8 (2010) 177-190. · Zbl 1335.17018 |
| [43] | Makhlouf, A., Hom-dendriform algebras and Rota-Baxter Hom-algebras, in Operads and Universal Algebra, eds. Bai, C., Guo, L. and Loday, J.-L., , Vol. 9 (World Scientific Publication, 2012), pp. 147-171. · Zbl 1351.17035 |
| [44] | A. Makhlouf and S. D. Silvestrov, Hom-algebra structures, J. Gen. Lie Theory Appl.2(2) (2008) 51-64; Preprints in Mathematical Sciences 2006:10, LUTFMA-5074-2006, Centre for Mathematical Sciences, Lund University (2006). · Zbl 1184.17002 |
| [45] | Makhlouf, A. and Silvestrov, S. D., Hom-algebras and Hom-coalgebras, J. Algebra Appl.9(4) (2010) 553-589. · Zbl 1259.16041 |
| [46] | A. Makhlouf and S. Silvestrov, Notes on 1-parameter formal deformations of Hom-associative and Hom-Lie algebras, Forum Math. 22(4) (2010) 715-739; 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 |
| [47] | Makhlouf, A. and Yau, D., Rota-Baxter Hom-Lie admissible algebras, Comm. Algebra23(3) (2014) 1231-1257. · Zbl 1367.17003 |
| [48] | Malcev, A. I., Analytic loops, Mat. Sb.36(3) (1955) 569-576. · Zbl 0065.00702 |
| [49] | Mishra, S. K. and Naolekar, A., \( \mathcal{O} \)-operator on Hom-Lie algebras, J. Math. Phys.61(12) (2020) 121701. · Zbl 1477.17070 |
| [50] | Myung, H. C., Malcev-Admissible Algebras, , Vol. 64 (Birkhäuser, 1986). · Zbl 0609.17009 |
| [51] | Nagy, P. T., Moufang loops and Malcev algebras, Sem. Sophus Lie3 (1993) 65-68. · Zbl 0786.22006 |
| [52] | Nijenhuis, A. and Richardson, R. W., Cohomology and deformations in graded Lie algebras, Bull. Amer. Math. Soc.72 (1966) 1-29. · Zbl 0136.30502 |
| [53] | Nijenhuis, A. and Richardson, R. W., Deformation of Malcev algebra structures, J. Math. Mech.17 (1967) 89-105. · Zbl 0166.30202 |
| [54] | Nijenhuis, A. and Richardson, R. W., Commutative algebra cohomology and deformations of Lie and associative algebras, J. Algebra9 (1968) 42-105. · Zbl 0175.31302 |
| [55] | Okubo, S., Introduction to Octonion and Other Non-Associative Algebras in Physics (Cambridge University Press, 1995). · Zbl 0841.17001 |
| [56] | L. Richard and S. D. Silvestrov, Quasi-Lie structure of \(\sigma \)-derivations of \(\mathbb{C}[ t^{\pm 1}]\), J. Algebra319(3) (2008) 1285-1304; Preprints in Mathematical Sciences, 2006:12, LUTFMA-5076-2006, Centre for Mathematical Sciences, Lund University (2006), arXiv:math/0608196[math.QA] (2006). · Zbl 1161.17017 |
| [57] | Richard, L. and Silvestrov, S., A note on quasi-Lie and Hom-Lie structures of \(\sigma \)-derivations of \(\mathbb{C}[ z_1^{\pm 1},\ldots, z_n^{\pm 1}]\), in Generalized Lie Theory in Mathematics, Physics and Beyond, eds. Silvestrov, S., Paal, E., Abramov, V. and Stolin, A. (Springer-Verlag, 2009), pp. 257-262. · Zbl 1169.17018 |
| [58] | N. Saadaou and S. Silvestrov, On \((\lambda,\mu,\gamma)\)-derivations of BiHom-Lie algebras, preprint (2020), arXiv:2010.09148 . |
| [59] | Sabinin, L. V., Smooth Quasigroups and Loops (Kluwer Academic Publication, 1999). · Zbl 0939.20064 |
| [60] | Sagle, A. A., Malcev algebras, Trans. Amer. Math. Soc.101 (1961) 426-458. · Zbl 0101.02302 |
| [61] | Sigurdsson, G. and Silvestrov, S. D., Graded quasi-Lie algebras of Witt type, CzechoslovakJ. Phys.56 (2006) 1287-1291. · Zbl 1109.17014 |
| [62] | Sigurdsson, G. and Silvestrov, S., Lie color and Hom-Lie algebras of Witt type and their central extensions, in Generalized Lie Theory in Mathematics, Physics and Beyond, eds. Silvestrov, S., Paal, E., Abramov, V. and Stolin, A. (Springer-Verlag, Berlin, Heidelberg, 2009), pp. 247-255. · Zbl 1169.17016 |
| [63] | Silvestrov, S., Paradigm of quasi-Lie and quasi-Hom-Lie algebras and quasi-defor-mations, in New Techniques in Hopf Algebras and Graded Ring Theory, ed. Vlaam, K. (Academic van Belgie Wet.Kunsten Brussels, 2007), pp. 165-177. · Zbl 1138.17301 |
| [64] | S. Silvestrov and C. Zargeh, HNN-extension of involutive multiplicative Hom-Lie algebras, preprint (2021), arXiv:2101.01319 [math.RA]. |
| [65] | Sun, Q., On Hom-Prealternative Bialgebras, Algebra Represent. Theory19 (2016) 657-677. · Zbl 1392.17003 |
| [66] | Sun, Q., Kupershmidt-(dual-)Nijenhuis structure on the alternative algebra with a representation.J. Algebra Appl.20(6) (2021) 2150097. · Zbl 1492.16037 |
| [67] | Yau, D., Hom-Malcev, Hom-alternative, and Hom-Jordan algebras, Int. Electron. J. Algebra11 (2012) 177-217. · Zbl 1258.17003 |
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