On the automorphism groups for some Leibniz algebras of low dimensions
Abstract
UDC 512.554
We study the automorphism groups of Leibniz algebras of low dimensions and obtain complete descriptions of the automorphism groups of Leibniz algebras of dimension 2 and some types of nilpotent Leibniz algebras of dimension 3.
References
Sh.~Ayupov, K.~Kudaybergenov, B.~Omirov, K.~Zhao, Semisimple Leibniz algebras, their derivations and automor-phisms, Linear and Multilinear Algebra, 68, № 10, 2005 – 2019 (2020); DOI:10.1080/03081087.2019.1567674. DOI: https://doi.org/10.1080/03081087.2019.1567674
Sh.~Ayupov, B.~Omirov, I.~Rakhimov, Leibniz algebras: structure and classification, CRC Press, Taylor & Francis Group (2020). DOI: https://doi.org/10.1201/9780429344336
A.~Blokh, On a generalization of the concept of Lie algebra, Dokl. Akad. Nauk SSSR, 165, № 3, 471 – 473 (1965) (in Russian).
J. M.~Casas, M. A.~Insua, M.~Ladra, S.~Ladra, An algorithm for the classification of $3$-dimensional complex Leibniz algebras, Linear Algebra and Appl., 436, № 9, 3747 – 3756 (2012); DOI:10.1016/j.laa.2011.11.039. DOI: https://doi.org/10.1016/j.laa.2011.11.039
C.~Cuvier, Alg`{e}bres de Leibnitz: d'{e}finitions, propri'{e}t'{e}s, Ann. Sci. '{E}c. Norm. Sup'{e}r. (4), 27, № 1, 1 – 45 (1994); DOI:10.24033/asens.1687. DOI: https://doi.org/10.24033/asens.1687
I.~Demir, K. C.~Misra, E.~Stitzinger, On some structures of Leibniz algebras, Recent Advances in Representation Theory, Quantum Groups, Algebraic Geometry, and Related Topics, Contemp. Math., 623, 41 – 54 (2014); DOI:10.1090/conm/623/12456. DOI: https://doi.org/10.1090/conm/623/12456
A. Kh.~Khudoyberdiyev, T. K.~Kurbanbaev, B. A.~Omirov, Classification of three-dimensional solvable $p$-adic Leibniz algebras, $p$-Adic Numbers Ultrametric Anal. Appl., 2, № 3, 207 – 221 (2010); DOI:10.1134/S2070046610030039. DOI: https://doi.org/10.1134/S2070046610030039
L. A.~Kurdachenko, J.~Otal, A. A.~Pypka, Relationships between the factors of the canonical central series of Leibniz algebras, Eur. J. Math., 2, № 2, 565 – 577 (2016); DOI:10.1007/s40879-016-0093-5. DOI: https://doi.org/10.1007/s40879-016-0093-5
L. A.~Kurdachenko, A. A.~Pypka, I. Ya.~Subbotin, On the automorphism groups of some Leibniz algebras, Int. J. Group Theory (to appear); DOI:10.22108/ IJGT.2021.130057.1735.
M.~Ladra, I. M.~Rikhsiboev, R. M.~Turdibaev, Automorphisms and derivations of Leibniz algebras, Ukr. Math. J., 68, № 7, 1062 – 1076 (2016); DOI:10.1007/s11253-016-1277-3. DOI: https://doi.org/10.1007/s11253-016-1277-3
J.-L.~Loday, Cyclic homology, Grundlehren Math. Wiss., 301, Springer Verlag, (1992); DOI:10.1007/978-3-662-11389-9. DOI: https://doi.org/10.1007/978-3-662-21739-9
J.-L.~Loday, Une version non commutative des alg`{e}bres de Lie: les alg`{e}bras de Leibniz, Enseign. Math., 39, 269 – 293 (1993).
J.-L.~Loday, T.~Pirashvili, Universal enveloping algebras of Leibniz algebras and (co)homology, Math. Ann., 296, № 1, 139 – 158 (1993); DOI:10.1007/ BF01445099. DOI: https://doi.org/10.1007/BF01445099
I. S.~Rakhimov, I. M.~Rikhsiboev, M. A.~Mohammed, An algorithm for classifications of three-dimensional Leibniz algebras over arbitrary fields, JP J. Algebra, Number Theory and Appl., 40, № 2, 181 – 198 (2018); DOI:10.17654/NT040020181. DOI: https://doi.org/10.17654/NT040020181
I. M.~Rikhsiboev, I. S.~Rakhimov, Classification of three dimensional complex Leibniz algebras, AIP Conf. Proc., 1450, № 1, 358 – 362 (2012); DOI:10.1063/1.4724168. DOI: https://doi.org/10.1063/1.4724168
V. S.~Yashchuk, On some Leibniz algebras, having small dimension, Algebra and Discrete Math., 27, № 2, 292 – 308 (2019).
Copyright (c) 2022 Леонід Андрійович Курдаченко, Олександр Олександрович Пипка, Тетяна Володимирівна Величко
This work is licensed under a Creative Commons Attribution 4.0 International License.
