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Residually solvable extensions of an infinite dimensional filiform Leibniz algebra. (English) Zbl 1487.17004

The fundamental Levi theorem presents the structure of any finite-dimensional Lie algebra. However, there does not exist a result about the classification of solvable and nilpotent Lie algebras, so several methods have been constructed for low-dimensional Lie algebras. In particular, the solvable extension method presented in [V. V. Morozov, Izv. Vyssh. Uchebn. Zaved., Mat. 1958, No. 4(5), 161–171 (1958; Zbl 0198.05501)] is the motivation to study infinite-dimensional Lie and Leibniz algebras in this work.
Concretely, the aim of the authors is to determine the class of all solvable extensions of infinite-dimensional filiform Lie and Leibniz algebras.
In Section 2 they mention the needed definitions (residually nilpotent, potentially solvable, pro-nilpotent, etc.) and show some illustrated examples and useful preliminary results. In the next section they consider infinite-dimensional Lie algebras and prove that the second cohomology group of the extension is trivial (Theorem 3.4). In the subsection 3.2 the extensions of the (non-Lie) Leibniz algebra introduced in [B. A. Omirov, Math. Notes 80, No. 2, 244–253 (2006; Zbl 1148.17002); translation from Mat. Zametki 80, No. 2, 251–261 (2006)] are studied and also a similar result to the previous theorem in Theorem 3.12 is obtained.

MSC:

17A32 Leibniz algebras
17B40 Automorphisms, derivations, other operators for Lie algebras and super algebras
17B56 Cohomology of Lie (super)algebras
17B65 Infinite-dimensional Lie (super)algebras
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[1] Abdurasulov, K. K.; Khalkulova, Kh. A., Solvable Lie algebras with maximal dimension of complementary space to nilradical, Uzbek. Mat. Zh., 1, 90-98 (2018) · Zbl 1474.17014
[2] Abdurasulov, K. K.; Solijanova, G. O., Maximal pro-solvable Lie algebras with maximal positively graded ideals of length \(\frac{ 3}{ 2} \), Bull. Inst. Math., 5, 25-32 (2020)
[3] Ancochea Bermudez, J. M.; Campoamor-Stursberg, R., Cohomologically rigid solvable Lie algebras with a nilradical of arbitrary characteristic sequence, Linear Algebra Appl., 488, 135-147 (2016) · Zbl 1328.17014
[4] Barnes, D. W., On Levi’s theorem for Leibniz algebras, Bull. Aust. Math. Soc., 86, 2, 184-185 (2012) · Zbl 1280.17002
[5] Bosko-Dunbar, L.; Dunbar, J. D.; Hird, J. T.; Stagg, K., Solvable Leibniz algebras with Heisenberg nilradical, Commun. Algebra, 43, 6, 2272-2281 (2015) · Zbl 1378.17004
[6] Casas, J. M.; Ladra, M.; Omirov, B. A.; Karimjanov, I., A classification of solvable Leibniz algebras with null-filiform nilradical, Linear Multilinear Algebra, 61, 6, 758-774 (2013) · Zbl 1317.17003
[7] Casas, J. M.; Ladra, M.; Omirov, B. A.; Karimjanov, I., Classification of solvable Leibniz algebras with naturally graded filiform nilradical, Linear Algebra Appl., 438, 7, 2973-3000 (2013) · Zbl 1300.17003
[8] Campoamor-Stursberg, R., Solvable Lie algebras with an \(\mathbb{N} \)-graded nilradical of maximum nilpotency degree and their invariants, J. Phys. A, 43, 14, 145-202 (2010) · Zbl 1197.17004
[9] Cañete, E. M.; Khudoyberdiyev, A. Kh., The classification of 4-dimensional Leibniz algebras, Linear Algebra Appl., 439, 273-288 (2013) · Zbl 1332.17005
[10] Fialowski, A., Classification of graded Lie algebras with two generators, Vestnik Moskov. Univ. Ser. I Mat. Mekh.. Vestnik Moskov. Univ. Ser. I Mat. Mekh., Mosc. Univ. Math. Bull., 38, 2, 76-79 (1983) · Zbl 0533.17008
[11] Gaybullaev, R. K., Outer derivations of complex solvable Lie algebras with nilradical of maximal rank, Uzbek. Mat. Zh., 4, 40-43 (2020) · Zbl 1488.17042
[12] Gaybullaev, R. K.; Khudoyberdiyev, A. Kh.; Pohl, K., Classification of solvable Leibniz algebras with Abelian nilradical and \((k - 1)\)-dimensional extension, Commun. Algebra, 48, 7, 3061-3078 (2020) · Zbl 1480.17004
[13] Karimjanov, I. A.; Khudoyberdiyev, A. Kh.; Omirov, B. A., Solvable Leibniz algebras with triangular nilradicals, Linear Algebra Appl., 466, 530-546 (2015) · Zbl 1393.17005
[14] Khakimdjanova, K.; Khakimdjanov, Yu., Sur une classe d’algebres de Lie de dimension infinie, Commun. Algebra, 29, 1, 177-191 (2001), (in French) · Zbl 0988.17015
[15] Kudoybergenov, K. K.; Ladra, M.; Omirov, B. A., On Levi-Malcev theorem for Leibniz algebras, Linear Multilinear Algebra, 67, 7, 1471-1482 (2019) · Zbl 1475.17004
[16] Khudoyberdiyev, A. Kh.; Ladra, M.; Omirov, B. A., On solvable Leibniz algebras whose nilradical is a direct sum of null-filiform algebras, Linear Multilinear Algebra, 62, 9, 1220-1239 (2014) · Zbl 1307.17003
[17] Khudoyberdiyev, A. Kh.; Rakhimov, I. S.; Said Husain, Sh. K., On classification of 5-dimensional solvable Leibniz algebras, Linear Algebra Appl., 457, 428-454 (2014) · Zbl 1352.17004
[18] Ladra, M.; Masutova, K. K.; Omirov, B. A., Corrigendum to “Classification of solvable Leibniz algebras with naturally graded filiform nilradical”, Linear Algebra Appl.. Linear Algebra Appl., Linear Algebra Appl., 507, 7, 513-517 (2016) · Zbl 1405.17004
[19] Leger, G.; Luks, E., Cohomology theorems for Borel-like solvable Lie algebras in arbitrary characteristic, Can. J. Math., 24, 1019-1026 (1972) · Zbl 0272.17004
[20] Mamadaliyev, U. X., A rigid solvable Leibniz algebra, Uzbek. Mat. Zh., 3, 70-78 (2013), (in Russian)
[21] Millionshchikov, D. V., Naturally graded Lie algebras of slow growth, Mat. Sb., 210, 6, 111-160 (2019), (in Russian) · Zbl 1435.17015
[22] Millionschikov, D. V., Cohomology of graded Lie algebras of maximal class with coefficients in the adjoint representation, Proc. Steklov Inst. Math., 263, 99-111 (2008) · Zbl 1196.17018
[23] Morozov, V. V., Classification of nilpotent Lie algebras of sixth order, Izv. Vysš. Učebn. Zaved., Mat., 5, 4, 161-171 (1958), (in Russian) · Zbl 0198.05501
[24] Mubarakzjanov, G. M., On solvable Lie algebras, Izv. Vysš. Učebn. Zaved., Mat., 32, 1, 114-123 (1963), (in Russian) · Zbl 0166.04104
[25] Mubarakzjanov, G. M., Classification of real structures of Lie algebras of fifth order, Izv. Vysš. Učebn. Zaved., Mat., 34, 3, 99-106 (1963), (in Russian) · Zbl 0166.04201
[26] Mubarakzjanov, G. M., Classification of solvable Lie algebras of sixth order with a non-nilpotent basis element, Izv. Vysš. Učebn. Zaved., Mat., 35, 4, 104-116 (1963), (in Russian) · Zbl 0166.04202
[27] Mubarakzjanov, G. M., Certain theorems on solvable Lie algebras, Izv. Vysš. Učebn. Zaved., Mat., 55, 6, 95-98 (1966), (in Russian) · Zbl 0204.04103
[28] Ndogmo, J.-C.; Winternitz, P., Generalized Casimir operators of solvable Lie algebras with Abelian nilradicals, J. Phys. A, 27, 8, 2787-2800 (1994) · Zbl 0835.17007
[29] Ndogmo, J.-C.; Winternitz, P., Solvable Lie algebras with Abelian nilradicals, J. Phys. A, 27, 2, 405-423 (1994) · Zbl 0828.17009
[30] Omirov, B. A., Thin Leibniz algebras, Math. Notes, 80, 2, 244-253 (2006) · Zbl 1148.17002
[31] Camacho, L. M.; Omirov, B. A.; Masutova, K. K., Solvable Leibniz algebras with filiform nilradical, Bull. Malays. Math. Sci. Soc., 39, 1, 283-303 (2016) · Zbl 1382.17002
[32] Rubin, J. L.; Winternitz, P., Solvable Lie algebras with Heisenberg ideals, J. Phys. A, 26, 5, 1123-1138 (1993) · Zbl 0773.17004
[33] Shabanskaya, A., Solvable indecomposable extensions of two nilpotent Lie algebras, Commun. Algebra, 44, 8, 3626-3667 (2016) · Zbl 1402.17020
[34] Shabanskaya, A.; Thompson, G., Six-dimensional Lie algebras with a five-dimensional nilradical, J. Lie Theory, 23, 2, 313-355 (2013) · Zbl 1280.17014
[35] Shabanskaya, A.; Thompson, G., Solvable extensions of a special class of nilpotent Lie algebras, Arch. Math. (Brno), 49, 3, 63-81 (2013)
[36] Shabanskaya, A., Right and left solvable extensions of an associative Leibniz algebra, Commun. Algebra, 45, 6, 2633-2661 (2017) · Zbl 1407.17005
[37] Šnobl, L.; Winternitz, P., Classification and Identification of Lie Algebras, CRM Monograph Series, vol. 33 (2014), AMS, 306 pp · Zbl 1331.17001
[38] Šnobl, L.; Winternitz, P., A class of solvable Lie algebras and their Casimir invariants, J. Phys. A, 38, 12, 2687-2700 (2005) · Zbl 1063.22023
[39] Šnobl, L.; Winternitz, P., All solvable extensions of a class of nilpotent Lie algebras of dimension n and degree of nilpotency \(n - 1\), J. Phys. A, 42, 10, Article 105201 pp. (2009) · Zbl 1178.17009
[40] Šnobl, L.; Winternitz, P., Solvable Lie algebras with Borel nilradicals, J. Phys. A, 45, 9, Article 095202 pp. (2012) · Zbl 1302.17020
[41] Weisfeiler, B. Ju., Infinite-dimensional filtered Lie algebras and their connection with graded Lie algebras, Funkc. Anal. Ego Prilož.. Funkc. Anal. Ego Prilož., Funct. Anal. Appl., 2, 1, 88-89 (1968), English translation: · Zbl 0245.17006
[42] Vergne, M., Cohomologie des algébres de Lie nilpotentes. Application á l’étude de la variété des algébres de Lie nilpotentes, Bull. Soc. Math. Fr., 98, 81-116 (1970), (in French) · Zbl 0244.17011
[43] Zel’manov, E. I., On some problems of group theory and Lie algebras, Math. USSR Sb., 66, 1, 159-168 (1990) · Zbl 0693.17006
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