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Necessary and sufficient conditions for a variety of Leibniz algebras to have polynomial growth. (English. Russian original) Zbl 1184.17003
J. Math. Sci., New York 152, No. 2, 282-287 (2008); translation from Fundam. Prikl. Mat. 12, No. 8, 207-215 (2006).
Summary: We study the behavior of the codimension sequence of polynomial identities of Leibniz algebras over a field of characteristic 0. We prove that a variety V has polynomial growth if and only if the condition \[ N_2 A,\widetilde{V_1 } \not\subset V \subset \widetilde{N_c A} \] holds, where \(N_2A\) is the variety of Lie algebras defined by the identity \((x_1 x_2)(x_3 x_4)(x_5 x_6)\equiv 0,\widetilde{V_1}\) is the variety of Leibniz algebras defined by the identity \(x_1(x_2 x_3)(x_4 x_5)\equiv 0\), and \(\widetilde{N_cA}\) is the variety of Leibniz algebras defined by the identity \((x_1 x_2)(x_{2c+1}x_{2c+2})\equiv 0\).

MSC:
17A32 Leibniz algebras
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[1] I. I. Benediktovic and A. E. Zalesskii, ”T-ideals of free Lie algebras with polynomial growth of the sequence of codimensions,” Izv. Akad. Nauk BSSR. Ser. Fiz.-Mat. Nauk, 3, 5–10 (1980).
[2] A. Blokh, ”A generalization of the concept of a Lie algebra,” Sov. Math. Dokl., 6, 1450–1452 (1966).
[3] S. P. Mishchenko, ”Varieties of polynomial growth of Lie algebras over a field of characteristic zero,” Math. Notes, 40, 901–905 (1986). · Zbl 0624.17007
[4] S. P. Mishchenko, ”Varieties of Lie algebras with two-step nilpotent commutant,” Izv. Akad. Nauk BSSR. Ser. Fiz.-Mat. Nauk, 6, 39–43 (1987). · Zbl 0646.17013
[5] S. P. Mishchenko, ”Growth in varieties of Lie algebras,” Russ. Math. Surv., 45, No. 6, 27–52 (1990). · Zbl 0718.17004 · doi:10.1070/RM1990v045n06ABEH002710
[6] S. P. Mishchenko and O. Cherevatenco, ”A Leibniz variety with weak growth,” Kazanskoe Mat. Obshch. Tr. Mat. Centra im. N. I. Lobachevskogo, 31, 103–104 (2005).
[7] S. P. Mishchenko and O. Cherevatenco, ”A Leibniz variety with weak growth,” Vestn. SamGU (2006).
[8] S. Mishchenko and A. Valenti, ”A Leibniz variety with almost polynomial growth,” J. Pure Appl. Algebra, 202, No. 1–3, 82–101 (2005). · Zbl 1179.17002 · doi:10.1016/j.jpaa.2005.01.013
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