# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
 [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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.