Geodesic
Necessary and sufficient conditions for a variety of Leibniz algebras to have polynomial growth
Fundamentalʹnaâ i prikladnaâ matematika, Tome 12 (2006) no. 8, pp. 207-215 
S. P. Mishchenko; O. I. Cherevatenko. Necessary and sufficient conditions for a variety of Leibniz algebras to have polynomial growth. Fundamentalʹnaâ i prikladnaâ matematika, Tome 12 (2006) no. 8, pp. 207-215. http://geodesic.mathdoc.fr/item/FPM_2006_12_8_a10/
@article{FPM_2006_12_8_a10,
     author = {S. P. Mishchenko and O. I. Cherevatenko},
     title = {Necessary and sufficient conditions for a~variety of {Leibniz} algebras to have polynomial growth},
     journal = {Fundamentalʹna\^a i prikladna\^a matematika},
     pages = {207--215},
     year = {2006},
     volume = {12},
     number = {8},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FPM_2006_12_8_a10/}
}
TY  - JOUR
AU  - S. P. Mishchenko
AU  - O. I. Cherevatenko
TI  - Necessary and sufficient conditions for a variety of Leibniz algebras to have polynomial growth
JO  - Fundamentalʹnaâ i prikladnaâ matematika
PY  - 2006
SP  - 207
EP  - 215
VL  - 12
IS  - 8
UR  - http://geodesic.mathdoc.fr/item/FPM_2006_12_8_a10/
LA  - ru
ID  - FPM_2006_12_8_a10
ER  - 
%0 Journal Article
%A S. P. Mishchenko
%A O. I. Cherevatenko
%T Necessary and sufficient conditions for a variety of Leibniz algebras to have polynomial growth
%J Fundamentalʹnaâ i prikladnaâ matematika
%D 2006
%P 207-215
%V 12
%N 8
%U http://geodesic.mathdoc.fr/item/FPM_2006_12_8_a10/
%G ru
%F FPM_2006_12_8_a10

Voir la notice de l'article provenant de la source Math-Net.Ru

We study the behaviour of the codimension sequence of polynomial identities of Leibniz algebras over a field of characteristic 0. We prove that a variety $\mathbf V$ has polynomial growth if and only if the condition $$ \mathbf N_2\mathbf A,\widetilde{\mathbf V_1}\not\subset\mathbf V\subset\widetilde{\mathbf N_c\mathbf A} $$ holds, where $\mathbf N_2\mathbf A$ is the variety of Lie algebras defined by the identity $$ (x_1x_2)(x_3x_4)(x_5x_6)\equiv 0, $$ $\widetilde{\mathbf V_1}$ is the variety of Leibniz algebras defined by the identity $$ x_1(x_2x_3)(x_4x_5)\equiv 0, $$ and $\widetilde{\mathbf N_c\mathbf A}$ is the variety of Leibniz algebras defined by the identity $$ (x_1x_2)\cdots(x_{2c+1}x_{2c+2})\equiv 0. $$

[1] Benediktovich I. I., Zalenskii A. E., “T-idealy svobodnykh algebr Li s polinomialnym rostom posledovatelnosti korazmernostei”, Izv. AN BSSR, 1980, no. 3, 5–10 | MR

[2] Blokh A. M., “Ob odnom obobschenii ponyatiya algebry Li”, DAN SSSR, 18:3 (1965), 471–473

[3] Mischenko S. P., “O mnogoobraziyakh polinomialnogo rosta algebr Li nad polem kharakteristiki nul”, Mat. zametki, 40:6 (1986), 713–721 | MR

[4] Mischenko S. P., “Mnogoobraziya algebr Li s dvustupenno nilpotentnym kommutantom”, Vestsi AN BSSR: Ser. fiz. mat. nauk, 1987, no. 6, 39–43

[5] Mischenko S. P., “Rost mnogoobrazii algebr Li”, Uspekhi mat. nauk, 45:6 (1990), 25–45

[6] Mischenko S. P., Cherevatenko O. I., “Mnogoobraziya algebr Leibnitsa slabogo rosta”, Kazanskoe matematicheskoe obschestvo. Trudy Matematicheskogo tsentra im. N. I. Lobachevskogo, 31 (2005), 103–104

[7] Mischenko S. P., Cherevatenko O. I., “Mnogoobraziya algebr Leibnitsa slabogo rosta”, Vestnik Samarskogo gosudarstvennogo universiteta, 2006

[8] Mishchenko S., Valenti A., “A Leibniz variety with almost polynomial growth”, J. Pure Appl. Algebra, 202:1–3 (2005), 82–101 | DOI | MR | Zbl