Geodesic
On almost nilpotent varieties of anticommutative metabelian algebras
Prikladnaâ diskretnaâ matematika, no. 4 (2017), pp. 35-48.

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Let $\Phi$ be a field of characteristic zero. We consider variety of anticommutative metabelian algebras, denoted $\mathbf{MA}$, in which the anticommutativity identity $x_1x_2\equiv-x_2x_1$ and the metabelian identity $(x_1x_2)(x_3x_4)\equiv0$ are satisfied. The associativity of multiplication is not assumed. Numerical invariants of the variety of all anticommutative metabelian algebras are obtained: the sequence of codimensions is $c_n(\mathbf{MA})={n!}/2$. An algorithm for computing the multiplicities of $m_\lambda(\mathbf{MA})$ for $n>2$ is presented. We define a series of anticommutative metabelian algebras $C_m$ for any integer $m\ge2$ and prove the existence of almost nilpotent variety with PI-exponent of $m$. Moreover, two almost nilpotent varieties of subexponential growth are studied. The first variety is the well-known variety of all metabelian Lie algebras, denoted $\mathbf A^2 $, the second – the almost nilpotent variety $\mathbf V_\mathrm{anti}$ generated by the anticommutative metabelian algebra $G$, $\mathbf V_\mathrm{anti}=\operatorname{var}(G)$, which is defined in our investigation. In case of varieties of anticommutative metabelian algebras, it is shown that there are only two almost nilpotent varieties of subexponential growth: $\mathbf A^2$ and $\mathbf V_\mathrm{anti}$. The proofs are based on the theory of irreducible modules, Young diagram and tableau, and some basic notions of the representation theory for the symmetric group. All results are obtained by means of combinatorial methods.
Keywords: polynomial identity, variety, almost nilpotent, codimension growth. 
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O. V. Shulezhko; N. P. Panov. On almost nilpotent varieties of anticommutative metabelian algebras. Prikladnaâ diskretnaâ matematika, no. 4 (2017), pp. 35-48. http://geodesic.mathdoc.fr/item/PDM_2017_4_a1/

[1] Bakhturin Yu. A., Identities of Lie Algebras, Nauka Publ., Moscow, 1985, 448 pp. (in Russian) | MR

[2] Giambruno A., Zaicev M., Polynomial Identities and Asymptotic Methods, AMS, Providence, RI, 2005, 352 pp. | MR | Zbl

[3] Frolova Yu. Yu., Shulezhko O. V., “Almost nilpotent varieties of Leibniz algebras”, Prikladnaya Diskretnaya Matematika, 2015, no. 2(28), 30–36 (in Russian) | DOI

[4] Mishchenko S., Valenti A., “An almost nilpotent variety of exponent 2”, Israel J. Mathematics, 199:1 (2014), 241–257 | DOI | MR | Zbl

[5] Mishchenko S. P., Shulezhko O. V., “On almost nilpotent varieties in the class of commutative metabelian algebras”, Vestnik Samarskogo Gosudarstvennogo Universiteta. Estestvenno-Nauchnaya Seriya, 2015, no. 3(125), 21–28 (in Russian)

[6] Chang N. T. K., Frolova Yu. Yu., “Almost nilpotent commutative metabelian varieties with not greater than exponential growth rate”, Proc. Intern. Conf. “Mal'tsevskie Chteniya”, Novosibirsk, 2014, 119 (in Russian)

[7] Mishchenko S., Valenti A., “On almost nilpotent varieties of subexponential growth”, J. Algebra, 423:1 (2015), 902–915 | DOI | MR | Zbl

[8] Shulezhko O. V., “On almost nilpotent varieties in different classes of linear algebras”, Chebyshevskiy Sbornik, 16:1 (2015), 67–88 (in Russian) | MR

[9] Mishchenko S. P., “Almost nilpotent metabelian varieties of polynomial growth”, Proc. Intern. Conf., Kazan, 2016, 247–248 (in Russian)

[10] Zaycev M. V., Mishchenko S. P., “Colength of varieties of linear algebras”, Mathematical Notes, 79:4 (2006), 511–517 | DOI | DOI | MR | Zbl

[11] Ratseev S. M., “The growth of varieties of Leibniz algebras with nilpotent commutator subalgebra”, Mathematical Notes, 82:1 (2007), 96–103 | DOI | DOI | MR | Zbl

[12] Giambruno A., Mishchenko S., “Degrees of irreducible characters of the symmetric group and exponential growth”, Proc. AMS, 144:3 (2016), 943–953 | DOI | MR | Zbl