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 Fundam. Prikl. Mat., 2016, Volume 21, Issue 1, Pages 93–104 (Mi fpm1705)

On the additive structure and asymptotics of codimensions $c_n$ in the algebra $F^{(5)}$

A. V. Grishin

Moscow State Pedagogical University

Abstract: In this paper, we investigate the additive structure of the algebra $F^{(5)}$, i.e., a relatively free, associative, countably-generated algebra with the identity $[x_1, …, x_5] = 0$ over an infinite field of characteristic ${\neq} 2,3$. We study the space of proper multilinear polynomials in this algebra and means of basis construction in one of its basic subspaces. As an additional result, we obtain estimations of codimensions $c_n = \operatorname{dim} P_n / P_n \cap T^{(5)}$, where $P_n$ is the space of multilinear polynomials of degree $n$ in $F^{(5)}$ and $T^{(5)}$ is the $T$-ideal generated by the long commutator $[x_1, …, x_5]$.

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English version:
Journal of Mathematical Sciences (New York), 2018, 233:5, 666–674

UDC: 512.552.4

Citation: A. V. Grishin, “On the additive structure and asymptotics of codimensions $c_n$ in the algebra $F^{(5)}$”, Fundam. Prikl. Mat., 21:1 (2016), 93–104; J. Math. Sci., 233:5 (2018), 666–674

Citation in format AMSBIB
\Bibitem{Gri16} \by A.~V.~Grishin \paper On the additive structure and asymptotics of codimensions~$c_n$ in the algebra~$F^{(5)}$ \jour Fundam. Prikl. Mat. \yr 2016 \vol 21 \issue 1 \pages 93--104 \mathnet{http://mi.mathnet.ru/fpm1705} \elib{http://elibrary.ru/item.asp?id=31015376} \transl \jour J. Math. Sci. \yr 2018 \vol 233 \issue 5 \pages 666--674 \crossref{https://doi.org/10.1007/s10958-018-3954-2} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85050938667} 

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This publication is cited in the following articles:
1. A. V. Grishin, “Asymptotics of the Codimensions $c_n$ in the Algebra $F^{(7)}$”, Math. Notes, 104:1 (2018), 22–28
2. A. V. Grishin, “On the measure of inclusion in relatively free algebras with the identity of Lie nilpotency of degree 3 or 4”, Sb. Math., 210:2 (2019), 234–244
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