Geodesic
On the additive structure and asymptotics of codimensions~$c_n$ in the algebra~$F^{(5)}$
Fundamentalʹnaâ i prikladnaâ matematika, Tome 21 (2016) no. 1, pp. 93-104.

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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 $[x_1, \dots, x_5] = 0$ 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 $[x_1, \dots, x_5]$. 
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A. V. Grishin. On the additive structure and asymptotics of codimensions~$c_n$ in the algebra~$F^{(5)}$. Fundamentalʹnaâ i prikladnaâ matematika, Tome 21 (2016) no. 1, pp. 93-104. http://geodesic.mathdoc.fr/item/FPM_2016_21_1_a8/

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