Fundamentalʹnaâ i prikladnaâ matematika, Tome 21 (2016) no. 1, pp. 93-104
Citer cet article
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/
@article{FPM_2016_21_1_a8,
author = {A. V. Grishin},
title = {On the additive structure and asymptotics of codimensions~$c_n$ in the algebra~$F^{(5)}$},
journal = {Fundamentalʹna\^a i prikladna\^a matematika},
pages = {93--104},
year = {2016},
volume = {21},
number = {1},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/FPM_2016_21_1_a8/}
}
TY - JOUR
AU - A. V. Grishin
TI - On the additive structure and asymptotics of codimensions $c_n$ in the algebra $F^{(5)}$
JO - Fundamentalʹnaâ i prikladnaâ matematika
PY - 2016
SP - 93
EP - 104
VL - 21
IS - 1
UR - http://geodesic.mathdoc.fr/item/FPM_2016_21_1_a8/
LA - ru
ID - FPM_2016_21_1_a8
ER -
%0 Journal Article
%A A. V. Grishin
%T On the additive structure and asymptotics of codimensions $c_n$ in the algebra $F^{(5)}$
%J Fundamentalʹnaâ i prikladnaâ matematika
%D 2016
%P 93-104
%V 21
%N 1
%U http://geodesic.mathdoc.fr/item/FPM_2016_21_1_a8/
%G ru
%F FPM_2016_21_1_a8
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 $[x_1, \dots, x_5]$.
[1] Volichenko I. B., $T$-ideal, porozhdennyi elementom $[x_1, x_2, x_3, x_4]$, Preprint, Institut matematiki AN BSSR, Minsk, 1978
[2] Grishin A. V., Tsybulya L. M., Shokola A. A., “O T-prostranstvakh i sootnosheniyakh v otnositelno svobodnykh lievski nilpotentnykh assotsiativnykh algebrakh”, Fundament. i prikl. matem., 16:3 (2010), 135–148 | MR
[3] Grishin A. V., Pchelintsev S. V., “O tsentrakh otnositelno svobodnykh assotsiativnykh algebr s tozhdestvom lievoi nilpotentnosti”, Matem. sb., 206:11 (2015), 113–130 | DOI | MR | Zbl
[4] Grishin A. V., Pchelintsev S. V., “Sobstvennye tsentralnye i yadernye mnogochleny otnositelno svobodnykh assotsiativnykh algebr s tozhdestvom lievoi nilpotentnosti stepeni $5$ i $6$”, Matem. sb., 207:12 (2016), 54–72 | DOI | MR | Zbl
[5] Giambruno A., Mishchenko S., Zaicev M., “Codimensions of algebras and growth functions”, Adv. Math., 217 (2008), 1027–1052 | DOI | MR | Zbl
