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 Mat. Sb., 2016, Volume 207, Number 12, Pages 54–72 (Mi msb8652)

Proper central and core polynomials of relatively free associative algebras with identity of Lie nilpotency of degrees 5 and 6

A. V. Grishina, S. V. Pchelintsevb

a Moscow State Pedagogical University
b Financial University under the Government of the Russian Federation, Moscow

Abstract: We study the centre of a relatively free associative algebra $F^{(n)}$ with the identity $[x_1,…,x_n]=0$ of Lie nilpotency of degree $n=5,6$ over a field of characteristic 0. It is proved that the core $Z^*(F^{(5)})$ of the algebra $F^{(5)}$ (the sum of all ideals of $F^{(5)}$ contained in its centre) is generated as a $\mathrm T$-ideal by the weak Hall polynomial $[[x,y]^{2},y]$. It is also proved that every proper central polynomial of $F^{(5)}$ is contained in the sum of $Z^*(F^{(5)})$ and the $\mathrm T$-space generated by $[[x,y]^{2}, z]$ and the commutator $[x_1,…, x_4]$ of degree 4. This implies that the centre of $F^{(5)}$ is contained in the $\mathrm T$-ideal generated by the commutator of degree 4.
Similar results are obtained for $F^{(6)}$; in particular, it is proved that the core $Z^{*}(F^{(6)})$ is generated as a $\mathrm T$-ideal by the commutator of degree 5.
Bibliography: 15 titles.

Keywords: identities of Lie nilpotency of degrees 5 and 6, centre, core, proper polynomial, extended Grassmann algebra, superalgebra, Grassmann hull, Hall polynomials.

 Funding Agency Grant Number Russian Foundation for Basic Research 16-01-00756-a This research was supported by the Russian Foundation for Basic Research (grant no. 16-01-00756-a).

Author to whom correspondence should be addressed

DOI: https://doi.org/10.4213/sm8652

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English version:
Sbornik: Mathematics, 2016, 207:12, 1674–1692

Bibliographic databases:

UDC: 512.552.4
MSC: Primary 16R10; Secondary 16R40

Citation: A. V. Grishin, S. V. Pchelintsev, “Proper central and core polynomials of relatively free associative algebras with identity of Lie nilpotency of degrees 5 and 6”, Mat. Sb., 207:12 (2016), 54–72; Sb. Math., 207:12 (2016), 1674–1692

Citation in format AMSBIB
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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. A. V. Grishin, “On the additive structure and asymptotics of codimensions $c_n$ in the algebra $F^{(5)}$”, J. Math. Sci., 233:5 (2018), 666–674
2. S. V. Pchelintsev, “Identities of metabelian alternative algebras”, Siberian Math. J., 58:4 (2017), 693–710
3. S. V. Pchelintsev, “Identities of the model algebra of multiplicity 2”, Siberian Math. J., 59:6 (2018), 1105–1124
4. 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
5. Murakami L.S.I., Pchelintsev S.V., Shashkov O.V., “Finite-Dimensional Right Alternative Superalgebras With Semisimple Strongly Alternative Even Part”, J. Algebra, 528 (2019), 150–176
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