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Sibirsk. Mat. Zh., 2016, Volume 57, Number 4, Pages 866–888 (Mi smj2789)  

This article is cited in 4 scientific papers (total in 4 papers)

The commutator width of some relatively free Lie algebras and nilpotent groups

V. A. Roman'kov

Omsk State University, Omsk, Russia

Abstract: We determine the exact values of the commutator width of absolutely free and free solvable Lie rings of finite rank, as well as free and free solvable Lie algebras of finite rank over an arbitrary field. We calculate the values of the commutator width of free nilpotent and free metabelian nilpotent Lie algebras of rank 2 or of nilpotency class 2 over an arbitrary field. We also find the values of the commutator width for free nilpotent and free metabelian nilpotent Lie algebras of finite rank at least 3 over an arbitrary field in the case that the nilpotency class exceeds the rank at least by 2. In the case of free nilpotent and free metabelian nilpotent Lie rings of arbitrary finite rank, as well as free nilpotent and free metabelian nilpotent Lie algebras of arbitrary finite rank over the field of rationals, we calculate the values of commutator width without any restrictions. It follows in particular that the free or nonabelian free solvable Lie rings of distinct finite ranks, as well as the free or nonabelian free solvable Lie algebras of distinct finite ranks over an arbitrary field are not elementarily equivalent to each other. We also calculate the exact values of the commutator width of free $\mathbb Q$-power nilpotent, free nilpotent, free metabelian, and free metabelian nilpotent groups of finite rank.

Keywords: free (solvable, metabelian, nilpotent, metabelian nilpotent) Lie algebra, free (solvable, metabelian, nilpotent, metabelian nilpotent) Lie ring, free ($\mathbb Q$-power nilpotent, metabelian, nilpotent, metabelian nilpotent) group, commutator width, elementary equivalence.

Funding Agency Grant Number
Russian Foundation for Basic Research 16.01.00577-а
The author was supported by the Russian Foundation for Basic Research (Grant 16.01.00577-a).


DOI: https://doi.org/10.17377/smzh.2016.57.410

Full text: PDF file (404 kB)
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English version:
Siberian Mathematical Journal, 2016, 57:4, 679–695

Bibliographic databases:

UDC: 512.54+512.55+512.57
Received: 22.08.2015

Citation: V. A. Roman'kov, “The commutator width of some relatively free Lie algebras and nilpotent groups”, Sibirsk. Mat. Zh., 57:4 (2016), 866–888; Siberian Math. J., 57:4 (2016), 679–695

Citation in format AMSBIB
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\by V.~A.~Roman'kov
\paper The commutator width of some relatively free Lie algebras and nilpotent groups
\jour Sibirsk. Mat. Zh.
\yr 2016
\vol 57
\issue 4
\pages 866--888
\mathnet{http://mi.mathnet.ru/smj2789}
\crossref{https://doi.org/10.17377/smzh.2016.57.410}
\elib{https://elibrary.ru/item.asp?id=27380081}
\transl
\jour Siberian Math. J.
\yr 2016
\vol 57
\issue 4
\pages 679--695
\crossref{https://doi.org/10.1134/S0037446616040108}
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\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84983657122}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. V. A. Roman'kov, “Solvability of equations in classes of solvable groups and Lie algebras”, Algebra and Logic, 56:3 (2017), 251–255  mathnet  crossref  crossref  mathscinet  isi
    2. E. N. Poroshenko, “Elementary equivalence of partially commutative Lie rings and algebras”, Algebra and Logic, 56:4 (2017), 348–352  mathnet  crossref  crossref  mathscinet  isi
    3. A. A. Konyrhanova, V. A. Romankov, “On solvability of commutator equations in Lie algebras”, Bull. Karaganda Univ-Math., 85:1 (2017), 57–64  isi
    4. D. L. Goncalves, T. Nasybullov, “On groups where the twisted conjugacy class of the unit element is a subgroup”, Commun. Algebr., 47:3 (2019), 930–944  crossref  mathscinet  zmath  isi  scopus
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