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 Sibirsk. Mat. Zh., 2007, Volume 48, Number 1, Pages 116–137 (Mi smj11)

Graded Lie algebras with few nontrivial components

N. Yu. Makarenko

Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences

Abstract: We prove that if a $(\mathbb Z/n\mathbb Z)$-graded Lie algebra $L=\bigoplus\limits_{i=0}^{n-1}L_i$ has $d$ nontrivial components $L_i$ and the null component $L_0$ has finite dimension $m$, then $L$ has a homogeneous solvable ideal of derived length bounded by a function of $d$ and of codimension bounded by a function of $m$ and $d$. An analogous result holds also for the $(\mathbb Z/n\mathbb Z)$-graded Lie rings $L=\bigoplus\limits_{i=0}^{n-1}L_i$ with few nontrivial components $L_i$ if the null component $L_0$ has finite order $m$. These results generalize Kreknin's theorem on the solvability of the $(\mathbb Z/n\mathbb Z)$-graded Lie rings $L=\bigoplus\limits_{i=0}^{n-1}L_i$ with trivial component $L_0$ and Shalev's theorem on the solvability of such Lie rings with few nontrivial components $L_i$. The proof is based on the method of generalized centralizers which was created by E. I. Khukhro for Lie rings and nilpotent groups with almost regular automorphisms of prime order [1], as well as on the technique developed in the work of N. Yu. Makarenko and E. I. Khukhro on the almost solvability of Lie algebras with an almost regular automorphism of finite order [2].

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English version:
Siberian Mathematical Journal, 2007, 48:1, 95–111

Bibliographic databases:

UDC: 512.5

Citation: N. Yu. Makarenko, “Graded Lie algebras with few nontrivial components”, Sibirsk. Mat. Zh., 48:1 (2007), 116–137; Siberian Math. J., 48:1 (2007), 95–111

Citation in format AMSBIB
\Bibitem{Mak07} \by N.~Yu.~Makarenko \paper Graded Lie algebras with few nontrivial components \jour Sibirsk. Mat. Zh. \yr 2007 \vol 48 \issue 1 \pages 116--137 \mathnet{http://mi.mathnet.ru/smj11} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2304883} \zmath{https://zbmath.org/?q=an:1164.17018} \elib{http://elibrary.ru/item.asp?id=15415038} \transl \jour Siberian Math. J. \yr 2007 \vol 48 \issue 1 \pages 95--111 \crossref{https://doi.org/10.1007/s11202-007-0011-7} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000244424100011} \elib{http://elibrary.ru/item.asp?id=13539161} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-33846611855} 

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This publication is cited in the following articles:
1. Khukhro EI, “Graded Lie rings with many commuting components and an application to 2-Frobenius groups”, Bulletin of the London Mathematical Society, 40:Part 5 (2008), 907–912
2. Khukhro EI, Makarenko NY, Shumyatsky P, “Nilpotent ideals in graded Lie algebras and at-most constant-free derivations”, Communications in Algebra, 36:5 (2008), 1869–1882
3. Khukhro EI, “Groups with an automorphism of prime order that is almost regular in the sense of rank”, Journal of the London Mathematical Society-Second Series, 77:Part 1 (2008), 130–148
4. E. I. Khukhro, “Koltsa Li s konechnoi tsiklicheskoi graduirovkoi, v kotoroi mnogo kommutiruyuschikh komponent”, Sib. elektron. matem. izv., 6 (2009), 243–250
5. Khukhro E.I., Shumyatsky P., “Nilpotency of finite groups with Frobenius groups of automorphisms”, Monatsh Math, 163:4 (2011), 461–470
6. Caldeira J. de Melo E. Shumyatsky P., “On Groups and Lie Algebras Admitting a Frobenius Group of Automorphisms”, J. Pure Appl. Algebr., 216:12 (2012), 2730–2736
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