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Algebra Discrete Math., 2012, Volume 14, Issue 1, Pages 49–70 (Mi adm84)  

This article is cited in 1 scientific paper (total in 1 paper)

RESEARCH ARTICLE

Inner automorphisms of Lie algebras related with generic $2\times 2$ matrices

Vesselin Drenskya, Şehmus Fındıkb

a Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, 1113 Sofia, Bulgaria
b Department of Mathematics, Çukurova University, 01330 Balcalı, Adana, Turkey

Abstract: Let $F_m=F_m(\mathrm{var}(sl_2(K)))$ be the relatively free algebra of rank $m$ in the variety of Lie algebras generated by the algebra $sl_2(K)$ over a field $K$ of characteristic $0$. Translating an old result of Baker from 1901 we present a multiplication rule for the inner automorphisms of the completion $\widehat{F_m}$ of $F_m$ with respect to the formal power series topology. Our results are more precise for $m=2$ when $F_2$ is isomorphic to the Lie algebra $L$ generated by two generic traceless $2\times 2$ matrices. We give a complete description of the group of inner automorphisms of $\widehat L$. As a consequence we obtain similar results for the automorphisms of the relatively free algebra $F_m/F_m^{c+1}=F_m(\mathrm{var}(sl_2(K))\cap {\mathfrak N}_c)$ in the subvariety of $\mathrm{var}(sl_2(K))$ consisting of all nilpotent algebras of class at most $c$ in $\mathrm{var}(sl_2(K))$.

Keywords: free Lie algebras, generic matrices, inner automorphisms, Baker–Campbell–Hausdorff formula.

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Bibliographic databases:
MSC: 17B01, 17B30, 17B40, 16R30
Received: 30.04.2012
Revised: 23.05.2012
Language:

Citation: Vesselin Drensky, Şehmus F{\i}nd{\i}k, “Inner automorphisms of Lie algebras related with generic $2\times 2$ matrices”, Algebra Discrete Math., 14:1 (2012), 49–70

Citation in format AMSBIB
\Bibitem{DreFin12}
\by Vesselin~Drensky, \c Sehmus~F{\i}nd{\i}k
\paper Inner automorphisms of Lie algebras related with generic $2\times 2$ matrices
\jour Algebra Discrete Math.
\yr 2012
\vol 14
\issue 1
\pages 49--70
\mathnet{http://mi.mathnet.ru/adm84}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=3052321}
\zmath{https://zbmath.org/?q=an:06256961}


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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. V. Drensky, P. Koshlukov, G. G. Machado, “GK-dimension of $2\times 2$ generic Lie matrices”, Publ. Math.-Debr., 89:1-2 (2016), 125–135  crossref  mathscinet  zmath  isi
  • Algebra and Discrete Mathematics
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