RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
 General information Latest issue Archive Impact factor Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 Algebra Discrete Math.: Year: Volume: Issue: Page: Find

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

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.

Full text: PDF file (246 kB)
References: PDF file   HTML file

Bibliographic databases:
MSC: 17B01, 17B30, 17B40, 16R30
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} 

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