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Algebra and Discrete Mathematics, 2012, том 14, выпуск 1, страницы 49–70
(Mi adm84)
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Эта публикация цитируется в 2 научных статьях (всего в 2 статьях)
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
Аннотация:
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 $\mathrm{var}(sl_2(K))$.
Ключевые слова:
free Lie algebras, generic matrices, inner automorphisms, Baker–Campbell–Hausdorff formula.
Поступила в редакцию: 30.04.2012 Исправленный вариант: 23.05.2012
Образец цитирования:
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
Образцы ссылок на эту страницу:
https://www.mathnet.ru/rus/adm84 https://www.mathnet.ru/rus/adm/v14/i1/p49
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