Inner automorphisms of Lie algebras related with generic \(2\times 2\) matrices
Let \(F_m=F_m(\text{var}(sl_2(K)))\) 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 t...
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| Дата: | 2018 |
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| Формат: | Стаття |
| Мова: | English |
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Lugansk National Taras Shevchenko University
2018
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| Онлайн доступ: | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/711 |
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| Назва журналу: | Algebra and Discrete Mathematics |
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oai:ojs.admjournal.luguniv.edu.ua:article-7112018-04-04T09:58:22Z Inner automorphisms of Lie algebras related with generic \(2\times 2\) matrices Drensky, Vesselin Fındık, Şehmus free Lie algebras, generic matrices, inner automorphisms,Baker-Campbell-Hausdorff formula 17B01, 17B30, 17B40, 16R30 Let \(F_m=F_m(\text{var}(sl_2(K)))\) 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 \(F_m/F_m^{c+1}=F_m(\text{var}(sl_2(K))\cap {\mathfrak N}_c)\) in the subvariety of \(\text{var}(sl_2(K))\) consisting of all nilpotent algebras of class at most \(c\) in \(\text{var}(sl_2(K))\). Lugansk National Taras Shevchenko University 2018-04-04 Article Article Peer-reviewed Article application/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/view/711 Algebra and Discrete Mathematics; Vol 14, No 1 (2012) 2415-721X 1726-3255 en https://admjournal.luguniv.edu.ua/index.php/adm/article/view/711/244 Copyright (c) 2018 Algebra and Discrete Mathematics |
| institution |
Algebra and Discrete Mathematics |
| collection |
OJS |
| language |
English |
| topic |
free Lie algebras generic matrices inner automorphisms,Baker-Campbell-Hausdorff formula 17B01 17B30 17B40 16R30 |
| spellingShingle |
free Lie algebras generic matrices inner automorphisms,Baker-Campbell-Hausdorff formula 17B01 17B30 17B40 16R30 Drensky, Vesselin Fındık, Şehmus Inner automorphisms of Lie algebras related with generic \(2\times 2\) matrices |
| topic_facet |
free Lie algebras generic matrices inner automorphisms,Baker-Campbell-Hausdorff formula 17B01 17B30 17B40 16R30 |
| format |
Article |
| author |
Drensky, Vesselin Fındık, Şehmus |
| author_facet |
Drensky, Vesselin Fındık, Şehmus |
| author_sort |
Drensky, Vesselin |
| title |
Inner automorphisms of Lie algebras related with generic \(2\times 2\) matrices |
| title_short |
Inner automorphisms of Lie algebras related with generic \(2\times 2\) matrices |
| title_full |
Inner automorphisms of Lie algebras related with generic \(2\times 2\) matrices |
| title_fullStr |
Inner automorphisms of Lie algebras related with generic \(2\times 2\) matrices |
| title_full_unstemmed |
Inner automorphisms of Lie algebras related with generic \(2\times 2\) matrices |
| title_sort |
inner automorphisms of lie algebras related with generic \(2\times 2\) matrices |
| description |
Let \(F_m=F_m(\text{var}(sl_2(K)))\) 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 \(F_m/F_m^{c+1}=F_m(\text{var}(sl_2(K))\cap {\mathfrak N}_c)\) in the subvariety of \(\text{var}(sl_2(K))\) consisting of all nilpotent algebras of class at most \(c\) in \(\text{var}(sl_2(K))\). |
| publisher |
Lugansk National Taras Shevchenko University |
| publishDate |
2018 |
| url |
https://admjournal.luguniv.edu.ua/index.php/adm/article/view/711 |
| work_keys_str_mv |
AT drenskyvesselin innerautomorphismsofliealgebrasrelatedwithgeneric2times2matrices AT fındıksehmus innerautomorphismsofliealgebrasrelatedwithgeneric2times2matrices |
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2024-04-12T06:26:16Z |
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2024-04-12T06:26:16Z |
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