On nilpotent Lie algebras of derivations of fraction fields
Let \(\mathbb K\) be an arbitrary field of characteristic zero and \(A\) an integral \(\mathbb K\)-domain. Denote by \(R\) the fraction field of \(A\) and by \(W(A)=RDer_{\mathbb K}A,\) the Lie algebra of \(\mathbb K\)-derivations on \(R\) obtained from \(Der_{\mathbb K}A\) via multiplication by...
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| Date: | 2016 |
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| Language: | English |
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Lugansk National Taras Shevchenko University
2016
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admjournalluguniveduua-article-2752017-03-06T18:33:31Z On nilpotent Lie algebras of derivations of fraction fields Petravchuk, Anatoliy P. Lie algebra, vector field, nilpotent algebra, derivation Primary 17B66; Secondary 17B05, 13N15 Let \(\mathbb K\) be an arbitrary field of characteristic zero and \(A\) an integral \(\mathbb K\)-domain. Denote by \(R\) the fraction field of \(A\) and by \(W(A)=RDer_{\mathbb K}A,\) the Lie algebra of \(\mathbb K\)-derivations on \(R\) obtained from \(Der_{\mathbb K}A\) via multiplication by elements of \(R.\) If \(L\subseteq W(A)\) is a subalgebra of \(W(A)\) denote by \(rk_{R}L\) the dimension of the vector space \(RL\) over the field \(R\) and by \(F=R^{L}\) the field of constants of \(L\) in \(R.\) Let \(L\) be a nilpotent subalgebra \(L\subseteq W(A)\) with \(rk_{R}L\leq 3\). It is proven that the Lie algebra \(FL\) (as a Lie algebra over the field \(F\)) is isomorphic to a finite dimensional subalgebra of the triangular Lie subalgebra \(u_{3}(F)\) of the Lie algebra \(Der F[x_{1}, x_{2}, x_{3}], \) where \(u_{3}(F)=\{f(x_{2}, x_{3})\frac{\partial}{\partial x_{1}}+g(x_{3})\frac{\partial}{\partial x_{2}}+c\frac{\partial}{\partial x_{3}}\}\) with \(f\in F[x_{2}, x_{3}], g\in F[x_3]\), \(c\in F.\) Lugansk National Taras Shevchenko University 2016-11-15 Article Article Peer-reviewed Article application/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/view/275 Algebra and Discrete Mathematics; Vol 22, No 1 (2016) 2415-721X 1726-3255 en https://admjournal.luguniv.edu.ua/index.php/adm/article/view/275/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/downloadSuppFile/275/112 Copyright (c) 2016 Algebra and Discrete Mathematics |
| institution |
Algebra and Discrete Mathematics |
| baseUrl_str |
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| datestamp_date |
2017-03-06T18:33:31Z |
| collection |
OJS |
| language |
English |
| topic |
Lie algebra vector field nilpotent algebra derivation Primary 17B66; Secondary 17B05 13N15 |
| spellingShingle |
Lie algebra vector field nilpotent algebra derivation Primary 17B66; Secondary 17B05 13N15 Petravchuk, Anatoliy P. On nilpotent Lie algebras of derivations of fraction fields |
| topic_facet |
Lie algebra vector field nilpotent algebra derivation Primary 17B66; Secondary 17B05 13N15 |
| format |
Article |
| author |
Petravchuk, Anatoliy P. |
| author_facet |
Petravchuk, Anatoliy P. |
| author_sort |
Petravchuk, Anatoliy P. |
| title |
On nilpotent Lie algebras of derivations of fraction fields |
| title_short |
On nilpotent Lie algebras of derivations of fraction fields |
| title_full |
On nilpotent Lie algebras of derivations of fraction fields |
| title_fullStr |
On nilpotent Lie algebras of derivations of fraction fields |
| title_full_unstemmed |
On nilpotent Lie algebras of derivations of fraction fields |
| title_sort |
on nilpotent lie algebras of derivations of fraction fields |
| description |
Let \(\mathbb K\) be an arbitrary field of characteristic zero and \(A\) an integral \(\mathbb K\)-domain. Denote by \(R\) the fraction field of \(A\) and by \(W(A)=RDer_{\mathbb K}A,\) the Lie algebra of \(\mathbb K\)-derivations on \(R\) obtained from \(Der_{\mathbb K}A\) via multiplication by elements of \(R.\) If \(L\subseteq W(A)\) is a subalgebra of \(W(A)\) denote by \(rk_{R}L\) the dimension of the vector space \(RL\) over the field \(R\) and by \(F=R^{L}\) the field of constants of \(L\) in \(R.\) Let \(L\) be a nilpotent subalgebra \(L\subseteq W(A)\) with \(rk_{R}L\leq 3\). It is proven that the Lie algebra \(FL\) (as a Lie algebra over the field \(F\)) is isomorphic to a finite dimensional subalgebra of the triangular Lie subalgebra \(u_{3}(F)\) of the Lie algebra \(Der F[x_{1}, x_{2}, x_{3}], \) where \(u_{3}(F)=\{f(x_{2}, x_{3})\frac{\partial}{\partial x_{1}}+g(x_{3})\frac{\partial}{\partial x_{2}}+c\frac{\partial}{\partial x_{3}}\}\) with \(f\in F[x_{2}, x_{3}], g\in F[x_3]\), \(c\in F.\) |
| publisher |
Lugansk National Taras Shevchenko University |
| publishDate |
2016 |
| url |
https://admjournal.luguniv.edu.ua/index.php/adm/article/view/275 |
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AT petravchukanatoliyp onnilpotentliealgebrasofderivationsoffractionfields |
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2025-12-02T15:40:00Z |
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2025-12-02T15:40:00Z |
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1850412155015266304 |