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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| Main Author: | |
| Format: | Article |
| Language: | English |
| Published: |
Lugansk National Taras Shevchenko University
2016
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| Subjects: | |
| Online Access: | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/275 |
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| Journal Title: | Algebra and Discrete Mathematics |
Institution
Algebra and Discrete Mathematics| Summary: | 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 \(f\in F[x_{2}, x_{3}], g\in F[x_3]\), \(c\in F.\) |
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