Lie algebras of derivations with large abelian ideals
Let \(\mathbb K\) be a field of characteristic zero, \(A=\mathbb{K}[x_{1},\ldots ,x_{n}]\) the polynomial ring and \(R=\mathbb{K}(x_{1},\dots,x_{n})\) the field of rational functions. The Lie algebra \({\widetilde W}_{n}(\mathbb{K}):=\operatorname{Der}_{\mathbb{K}}R\) of all \(\mathbb{K}\)-derivatio...
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| Видавець: | Lugansk National Taras Shevchenko University |
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| Дата: | 2019 |
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| Формат: | Стаття |
| Мова: | English |
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Lugansk National Taras Shevchenko University
2019
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oai:ojs.admjournal.luguniv.edu.ua:article-2732019-10-20T08:14:09Z Lie algebras of derivations with large abelian ideals Klymenko, I. S. Lysenko, S. V. Petravchuk, Anatoliy Lie algebra, vector field, polynomial ring, abelian ideal, derivation 17B66; 17B05, 13N15 Let \(\mathbb K\) be a field of characteristic zero, \(A=\mathbb{K}[x_{1},\ldots ,x_{n}]\) the polynomial ring and \(R=\mathbb{K}(x_{1},\dots,x_{n})\) the field of rational functions. The Lie algebra \({\widetilde W}_{n}(\mathbb{K}):=\operatorname{Der}_{\mathbb{K}}R\) of all \(\mathbb{K}\)-derivation on \(R\) is a vector space (of dimension n) over \(R\) and every its subalgebra \(L\) has rank \(\operatorname{rk}_{R}L=\dim_{R}RL\). We study subalgebras \(L\) of rank m over \(R\) of the Lie algebra \(\widetilde{W}_{n}(\mathbb{K})\) with an abelian ideal \(I\subset L\) of the same rank \(m\) over \(R\). Let \(F\) be the field of constants of \(L\) in \(R\). It is proved that there exist a basis \(D_1, \ldots, D_m\) of \(FI\) over \(F\), elements \(a_1, \ldots, a_k\in R\) such that \(D_i(a_j)=\delta_{ij}\), \(i=1, \ldots, m\), \(j=1,\ldots, k\), and every element \(D\in FL\) is of the form \(D=\sum_{i=1}^{m}f_i(a_1, \ldots, a_k)D_i\) for some \(f_i\in F[t_1, \ldots t_k]\), \(\deg f_i\leq 1\). As a consequence it is proved that \(L\) is isomorphic to a subalgebra (of a very special type) of the general affine Lie algebra \({\rm aff}_{m}(F)\). Lugansk National Taras Shevchenko University 2019-10-20 Article Article Peer-reviewed Article application/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/view/273 Algebra and Discrete Mathematics; Vol 28, No 1 (2019) 2415-721X 1726-3255 en https://admjournal.luguniv.edu.ua/index.php/adm/article/view/273/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/downloadSuppFile/273/557 Copyright (c) 2019 Algebra and Discrete Mathematics |
| institution |
Algebra and Discrete Mathematics |
| collection |
OJS |
| language |
English |
| topic |
Lie algebra vector field polynomial ring abelian ideal derivation 17B66; 17B05 13N15 |
| spellingShingle |
Lie algebra vector field polynomial ring abelian ideal derivation 17B66; 17B05 13N15 Klymenko, I. S. Lysenko, S. V. Petravchuk, Anatoliy Lie algebras of derivations with large abelian ideals |
| topic_facet |
Lie algebra vector field polynomial ring abelian ideal derivation 17B66; 17B05 13N15 |
| format |
Article |
| author |
Klymenko, I. S. Lysenko, S. V. Petravchuk, Anatoliy |
| author_facet |
Klymenko, I. S. Lysenko, S. V. Petravchuk, Anatoliy |
| author_sort |
Klymenko, I. S. |
| title |
Lie algebras of derivations with large abelian ideals |
| title_short |
Lie algebras of derivations with large abelian ideals |
| title_full |
Lie algebras of derivations with large abelian ideals |
| title_fullStr |
Lie algebras of derivations with large abelian ideals |
| title_full_unstemmed |
Lie algebras of derivations with large abelian ideals |
| title_sort |
lie algebras of derivations with large abelian ideals |
| description |
Let \(\mathbb K\) be a field of characteristic zero, \(A=\mathbb{K}[x_{1},\ldots ,x_{n}]\) the polynomial ring and \(R=\mathbb{K}(x_{1},\dots,x_{n})\) the field of rational functions. The Lie algebra \({\widetilde W}_{n}(\mathbb{K}):=\operatorname{Der}_{\mathbb{K}}R\) of all \(\mathbb{K}\)-derivation on \(R\) is a vector space (of dimension n) over \(R\) and every its subalgebra \(L\) has rank \(\operatorname{rk}_{R}L=\dim_{R}RL\). We study subalgebras \(L\) of rank m over \(R\) of the Lie algebra \(\widetilde{W}_{n}(\mathbb{K})\) with an abelian ideal \(I\subset L\) of the same rank \(m\) over \(R\). Let \(F\) be the field of constants of \(L\) in \(R\). It is proved that there exist a basis \(D_1, \ldots, D_m\) of \(FI\) over \(F\), elements \(a_1, \ldots, a_k\in R\) such that \(D_i(a_j)=\delta_{ij}\), \(i=1, \ldots, m\), \(j=1,\ldots, k\), and every element \(D\in FL\) is of the form \(D=\sum_{i=1}^{m}f_i(a_1, \ldots, a_k)D_i\) for some \(f_i\in F[t_1, \ldots t_k]\), \(\deg f_i\leq 1\). As a consequence it is proved that \(L\) is isomorphic to a subalgebra (of a very special type) of the general affine Lie algebra \({\rm aff}_{m}(F)\). |
| publisher |
Lugansk National Taras Shevchenko University |
| publishDate |
2019 |
| url |
https://admjournal.luguniv.edu.ua/index.php/adm/article/view/273 |
| work_keys_str_mv |
AT klymenkois liealgebrasofderivationswithlargeabelianideals AT lysenkosv liealgebrasofderivationswithlargeabelianideals AT petravchukanatoliy liealgebrasofderivationswithlargeabelianideals |
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2024-04-12T06:25:16Z |
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2024-04-12T06:25:16Z |
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1796109249305116672 |