Solvable Lie algebras of derivations of polynomial rings in three variables
Let $\mathbb K$ be an algebraically closed field of characteristic zero, A=$\mathbb K$[x1,x2,x3] be the polynomial ring in three variables and R$=\mathbb K$(x1,x2,x3) be the field of rational functions. If L is a subalgebra of the Lie algebra W3($\mathbb K$) of all $\mathbb K$-derivations of A, then...
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| Дата: | 2018 |
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| Автори: | , , |
| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Pidstryhach Institute for Applied Problems of Mechanics and Mathematics of NAS of Ukraine
2018
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| Теми: | |
| Онлайн доступ: | http://journals.iapmm.lviv.ua/ojs/index.php/APMM/article/view/apmm2018.16.7-13 |
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| Назва журналу: | Prykladni Problemy Mekhaniky i Matematyky |
Репозиторії
Prykladni Problemy Mekhaniky i Matematyky| Резюме: | Let $\mathbb K$ be an algebraically closed field of characteristic zero, A=$\mathbb K$[x1,x2,x3] be the polynomial ring in three variables and R$=\mathbb K$(x1,x2,x3) be the field of rational functions. If L is a subalgebra of the Lie algebra W3($\mathbb K$) of all $\mathbb K$-derivations of A, then RL is a Lie algebra over $\mathbb K$ and dimRRL will be called the rank of L over R. We study solvable subalgebras L of W3($\mathbb K$) of rank 3 over R. It is proved that L is isomorphic to a subalgebra of the general affine Lie algebra aff3($\mathbb K$) if L contains an abelian ideal I of rank 3 over R. If L has an ideal I with rkRI=2, then L is contained in a subalgebra $\bar{\it L}$ of $\tilde{W}_3(\mathbb K)=\it{Der}_{\mathbb K}\it R$ such that $\bar{\it L}$ is an extension of a subalgebra of aff2(F) by a subalgebra of dimension ≤2, where F is the field of constants of I in R. Cite as: Ie. Yu. Chapovskyi, D. I. Efimov, A. P. Petravchuk, "Solvable Lie algebras of derivations of polynomial rings in three variables," Prykl. Probl. Mekh. Mat., Issue 16, 7–13 (2018), https://doi.org/10.15407/apmm2018.16.7-13 |
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