Centralizers of Jacobian derivations
Let \(\mathbb K\) be an algebraically closed field of characteristic zero, \(\mathbb K[x, y]\) the polynomial ring in variables \(x\), \(y\) and let \(W_2(\mathbb K)\) be the Lie algebra of all \(\mathbb K\)-derivations on \(\mathbb K[x, y]\). A derivation \(D \in W_2(\mathbb K)\) is called a Jacobi...
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Lugansk National Taras Shevchenko University
2023
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oai:ojs.admjournal.luguniv.edu.ua:article-21862023-12-11T16:21:07Z Centralizers of Jacobian derivations Efimov, D. I. Petravchuk, A. P. Sydorov, M. S. Lie algebra, Jacobian derivation, differential equation, centralizer, integrable system Primary 17B66; Secondary 17B80 Let \(\mathbb K\) be an algebraically closed field of characteristic zero, \(\mathbb K[x, y]\) the polynomial ring in variables \(x\), \(y\) and let \(W_2(\mathbb K)\) be the Lie algebra of all \(\mathbb K\)-derivations on \(\mathbb K[x, y]\). A derivation \(D \in W_2(\mathbb K)\) is called a Jacobian derivation if there exists \(f \in \mathbb K[x, y]\) such that \(D(h) = \det J(f, h)\) for any \(h \in \mathbb K[x, y]\) (here \(J(f, h)\) is the Jacobian matrix for \(f\) and \(h\)). Such a derivation is denoted by \(D_f\) . The kernel of \(D_f\) in \(\mathbb K[x, y]\) is a subalgebra \(\mathbb K[p]\) where \(p=p(x, y)\) is a polynomial of smallest degree such that \(f(x, y) = \varphi (p(x, y))\) for some \(\varphi (t) \in \mathbb K[t]\). Let \(C = C_{W_2(\mathbb K)} (D_f)\) be the centralizer of \(D_f\) in \(W_2(\mathbb K)\). We prove that \(C\) is the free \(\mathbb K[p]\)-module of rank \(1\) or \(2\) over \(\mathbb K[p]\) and point out a criterion of being a module of rank \(2\). These results are used to obtain a class of integrable autonomous systems of differential equations. Lugansk National Taras Shevchenko University 2023-12-11 Article Article Peer-reviewed Article application/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/view/2186 10.12958/adm2186 Algebra and Discrete Mathematics; Vol 36, No 1 (2023) 2415-721X 1726-3255 en https://admjournal.luguniv.edu.ua/index.php/adm/article/view/2186/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/downloadSuppFile/2186/1138 Copyright (c) 2023 Algebra and Discrete Mathematics |
| institution |
Algebra and Discrete Mathematics |
| collection |
OJS |
| language |
English |
| topic |
Lie algebra Jacobian derivation differential equation centralizer integrable system Primary 17B66; Secondary 17B80 |
| spellingShingle |
Lie algebra Jacobian derivation differential equation centralizer integrable system Primary 17B66; Secondary 17B80 Efimov, D. I. Petravchuk, A. P. Sydorov, M. S. Centralizers of Jacobian derivations |
| topic_facet |
Lie algebra Jacobian derivation differential equation centralizer integrable system Primary 17B66; Secondary 17B80 |
| format |
Article |
| author |
Efimov, D. I. Petravchuk, A. P. Sydorov, M. S. |
| author_facet |
Efimov, D. I. Petravchuk, A. P. Sydorov, M. S. |
| author_sort |
Efimov, D. I. |
| title |
Centralizers of Jacobian derivations |
| title_short |
Centralizers of Jacobian derivations |
| title_full |
Centralizers of Jacobian derivations |
| title_fullStr |
Centralizers of Jacobian derivations |
| title_full_unstemmed |
Centralizers of Jacobian derivations |
| title_sort |
centralizers of jacobian derivations |
| description |
Let \(\mathbb K\) be an algebraically closed field of characteristic zero, \(\mathbb K[x, y]\) the polynomial ring in variables \(x\), \(y\) and let \(W_2(\mathbb K)\) be the Lie algebra of all \(\mathbb K\)-derivations on \(\mathbb K[x, y]\). A derivation \(D \in W_2(\mathbb K)\) is called a Jacobian derivation if there exists \(f \in \mathbb K[x, y]\) such that \(D(h) = \det J(f, h)\) for any \(h \in \mathbb K[x, y]\) (here \(J(f, h)\) is the Jacobian matrix for \(f\) and \(h\)). Such a derivation is denoted by \(D_f\) . The kernel of \(D_f\) in \(\mathbb K[x, y]\) is a subalgebra \(\mathbb K[p]\) where \(p=p(x, y)\) is a polynomial of smallest degree such that \(f(x, y) = \varphi (p(x, y))\) for some \(\varphi (t) \in \mathbb K[t]\). Let \(C = C_{W_2(\mathbb K)} (D_f)\) be the centralizer of \(D_f\) in \(W_2(\mathbb K)\). We prove that \(C\) is the free \(\mathbb K[p]\)-module of rank \(1\) or \(2\) over \(\mathbb K[p]\) and point out a criterion of being a module of rank \(2\). These results are used to obtain a class of integrable autonomous systems of differential equations. |
| publisher |
Lugansk National Taras Shevchenko University |
| publishDate |
2023 |
| url |
https://admjournal.luguniv.edu.ua/index.php/adm/article/view/2186 |
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2024-04-12T06:26:09Z |
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2024-04-12T06:26:09Z |
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