Wronskians as $N$-ary brackets in finite-dimensional analogues of $sl(2)$
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866913006329266176 |
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| author | Kiselev, Arthemy V. |
| author_facet | Kiselev, Arthemy V. |
| contents | The Wronskian determinants (for coefficients of higher-order differential operators on the affine real line or circle) satisfy the table of Jacobi-type quadratic identities for strong homotopy Lie algebras -- i.e. for a particular case of $L_\infty$-deformations -- for the Lie algebra of vector fields on that one-dimensional affine manifold. We show that the standard realisation of $\mathfrak{sl}(2)$ by quadratic-coefficient vector fields is the bottom structure in a sequence of finite-dimensional polynomial algebras $\Bbbk_N[x]$ with the Wronskians as $N$-ary brackets; the structure constants are calculated explicitly.
Key words: Wronskian determinant, $N$-ary bracket, $L_\infty$-\/algebra, strong homotopy Lie algebra, $sl(2)$, Witt algebra, Vandermonde determinant. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_02145 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Wronskians as $N$-ary brackets in finite-dimensional analogues of $sl(2)$ Kiselev, Arthemy V. Rings and Algebras Commutative Algebra Quantum Algebra 15A15, 17A42, 17B66, 17B01, 13D10 The Wronskian determinants (for coefficients of higher-order differential operators on the affine real line or circle) satisfy the table of Jacobi-type quadratic identities for strong homotopy Lie algebras -- i.e. for a particular case of $L_\infty$-deformations -- for the Lie algebra of vector fields on that one-dimensional affine manifold. We show that the standard realisation of $\mathfrak{sl}(2)$ by quadratic-coefficient vector fields is the bottom structure in a sequence of finite-dimensional polynomial algebras $\Bbbk_N[x]$ with the Wronskians as $N$-ary brackets; the structure constants are calculated explicitly. Key words: Wronskian determinant, $N$-ary bracket, $L_\infty$-\/algebra, strong homotopy Lie algebra, $sl(2)$, Witt algebra, Vandermonde determinant. |
| title | Wronskians as $N$-ary brackets in finite-dimensional analogues of $sl(2)$ |
| topic | Rings and Algebras Commutative Algebra Quantum Algebra 15A15, 17A42, 17B66, 17B01, 13D10 |
| url | https://arxiv.org/abs/2510.02145 |