Jacobi identities for Wronskian determinants over multidimension
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866915691396857856 |
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| author | Kiselev, Arthemy V. |
| author_facet | Kiselev, Arthemy V. |
| contents | The generalised Wronskian of differential order $k\geqslant 1$ for $N$ functions $f_1$, $\ldots$, $f_N$ in $d\geqslant 1$ independent variables $x^1$, $\ldots$, $x^d$ is the determinant of the matrix with these functions' derivatives $\partial^{|σ_i|} f_j / \partial (x^1)^{σ_i^1}\cdots \partial (x^d)^{σ_i^d}$ (of orders $0 \leqslant |σ_i| \leqslant k$), where the multi-indices $σ_i$ mark (all or part of) fibre variables $u_{σ_i}$ in the $k$th jet space $J^k\bigl(\mathbb{R}^d\to\mathbb{R}\bigr)$. We prove that these (in)complete Wronskians -- provided that their lowest-order parts are complete at differential orders $\ell\leqslant 1$ -- over the $d$-dimensional base satisfy the table of bi-linear, Jacobi-type identities for Schlessinger--Stasheff's strongly homotopy Lie algebras. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_03848 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Jacobi identities for Wronskian determinants over multidimension Kiselev, Arthemy V. Rings and Algebras Mathematical Physics Commutative Algebra Quantum Algebra 13D10, 15A15, 17A42, 17B01, 17B66 The generalised Wronskian of differential order $k\geqslant 1$ for $N$ functions $f_1$, $\ldots$, $f_N$ in $d\geqslant 1$ independent variables $x^1$, $\ldots$, $x^d$ is the determinant of the matrix with these functions' derivatives $\partial^{|σ_i|} f_j / \partial (x^1)^{σ_i^1}\cdots \partial (x^d)^{σ_i^d}$ (of orders $0 \leqslant |σ_i| \leqslant k$), where the multi-indices $σ_i$ mark (all or part of) fibre variables $u_{σ_i}$ in the $k$th jet space $J^k\bigl(\mathbb{R}^d\to\mathbb{R}\bigr)$. We prove that these (in)complete Wronskians -- provided that their lowest-order parts are complete at differential orders $\ell\leqslant 1$ -- over the $d$-dimensional base satisfy the table of bi-linear, Jacobi-type identities for Schlessinger--Stasheff's strongly homotopy Lie algebras. |
| title | Jacobi identities for Wronskian determinants over multidimension |
| topic | Rings and Algebras Mathematical Physics Commutative Algebra Quantum Algebra 13D10, 15A15, 17A42, 17B01, 17B66 |
| url | https://arxiv.org/abs/2511.03848 |