Normal forms for ordinary differential operators, I
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866909887195250688 |
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| author | Guo, J. Zheglov, A. B. |
| author_facet | Guo, J. Zheglov, A. B. |
| contents | In this paper we develop the generalised Schur theory offered in the recent paper by the second author in dimension one case, and apply it to obtain a new explicit parametrisation of torsion free rank one sheaves on projective irreducible curves with vanishing cohomology groups.
This parametrisation is obtained with the help of normal forms - a notion we introduce in this paper. Namely, considering the ring of ordinary differential operators $D_1=K[[x]][\partial ]$ as a subring of a certain complete non-commutative ring $\hat{D}_1^{sym}$, the normal forms of differential operators mentioned here are obtained after conjugation by some invertible operator ("Schur operator"), calculated using one of the operators in a ring. Normal forms of commuting operators are polynomials with constant coefficients in the differentiation, integration and shift operators, which have a restricted finite order in each variable, and can be effectively calculated for any given commuting operators. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_14414 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Normal forms for ordinary differential operators, I Guo, J. Zheglov, A. B. Algebraic Geometry Mathematical Physics Quantum Algebra Rings and Algebras In this paper we develop the generalised Schur theory offered in the recent paper by the second author in dimension one case, and apply it to obtain a new explicit parametrisation of torsion free rank one sheaves on projective irreducible curves with vanishing cohomology groups. This parametrisation is obtained with the help of normal forms - a notion we introduce in this paper. Namely, considering the ring of ordinary differential operators $D_1=K[[x]][\partial ]$ as a subring of a certain complete non-commutative ring $\hat{D}_1^{sym}$, the normal forms of differential operators mentioned here are obtained after conjugation by some invertible operator ("Schur operator"), calculated using one of the operators in a ring. Normal forms of commuting operators are polynomials with constant coefficients in the differentiation, integration and shift operators, which have a restricted finite order in each variable, and can be effectively calculated for any given commuting operators. |
| title | Normal forms for ordinary differential operators, I |
| topic | Algebraic Geometry Mathematical Physics Quantum Algebra Rings and Algebras |
| url | https://arxiv.org/abs/2406.14414 |