An Algebraic Geometric Foundation for a Classification of Superintegrable Systems in Arbitrary Dimension
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arXiv
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| Format: | Preprint |
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2019
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| author | Kress, Jonathan Schöbel, Konrad Vollmer, Andreas |
| author_facet | Kress, Jonathan Schöbel, Konrad Vollmer, Andreas |
| contents | Second-order superintegrable systems in dimensions two and three are essentially classified. With increasing dimension, however, the non-linear partial differential equations employed in current methods become unmanageable. Here we propose a new, algebraic-geometric approach to the classification problem - based on a proof that the classification space for irreducible non-degenerate second-order superintegrable systems is naturally endowed with the structure of a quasi-projective variety with a linear isometry action. On constant curvature manifolds our approach leads to a single, simple and explicit algebraic equation defining the variety classifying superintegrable Hamiltonians that satisfy all relevant integrability conditions generically. In particular, this includes all non-degenerate superintegrable systems known to date and shows that our approach is manageable in arbitrary dimension. Our work establishes the foundations for a complete classification of second-order superintegrable systems in arbitrary dimension, derived from the geometry of the classification space, with many potential applications to related structures such as quadratic symmetry algebras and special functions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1911_11925 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | An Algebraic Geometric Foundation for a Classification of Superintegrable Systems in Arbitrary Dimension Kress, Jonathan Schöbel, Konrad Vollmer, Andreas Differential Geometry Mathematical Physics 14H70, 70H06, 70H33, 35N10 Second-order superintegrable systems in dimensions two and three are essentially classified. With increasing dimension, however, the non-linear partial differential equations employed in current methods become unmanageable. Here we propose a new, algebraic-geometric approach to the classification problem - based on a proof that the classification space for irreducible non-degenerate second-order superintegrable systems is naturally endowed with the structure of a quasi-projective variety with a linear isometry action. On constant curvature manifolds our approach leads to a single, simple and explicit algebraic equation defining the variety classifying superintegrable Hamiltonians that satisfy all relevant integrability conditions generically. In particular, this includes all non-degenerate superintegrable systems known to date and shows that our approach is manageable in arbitrary dimension. Our work establishes the foundations for a complete classification of second-order superintegrable systems in arbitrary dimension, derived from the geometry of the classification space, with many potential applications to related structures such as quadratic symmetry algebras and special functions. |
| title | An Algebraic Geometric Foundation for a Classification of Superintegrable Systems in Arbitrary Dimension |
| topic | Differential Geometry Mathematical Physics 14H70, 70H06, 70H33, 35N10 |
| url | https://arxiv.org/abs/1911.11925 |