Second-order superintegrable systems from semi-simple and nilpotent Frobenius structures
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| Format: | Preprint |
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2026
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| _version_ | 1866918490285277184 |
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| author | Vollmer, Andreas |
| author_facet | Vollmer, Andreas |
| contents | Recently, it was shown that a rich class of second-order (maximally) superintegrable systems has an underpinning Hesse-Frobenius structure, i.e.\ a Frobenius structure that is compatible with a Hessian structure such that the Hessian pre-potential is also a Frobenius pre-potential. Hence, these superintegrable systems arise, locally, from (possibly non-unital) Frobenius algebras. We use a conification to lift systems of non-zero constant sectional curvature to flat ones and we employ a direct product construction to generate higher-dimensional second-order maximally superintegrable systems on pseudo-Euclidean spaces. We apply the method to very basic semi-simple and nilpotent algebras and we explicitly construct the arising second-order superintegrable systems. All non-degenerate second-order maximally superintegrable systems in three dimensions arise from these examples. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_01978 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Second-order superintegrable systems from semi-simple and nilpotent Frobenius structures Vollmer, Andreas Mathematical Physics Differential Geometry 70H06, 53D45 Recently, it was shown that a rich class of second-order (maximally) superintegrable systems has an underpinning Hesse-Frobenius structure, i.e.\ a Frobenius structure that is compatible with a Hessian structure such that the Hessian pre-potential is also a Frobenius pre-potential. Hence, these superintegrable systems arise, locally, from (possibly non-unital) Frobenius algebras. We use a conification to lift systems of non-zero constant sectional curvature to flat ones and we employ a direct product construction to generate higher-dimensional second-order maximally superintegrable systems on pseudo-Euclidean spaces. We apply the method to very basic semi-simple and nilpotent algebras and we explicitly construct the arising second-order superintegrable systems. All non-degenerate second-order maximally superintegrable systems in three dimensions arise from these examples. |
| title | Second-order superintegrable systems from semi-simple and nilpotent Frobenius structures |
| topic | Mathematical Physics Differential Geometry 70H06, 53D45 |
| url | https://arxiv.org/abs/2601.01978 |