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arXiv · 2601.01978

Second-order superintegrable systems from semi-simple and nilpotent Frobenius structures

Andreas Vollmer
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Abstract

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 present a conification to lift systems of non-zero constant sectional curvature to flat ones, thus reducing the problem on the Frobenius algebra side to the associative case. Furthermore, we employ a direct product construction to generate higher-dimensional second-order maximally superintegrable systems on flat spaces. We apply this approach to some basic semi-simple and nilpotent algebras and explicitly construct the arising second-order superintegrable systems using a Neumann series expansion. We find that all non-degenerate second-order maximally superintegrable systems in three dimensions arise from these examples.

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Andreas Vollmer. 2026-01-05. Second-order superintegrable systems from semi-simple and nilpotent Frobenius structures. https://arxiv.org/abs/2601.01978

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