Abstract
We construct the Lie algebras of systems of \(2g\) graded heat operators \(Q_0,Q_2,\dots,Q_{4g-2}\) that determine the sigma functions \(\sigma(z,\lambda)\) of hyperelliptic curves of genera \(g=1\), \(2\), and \(3\). As a corollary, we find that the system of three operators \(Q_0\), \(Q_2\), and \(Q_4\) is already sufficient for determining the sigma functions. The operator \(Q_0\) is the Euler operator, and each of the operators \(Q_{2k}\), \(k>0\), determines a \(g\)-dimensional Schrödinger equation with potential quadratic in \(z\) for a nonholonomic frame of vector fields in the space \(\mathbb C^{2g}\) with coordinates \(\lambda\). For any solution \(\varphi(z,\lambda)\) of the system of heat equations, we introduce the graded ring \(\mathscr R_\varphi\) generated by the logarithmic derivatives of \(\varphi(z,\lambda)\) of order \(\ge 2\) and present the Lie algebra of derivations of \(\mathscr R_\varphi\) explicitly. We show how this Lie algebra is related to our system of nonlinear equations. For \(\varphi(z,\lambda)=\sigma(z,\lambda)\), this leads to a well-known result on how to construct the Lie algebra of differentiations of hyperelliptic functions of genus \(g=1,2,3\).
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Acknowledgments
The authors wish to express gratitude to Dmitrii Vladimirovich Millionshchikov for valuable remarks and comments, which have helped us to improve the exposition of our results.
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Buchstaber, V.M., Bunkova, E.Y. Lie Algebras of Heat Operators in a Nonholonomic Frame. Math Notes 108, 15–28 (2020). https://doi.org/10.1134/S0001434620070020
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DOI: https://doi.org/10.1134/S0001434620070020