Lie algebras of heat operators in a nonholonomic frame. (English. Russian original) Zbl 1450.37064
Math. Notes 108, No. 1, 15-28 (2020); translation from Mat. Zametki 108, No. 1, 17-32 (2020).
Summary: 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 \(g=1,2,3\).
MSC:
| 37K20 | Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions |
| 37K30 | Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with infinite-dimensional Lie algebras and other algebraic structures |
| 14H52 | Elliptic curves |
| 14H42 | Theta functions and curves; Schottky problem |
| 14H45 | Special algebraic curves and curves of low genus |
| 14H70 | Relationships between algebraic curves and integrable systems |
Keywords:
heat operator; grading; polynomial Lie algebra; differentiation; abelian functions over parametersReferences:
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