Lie Algebras of Heat Operators in a Nonholonomic Frame

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 $g=1,2,3$.