Lie algebras of systems of $2 g$ graded heat conduction operators $Q_{2k}$, where $k = 0,1, \ldots,2 g-1$, determining sigma functions $\sigma(z, \lambda)$ of genus $g = 1,2$, and $3$ hyperelliptic curves are constructed. As a corollary, it is found that a system of three operators $Q_0, Q_2$ and $Q_4$ is already sufficient to determine the sigma functions. The operator $Q_0$ is the Euler operator, and each of the operators $Q_{2k}$, $k>0$, determines a $g$-dimensional Schrodinger equation with quadratic potential in $z$ for a nonholonomic frame of vector fields in $\mathbb{C}^{2g}$ with coordinates $\lambda$. An analogy of the Cole--Hopf transformation is considered. It associates with each solution $\varphi(z, \lambda)$ of a linear system of heat equations a system of nonlinear equations for the vector function $\nabla \ln \varphi(z, \lambda)$, where $\nabla$ is the gradient of the function in $z$. For any solution $\varphi(z, \lambda)$ of the system of heat equations the graded ring $\mathcal{R}_{\varphi}$ is introduced. It is generated by the logarithmic derivatives of the function $\varphi(z, \lambda)$ of order of at least $2$. The Lie algebra of derivations of the ring $\mathcal{R}_{\varphi}$ is presented explicitly. The interrelation of this Lie algebra with the system of nonlinear equations is shown. In the case when $\varphi(z, \lambda) = \sigma(z, \lambda)$, this leads to a known result of constructing Lie algebras of derivations of hyperellitic functions of genus $g = 1,2,3$.

中文翻译：

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非完整坐标系中热算符的李代数

系统的李代数 $2 g$ 分级热传导算子 $Q_{2k}$，其中 $k = 0,1, \ldots,2 g-1$，确定 sigma 函数 $\sigma(z, \lambda)$属 $g = 1,2$ 和 $3$ 超椭圆曲线被构建。作为推论，发现三个运算符 $Q_0、Q_2$ 和 $Q_4$ 的系统已经足以确定 sigma 函数。算子$Q_0$是欧拉算子，每个算子$Q_{2k}$，$k>0$，确定一个$g$维薛定谔方程，在$z$中具有二次势，用于向量的非完整坐标系$\mathbb{C}^{2g}$ 中的字段，坐标为 $\lambda$。考虑了 Cole--Hopf 变换的类比。它与线性热方程系统的每个解 $\varphi(z, \lambda)$ 相关联 向量函数的非线性方程组 $\nabla \ln \varphi(z, \lambda)$，其中 $\nabla$ 是函数在 $z$ 中的梯度。对于热方程组的任何解 $\varphi(z, \lambda)$，引入了分级环 $\mathcal{R}_{\varphi}$。它由函数 $\varphi(z, \lambda)$ 的对数导数生成，阶数至少为 $2$。环 $\mathcal{R}_{\varphi}$ 导数的李代数被明确地给出。显示了该李代数与非线性方程组的相互关系。在 $\varphi(z, \lambda) = \sigma(z, \lambda)$ 的情况下，这会导致构造 $g = 1,2,3$ 属超椭圆函数导数的李代数的已知结果. 它由函数 $\varphi(z, \lambda)$ 的对数导数生成，阶数至少为 $2$。环 $\mathcal{R}_{\varphi}$ 导数的李代数被明确地给出。显示了该李代数与非线性方程组的相互关系。在 $\varphi(z, \lambda) = \sigma(z, \lambda)$ 的情况下，这会导致构造 $g = 1,2,3$ 属超椭圆函数导数的李代数的已知结果. 它由函数 $\varphi(z, \lambda)$ 的对数导数生成，阶数至少为 $2$。环 $\mathcal{R}_{\varphi}$ 导数的李代数被明确地给出。显示了该李代数与非线性方程组的相互关系。在 $\varphi(z, \lambda) = \sigma(z, \lambda)$ 的情况下，这会导致构造 $g = 1,2,3$ 属超椭圆函数导数的李代数的已知结果.