Lie Algebras of Heat Operators in Nonholonomic Frame

Abstract

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 Schr\"odinger 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 $g = 1,2,3$.