Lie Algebras of Heat Operators in Nonholonomic Frame

Abstract

Lie algebras of systems of $2g$ graded heat conduction operators $Q_{2k}$, where $k=0,1,\dots ,2g-1$, 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 Schrödinger 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 $\phi (z,\lambda )$ of a linear system of heat equations a system of nonlinear equations for the vector function $\nabla \ln \phi (z,\lambda )$, where $\nabla$ is the gradient of the function in $z$. For any solution $\phi (z,\lambda )$ of the system of heat equations the graded ring ${\mathcal{R}}_{\phi }$ is introduced. It is generated by the logarithmic derivatives of the function $\phi (z,\lambda )$ of order of at least $2$. The Lie algebra of derivations of the ring ${\mathcal{R}}_{\phi }$ is presented explicitly. The interrelation of this Lie algebra with the system of nonlinear equations is shown. In the case when $\phi (z,\lambda )=\sigma (z,\lambda )$, this leads to a known result of constructing Lie algebras of derivations of hyperellitic functions of genus $g=1,2,3$.

15 pages

Cited by in corpus (2)

• Differentiation of Genus $4$ Hyperelliptic Functions
• Sigma functions and Lie algebras of Schrödinger operators