Lie Algebras of Heat Operators in a Nonholonomic Frame

Abstract

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 Schrodinger 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 $$g=1,2,3$$ .