2019/11/19 by Buchstaber, V. M., Bunkova, E. Yu.
#11G05 #14H45 #37K20 #Algebraic Geometry (math.AG) #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Number Theory (math.NT)
paper · doi:10.48550/arxiv.1911.08266
Lie algebras of systems of 2 g graded heat conduction operators Q2k, where k = 0,1, …,2 g-1, determining sigma functions σ(z, λ) of genus g = 1,2, and 3 hyperelliptic curves are constructed. As a corollary, it is found that a system of three operators Q0, Q2 and Q4 is already sufficient to determine the sigma functions. The operator Q0 is the Euler operator, and each of the operators Q2k, k>0, determines a g-dimensional Schrödinger equation with quadratic potential in z for a nonholonomic frame of vector fields in ℂ2g with coordinates λ. An analogy of the Cole--Hopf transformation is considered. It associates with each solution φ(z, λ) of a linear system of heat equations a system of nonlinear equations for the vector function ∇ ln φ(z, λ), where ∇ is the gradient of the function in z. For any solution φ(z, λ) of the system of heat equations the graded ring Rφ is introduced. It is generated by the logarithmic derivatives of the function φ(z, λ) of order of at least 2. The Lie algebra of derivations of the ring Rφ is presented explicitly. The interrelation of this Lie algebra with the system of nonlinear equations is shown. In the case when φ(z, λ) = σ(z, λ), this leads to a known result of constructing Lie algebras of derivations of hyperellitic functions of genus g = 1,2,3.