2005/05/02 by Kanehisa Takasaki, Takasaki, Kanehisa
Mathematics · Physics and Astronomy · #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic structures and combinatorial models #hep-th #math-ph #math.AG #math.MP #nlin.SI
paper · pdf · doi:10.48550/arxiv.nlin/0505005
latex2e, use amssymb and amsmath packages, contribution to proceedings of RIMS workshop "Elliptic Integrable Systems" (RIMS, 2004)
arxiv created 2005/05/02 · arxiv updated 2009/12/01
Commuting pairs of ordinary differential operators are classified by a set of algebro-geometric data called ``algebraic spectral data''. These data consist of an algebraic curve (``spectral curve'') Γ with a marked point γ_∞, a holomorphic vector bundle E on Γ and some additional data related to the local structure of Γ and E in a neighborhood of γ_∞. If the rank r of E is greater than 1, one can use the so called ``Tyurin parameters'' in place of E itself. The Tyurin parameters specify the pole structure of a basis of joint eigenfunctions of the commuting pair. These data can be translated to the language of an infinite dimensional Grassmann manifold. This leads to a dynamical system of the standard exponential flows on the Grassmann manifold, in which the role of Tyurin parameters and some other parameters is made clear.