Quadratic algebras and integrable chains

Abstract

Using Krichever-Phong's universal formula, we show that a multiplicative representation linearizes Sklyanin quadratic brackets for a multi-pole Lax function with a spectral parameter. The spectral parameter can be either rational or elliptic. As a by-product, we obtain an extension of a Sklyanin algebra in the elliptic case. Krichever-Phong's formula provides a hierarchy of symplectic structures, and we show that there exists a non-trivial cubic bracket in Sklyanin's case.

24 pages

References in corpus (2)

• Vector bundles and Lax equations on algebraic curves
• Surfaces and the Sklyanin bracket

Cited by in corpus (1)

• Factorizations of Rational Matrix Functions with Application to Discrete Isomonodromic Transformations and Difference Painlevé Equations