Quadratic algebras and integrable chains

TL;DR

This paper demonstrates how Krichever-Phong's universal formula linearizes Sklyanin quadratic brackets for multi-pole Lax functions with rational or elliptic spectral parameters, extending Sklyanin algebra and revealing new symplectic structures.

Contribution

It introduces a multiplicative representation that simplifies Sklyanin quadratic brackets and extends Sklyanin algebra in the elliptic case, uncovering a new cubic bracket.

Findings

Linearization of quadratic brackets via multiplicative representation

Extension of Sklyanin algebra in elliptic case

Existence of a non-trivial cubic bracket in Sklyanin's case

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.