Generalization of the linear r-matrix formulation through Loop coproducts

TL;DR

This paper introduces a novel loop coproduct framework that generalizes the linear r-matrix formulation, unifying various algebraic structures in classical integrable systems and suggesting potential extensions.

Contribution

It presents a new loop coproduct method that encompasses the linear r-matrix, Sklyanin, and reflection algebras, expanding the theoretical landscape of integrable systems.

Findings

Unified algebraic framework for integrable systems

Derivation of known structures as special cases

Discussion on potential generalizations of r-matrix formalism

Abstract

A new method for the construction of classical integrable systems, that we call loop coproduct formulation, is presented. We show that the linear r-matrix formulation, the Sklyanin algebras and the reflection algebras can be obtained as particular subcases of this framework. We comment on the possible generalizations of the r-matrix formalism introduced through this approach.