Loop coproducts

TL;DR

This paper introduces loop coproducts in Poisson algebras, demonstrating their role in generating involutive Casimir functions and unifying various integrable models, including Gaudin models, with potential for broader generalizations.

Contribution

It presents a novel loop coproduct framework that generalizes the coproduct method, enabling new insights into integrable models and extending Gaudin algebra constructions.

Findings

Loop coproducts produce involutive Casimir functions.

Recovering Gaudin models as special cases.

Potential to generalize Gaudin algebras to Poisson algebras.

Abstract

In this paper we show that if $A$ is a Poisson algebra equipped with a set of maps $\Delta^{(i)}_\la:A \to A^{\otimes N}$ satisfying suitable conditions, then the images of the Casimir functions of $A$ under the maps $\Delta^{(i)}_\la$ (that we call "loop coproducts") are in involution. Rational, trigonometric and elliptic Gaudin models can be recovered as particular cases of this result, and we show that the same happens for the integrable (or partially integrable) models that can be obtained through the so called coproduct method. On the other hand, this loop coproduct approach is potentially much more general, and could allow the generalization of the Gaudin algebras from the Lie-Poisson to the Poisson algebras context and, hopefully, the definition of new integrable models.