Loop coproducts

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.