A stochastic Poisson structure associated to a Yang-Baxter equation

We consider a simple solution of a Yang-Baxter equation on loop algebra and deduce from it a Sklyanin Poisson structure which operates continuously on a Sobolev test algebra on the Wiener space of the Lie algebra. It is very classical that the solution of the classical Yang-Baxter equation on a finite dimensional algebra gives a Poisson structure on the algebra of smooth function on the finite dimensional Lie algebra [1]. Solution of Yang-Baxter equation in infinite dimension were studied by Belavin-Drinfeld in [2]. So it should be a Poisson structure associated to the solution of this Yang-Baxter equation.We refer to the review [3] Our goal in this communication is to define functional spaces where this Poisson structure act continuously in a simple case. For that we use the apparatus of infinite dimensional analysis. Poisson structure in infinite dimensional analysis were studied by Dito-Léandre [4] and after by Léandre in [5], [6], [7], [8], [9], [10], [11]. Especially Léandre studied the case of hydrodynamic Poisson structure in [9], [10], [11]. We consider another type of Poisson structure on the Wiener space. 1. The deterministic case Let so(n) be the Lie algebra of the orthogonal group endowed with its biinvariant Killing form Tr. We consider several complexified Banach spaces: -)The Banach space Hp of maps s→ B(s) from S1 into so(n) such that