Projective limits of local shift morphism

Abstract

We define the notion of projective limit of local shift morphisms of type $\left(r,s\right)$ and endow the space of such mathematical objects with an adapted differential structure. The notion of shift Poisson tensor $P$ on a Hilbert tower corresponds to such morphisms which are antisymmetric and whose Schouten bracket $\left[P,P\right]$ vanishes. We illustrate this notion with the example of the famous KdV equation on the circle ${\mathbb{S}}^1$ for which one can associate a couple of compatible Poisson tensors of this type on the Hilbert tower $\left( H^{n}(\mathbb{S}^{1})\right) _{n\in\mathbb{N}% ^{\ast}}$.