Poisson structures on loop spaces of [formula omitted] and an [formula omitted]-matrix associated with the universal elliptic curve

Abstract

We construct a family of Poisson structures of hydrodynamic type on the loop space of ℂPn−1. This family is parametrized by the moduli space of elliptic curves or, in other words, by the modular parameter τ. This family can be lifted to a homogeneous Poisson structure on the loop space of ℂn but in order to do that we need to upgrade the modular parameter τ to an additional field τ(x) with Poisson brackets {τ(x),τ(y)}=0,{τ(x),za(y)}=2πiza(y)δ′(x−y) where z1,…,zn are coordinates on ℂn. These homogeneous Poisson structures can be written in terms of an elliptic r-matrix of hydrodynamic type.