Poisson structures on loop spaces of $\mathbb{C}{P}^n$ and an $r$-matrix associated with the universal elliptic curve

Abstract

We construct a family of Poisson structures of hydrodynamic type on the loop space of $\mathbb{C}{P}^{n-1}$. This family is parametrized by the moduli space of elliptic curves or, in other words, by the modular parameter $\tau$. This family can be lifted to a homogeneous Poisson structure on the loop space of ${\mathbb{C}}^n$ but in order to do that we need to upgrade the modular parameter $\tau$ to an additional field $\tau (x)$ with Poisson brackets $\{\tau (x),\tau (y)\}=0,\{\tau (x),z_a(y)\}=2\pi i{z}_a(y){\delta }^{\prime }(x-y)$ where $z_1,...,z_n$ are coordinates on ${\mathbb{C}}^n$. These homogeneous Poisson structures can be written in terms of an elliptic $r$-matrix of hydrodynamic type.

15 pages, latex

References in corpus (1)

• Constructions and classifications of projective Poisson varieties