Journal of Geometry and Physics | 2019
Poisson structures on loop spaces of ℂPn and an r-matrix associated with the universal elliptic curve
Abstract
Abstract We construct a family of Poisson structures of hydrodynamic type on the loop space of ℂ P n − 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 ℂ n 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 ) , z a ( y ) } = 2 π i z a ( y ) δ ′ ( x − y ) where z 1 , … , z n are coordinates on ℂ n . These homogeneous Poisson structures can be written in terms of an elliptic r -matrix of hydrodynamic type.