Compatible quadratic Poisson brackets related to a family of elliptic curves

TL;DR

This paper constructs nine compatible quadratic Poisson structures linked to elliptic curves, providing explicit Casimir elements and showing their linear combinations relate to elliptic algebras.

Contribution

It introduces a family of nine compatible quadratic Poisson brackets connected to elliptic curves, with explicit Casimir elements, advancing the understanding of elliptic Poisson structures.

Findings

Nine compatible quadratic Poisson structures constructed

Explicit Casimir elements obtained for the elliptic Poisson structure

Linear combinations relate to elliptic algebras in n generators

Abstract

We construct nine pairwise compatible quadratic Poisson structures such that a generic linear combination of them is associated with an elliptic algebra in n generators. Explicit formulas for Casimir elements of this elliptic Poisson structure are obtained.