Nonabelian elliptic Poisson structures on projective spaces

TL;DR

This paper reviews nonabelian Poisson structures on affine and projective spaces over complex numbers, introduces new examples on complex projective spaces, and explores their dependence on modular and discrete parameters.

Contribution

It constructs a new class of nonabelian Poisson structures on complex projective spaces depending on modular and discrete parameters.

Findings

Constructed nonabelian Poisson structures on $\\mathbb{C} P^{n-1}$ for $n>2$.

Demonstrated dependence on modular parameter $\tau$ and integer $k$.

Linked these structures to quadratic elliptic Poisson algebras $q_{n,k}(\tau)$.

Abstract

We review nonabelian Poisson structures on affine and projective spaces over $\mathbb{C}$. We also construct a class of examples of nonabelian Poisson structures on $\mathbb{C} P^{n-1}$ for $n>2$. These nonabelian Poisson structures depend on a modular parameter $\tau\in\mathbb{C}$ and an additional descrete parameter $k\in\mathbb{Z}$, where $1\leq k<n$ and $k,n$ are coprime. The abelianization of these Poisson structures can be lifted to the quadratic elliptic Poisson algebras $q_{n,k}(\tau)$.