Nonabelian elliptic Poisson structures on projective spaces

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 $1\le k<n$ and $k,n$ are coprime. The abelianization of these Poisson structures can be lifted to the quadratic elliptic Poisson algebras $q_{n,k}(\tau )$.