Noncommutative elliptic Poisson structures on projective spaces

We review noncommutative Poisson structures on affine and projective spaces over C. We also construct a class of examples of noncommutative Poisson structures on C^{P n - 1} for n > 2. These noncommutative Poisson structures depend on a modular parameter τ ∈ C and an additional discrete parameter k ∈ Z, where 1 ≤ 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 (τ).