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). [ABSTRACT FROM AUTHOR]