Description of noncommutative symplectic manifolds for self-injective Nakayama algebras

We determine all possible exact noncommutative symplectic structures for certain path algebras with relations. These algebras are the path algebras of a cyclic quiver with r+1 arrows quotiented by the ideal generated by the paths of length l. The main result is that there are exact noncommutative symplectic structures only when l=s(r+1)+1 with s≥1. In this case a description of the open subset of one-forms z such that dz is a non-degenerate symplectic form is given, by reducing it to the case s=1.