On double Poisson structures on commutative algebras

Double Poisson structures (à la Van den Bergh) on commutative algebras are considered. The main result shows that there are no non-trivial such structures on polynomial algebras of Krull dimension greater than one. For an arbitrary commutative algebra AA, this places significant restrictions on possible double Poisson structures. Exotic double Poisson structures are exhibited by the case of the polynomial algebra on a single generator, previously considered by Van den Bergh.