  Double Poisson structures (a la Van den Bergh) on commutative algebras are
studied; the main result shows that there are no non-trivial such structures on
polynomial algebras of Krull dimension greater than one. For a general
commutative algebra A, 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.
