On double Poisson structures on commutative algebras

TL;DR

This paper investigates double Poisson structures on commutative algebras, proving their non-existence on polynomial algebras of dimension greater than one and exploring special cases with unique structures.

Contribution

It establishes that non-trivial double Poisson structures do not exist on polynomial algebras of Krull dimension greater than one, clarifying their limitations.

Findings

No non-trivial double Poisson structures on polynomial algebras of dimension > 1

Restrictions on double Poisson structures for general commutative algebras

Existence of exotic structures on polynomial algebra with one generator

Abstract

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.