On double Poisson structures on commutative algebras

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.

12 pages; very minor revision

References in corpus (1)

• Lectures on Noncommutative Geometry

Cited by in corpus (9)

• Recursion and Hamiltonian operators for integrable nonabelian difference equations
• Modified Double Poisson Brackets
• Morphisms of double (quasi-)Poisson algebras and action-angle duality of integrable systems
• Hamiltonian structures for integrable nonabelian difference equations
• Double Multiplicative Poisson Vertex Algebras
• Double quasi-Poisson brackets : fusion and new examples
• Around Van den Bergh's double brackets for different bimodule structures
• Functorial constructions related to double Poisson vertex algebras
• Shifted double Lie-Rinehart algebras