Double Poisson vertex algebras and non-commutative Hamiltonian equations

Abstract

We develop the formalism of double Poisson vertex algebras (local and non-local) aimed at the study of non-commutative Hamiltionan PDEs. This is a generalization of the theory of double Poisson algebras, developed by Van den Bergh, which is used in the study of Hamiltonian ODEs. We apply our theory of double Poisson vertex algebras to non-commutative KP and Gelfand-Dickey hierarchies. We also construct the related non-commutative de Rham and variational complexes.

56 pages. Minor editing and corrections and references added following the referee suggestions

References in corpus (1)

• Adler-Gelfand-Dickey approach to classical W-algebras within the theory of Poisson vertex algebras

Cited by in corpus (14)

• Simple finite-dimensional double algebras
• Recursion and Hamiltonian operators for integrable nonabelian difference equations
• A new scheme of integrability for (bi)Hamiltonian PDE
• Spin versions of the complex trigonometric Ruijsenaars-Schneider model from cyclic quivers
• Symmetries and reductions on the noncommutative Kadomtsev-Petviashvili and Gelfand-Dickey hierarchies
• Hamiltonian structures for integrable nonabelian difference equations
• Double Multiplicative Poisson Vertex Algebras
• Classical affine W-superalgebras via generalized Drinfeld-Sokolov reductions and related integrable systems
• Around Van den Bergh's double brackets for different bimodule structures
• Functorial constructions related to double Poisson vertex algebras
• Poisson structures in the Banach setting: comparison of different approaches
• Non-commutative Courant algebroids and Quiver algebras
• Symmetry approach to integrability and non-associative algebraic structures
• Shifted double Lie-Rinehart algebras