Local and non-local multiplicative Poisson vertex algebras and differential-difference equations

Abstract

We develop the notions of multiplicative Lie conformal and Poisson vertex algebras, local and non-local, and their connections to the theory of integrable differential-difference Hamiltonian equations. We establish relations of these notions to $q$-deformed $W$-algebras and lattice Poisson algebras. We introduce the notion of Adler type pseudodifference operators and apply them to integrability of differential-difference Hamiltonian equations.

arXiv comment42 pages

References in corpus (15)

• Quantum Affine Algebras and Deformations of the Virasoro and W-algebras
• Poisson vertex algebras in the theory of Hamiltonian equations
• Vertex operator solutions to the discrete KP-hierarchy
• The Extended Bigraded Toda hierarchy
• Darboux transformations and Recursion operators for differential--difference equations
• $\Gamma$-Conformal Algebras
• Non-local Poisson structures and applications to the theory of integrable systems
• The algebra of q-pseudodifferential symbols and the q-W_{KP}^n algebra
• The Hamiltonian structures of the two-dimensional Toda lattice and R-matrices
• Adler-Gelfand-Dickey approach to classical W-algebras within the theory of Poisson vertex algebras
• Classical affine W-algebras and the associated integrable Hamiltonian hierarchies for classical Lie algebras
• Dirac reduction for Poisson vertex algebras
• Lattice Analogue of W-infinity Algebra and Discrete KP-Hierarchy
• Hamiltonian evolutions of twisted gons in $\RP^n$
• A new scheme of integrability for (bi)Hamiltonian PDE

Cited by in corpus (4)

• Recursion and Hamiltonian operators for integrable nonabelian difference equations
• Hamiltonian structures for integrable nonabelian difference equations
• Double Multiplicative Poisson Vertex Algebras
• A Darboux-Getzler theorem for scalar difference Hamiltonian operators