Local and Non-local Multiplicative Poisson Vertex Algebras and Differential-Difference Equations

  title={Local and Non-local Multiplicative Poisson Vertex Algebras and Differential-Difference Equations},
  author={Alberto De Sole and Victor G. Kac and Daniele Valeri and Minoru Wakimoto},
  journal={Communications in Mathematical Physics},
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. 
Double Multiplicative Poisson Vertex Algebras
We develop the theory of double multiplicative Poisson vertex algebras. These structures, defined at the level of associative algebras, are shown to be such that they induce a classical structure of
Hamiltonian Structures for Integrable Nonabelian Difference Equations
In this paper we extensively study the notion of Hamiltonian structure for nonabelian differential-difference systems, exploring the link between the different algebraic (in terms of double Poisson
Classical and Quantum $${\mathcal {W}}$$-Algebras and Applications to Hamiltonian Equations
  • A. Sole
  • Mathematics
    Springer INdAM Series
  • 2019
We start by giving an overview of the four fundamental physical theories, namely classical mechanics, quantum mechanics, classical field theory and quantum field theory, and the corresponding
A Darboux–Getzler Theorem for Scalar Difference Hamiltonian Operators
In this paper we extend the notion of Poisson–Lichnerowicz cohomology, an object encapsulating the building blocks for the theory of deformations of Hamiltonian operators, to the difference case. A
Recursion and Hamiltonian operators for integrable nonabelian difference equations
In this paper, we carry out the algebraic study of integrable differential-difference equations whose field variables take values in an associative (but not commutative) algebra. We adapt the


Non-local Poisson structures and applications to the theory of integrable systems
We develop a rigorous theory of non-local Poisson structures, built on the notion of a non-local Poisson vertex algebra. As an application, we find conditions that guarantee applicability of the
Dirac Reduction for Poisson Vertex Algebras
We construct an analogue of Dirac’s reduction for an arbitrary local or non-local Poisson bracket in the general setup of non-local Poisson vertex algebras. This leads to Dirac’s reduction of an
Poisson Λ-brackets for Differential–Difference Equations
We introduce the notion of a multiplicative Poisson $\lambda$-bracket, which plays the same role in the theory of Hamiltonian differential–difference equations as the usual Poisson
Quantum affine algebras and deformations of the Virasoro and 237-1237-1237-1
Using the Wakimoto realization of quantum affine algebras we define new Poisson algebras, which areq-deformations of the classicalW. We also define their free field realizations, i.e. homomorphisms
Darboux transformations and recursion operators for differential-difference equations
We review two concepts directly related to the Lax representations of integrable systems: Darboux transformations and recursion operators. We present an extensive list of integrable
A New Scheme of Integrability for (bi)Hamiltonian PDE
We develop a new method for constructing integrable Hamiltonian hierarchies of Lax type equations, which combines the fractional powers technique of Gelfand and Dickey, and the classical Hamiltonian
Γ-conformal algebras
Γ-conformal algebra is an axiomatic description of the operator product expansion of chiral fields with simple poles at finitely many points. We classify these algebras and their representations in
Poisson vertex algebras in the theory of Hamiltonian equations
Abstract.We lay down the foundations of the theory of Poisson vertex algebras aimed at its applications to integrability of Hamiltonian partial differential equations. Such an equation is called
Classical lattice W algebras and integrable systems
Several aspects of the lattice algebra are studied. Motivated by the fact that the Lotka - Volterra model can be written in terms of a current of the lattice Virasoro algebra (the Faddeev - Takhtajan