Hamiltonian structures for integrable nonabelian difference equations

Abstract

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 algebras and vertex algebras) and geometric (in terms of nonabelian Poisson bivectors) definitions. We introduce multiplicative double Poisson vertex algebras (PVAs) as the suitable noncommutative counterpart to multiplicative PVAs, used to describe Hamiltonian differential-difference equations in the commutative setting, and prove that these algebras are in one-to-one correspondence with the Poisson structures defined by difference operators, providing a sufficient condition for the fulfilment of the Jacobi identity. Moreover, we define nonabelian polyvector fields and their Schouten brackets, for both finitely generated noncommutative algebras and infinitely generated difference ones: this allows us to provide a unified characterisation of Poisson bivectors and double quasi-Poisson algebra structures. Finally, as an application we obtain some results towards the classification of local scalar Hamiltonian difference structures and construct the Hamiltonian structures for the nonabelian Kaup, Ablowitz-Ladik and Chen-Lee-Liu integrable lattices.

Double Multiplicative Poisson Vertex Algebras

“…In this section we formalize the theory of non-local and rational double multiplicative Poisson vertex algebras. They play a crucial role in the context of non-commutative Hamiltonian differential-difference equations, see [CW1,CW2]. The exposition follows [DSKVW1] where the commutative case is treated.…”

“…Relation to the work of Casati-Wang. While we were working on this project, we became aware that a parallel investigation on double multiplicative Poisson vertex algebras was carried out independently by Casati and Wang [CW2]. For the reader's convenience, let us outline the main differences between these two works.…”

“…For the reader's convenience, let us outline the main differences between these two works. Firstly, Casati and Wang [CW2] introduced double multiplicative Poisson vertex algebras on the space of non-commutative Laurent polynomials with an infinite order automorphism, while we work in the more general setup of algebras of non-commutative difference functions and we provide classification results in Section 4. Their motivation stems from the integrability of non-abelian difference equations, which we also consider in Section 6, though we do not study the several non-local examples gathered in [CW2,Sect.…”

“…We wish to thank Sylvain Carpentier for useful and stimulating discussions. We are also grateful to Matteo Casati for sharing with us a preliminary version of the work [CW2] and for enlightening discussions on this topic. M.F.…”

Around Van den Bergh's double brackets for different bimodule structures

“…In particular, this principle emphasises an important reason for the development of the study of double Poisson brackets. Some other useful aspects of this theory include classification results [5,21,22,24,27], double Poisson cohomology [1,23,27], connection to (pre-)Calabi-Yau algebras [6,13,15,19], study from the point of view of properads [18,19] and relation to integrable systems [7,10,11,12].…”