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Hamiltonian structures for integrable nonabelian difference equations

Casati, Matteo, Wang, Jing Ping (2022) Hamiltonian structures for integrable nonabelian difference equations. Communications in Mathematical Physics, 392 . pp. 219-278. ISSN 0010-3616. (doi:10.1007/s00220-022-04348-3) (KAR id:93235)


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.

Item Type: Article
DOI/Identification number: 10.1007/s00220-022-04348-3
Subjects: Q Science > QA Mathematics (inc Computing science)
Divisions: Divisions > Division of Computing, Engineering and Mathematical Sciences > School of Mathematics, Statistics and Actuarial Science
Depositing User: Jing Ping Wang
Date Deposited: 21 Feb 2022 09:39 UTC
Last Modified: 31 Mar 2023 23:00 UTC
Resource URI: (The current URI for this page, for reference purposes)

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