| Abstract: |
| The Hamiltonian structures of (integrable) differential--difference systems, usually obtained in terms of difference (\emph{shift}) operators, has some remarkable algebraic properties -- some cases have been investigated by Dubrovin many years ago,\footnote{B. A. Dubrovin. Differential-geometric Poisson brackets on a lattice. \emph{Funct. Anal. Appl.}, 23(2):57, 1989.} and more recently it has been shown that they can all be recomprised under the notion of multiplicative Poisson vertex algebras.\footnote{A. De Sole, V. G. Kac, D. Valeri, and M. Wakimoto. Local and Non-local Multiplicative Poisson Vertex Algebras and Differential-Difference Equations. \emph{Commun. Math. Phys.}, 370(3):1019, 2019.}
Following a differential-geometrical approach, we introduced the corresponding notion of Poisson bivector and Poisson cohomology.\footnote{MC and J. P. Wang. A Darboux-Getzler theorem for scalar difference Hamiltonian operators. \emph{Commun. Math. Phys.}, 374:1497, 2020.} This allowed us to better understand the classification of bi-Hamiltonian pairs present in De Sole et al., extend the notion to include noncommutative differential-difference systems,\footnote{MC and J. P. Wang. Hamiltonian structures for Integrable Nonabelian Difference Equations. \emph{Commun. Math. Phys.}, 392:219, 2022.} and most recently expand the classification of such structures to multi-component cases.\footnote{MC and D. Valeri, Multi-component Hamiltonian difference operators, \emph{in preparation}} |
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