Adler–Oevel-Ragnisco type operators and Poisson vertex algebras

The theory of triples of Poisson brackets and related integrable systems, based on a classical $R$-matrix $R \in \mathrm{End}_\mathbb{F}(\mathfrak{g})$, where $\mathfrak{g}$ is a finite dimensional associative algebra over a field F viewed as a Lie algebra, was developed by Oevel–Ragnisco and Li–Parmentier [$\href{https://doi.org/10.1016/0378-4371(89)90398-1}{\textrm{OR89}}$, $\href{https://doi.org/10.1007/BF01228340}{\textrm{LP89}}$]. In the present paper we develop an “affine” analogue of this theory by introducing the notion of a continuous Poisson vertex algebra and constructing triples of Poisson $\lambda$-brackets. We introduce the corresponding Adler type identities and apply them to integrability of hierarchies of Hamiltonian PDEs.

Keywords

Poisson vertex algebra, $R$-matrix, integrable hierarchy, Adler identity

2010 Mathematics Subject Classification

Primary 17B63. Secondary 17B08, 17B69, 17B80, 37K30.

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To Claudio Procesi, a friend and a source of inspiration

The first-named author is supported by the national PRIN project n. 2017YRA3LK, the PNRR MUR project PE0000023-NQSTI, the project MMNLP (Mathematical Methods in Non Linear Physics) of the INFN.

The second-named author is supported by the Simons collaboration grant and the Bert and Ann Kostant fund.

The third-named author is supported by the project MMNLP (Mathematical Methods in Non Linear Physics) of the INFN.

Received 3 August 2022

Received revised 24 January 2023

Accepted 13 February 2023

Published 15 May 2024