Boundary Lax pairs from non-ultra local Poisson algebras

TL;DR

This paper introduces new boundary extensions of non-ultra local Poisson algebras, providing a systematic construction of classical Lax pairs and exploring their physical applications, notably in the classical boundary PCM model.

Contribution

It develops a framework for boundary extensions of non-ultra local Poisson algebras, linking classical and quantum boundary algebras, and constructs associated Lax pairs.

Findings

Boundary extensions depend on a boundary scalar matrix and an anti-automorphism.

The non-ultra local part of the original algebra vanishes for any parameter choice.

The classical boundary PCM model is analyzed as an example.

Abstract

We consider non-ultra local linear Poisson algebras on a continuous line . Suitable combinations of representations of these algebras yield representations of novel generalized linear Poisson algebras or "boundary" extensions. They are parametrized by a "boundary" scalar matrix and depend in addition on the choice of an anti-automorphism. The new algebras are the classical-linear counterparts of known quadratic quantum boundary algebras. For any choice of parameters the non-ultra local contribution of the original Poisson algebra disappears. We also systematically construct the associated classical Lax pair. The classical boundary PCM model is examined as a physical example.