Matrix biorthogonal polynomials on the unit circle and non-Abelian Ablowitz-Ladik hierarchy

TL;DR

This paper extends the connection between biorthogonal polynomials on the unit circle and the Ablowitz-Ladik hierarchy to the matrix case, deriving semidiscrete zero-curvature equations for the non-Abelian version.

Contribution

It generalizes the semidiscrete zero-curvature equations from scalar to matrix biorthogonal polynomials, establishing a non-Abelian Ablowitz-Ladik hierarchy.

Findings

Derived semidiscrete zero-curvature equations for matrix case

Extended the link between biorthogonal polynomials and integrable hierarchies

Established non-Abelian Ablowitz-Ladik equations

Abstract

Adler and van Moerbeke \cite{AVM} described a reduction of 2D-Toda hierarchy called Toeplitz lattice. This hierarchy turns out to be equivalent to the one originally described by Ablowitz and Ladik \cite{AL} using semidiscrete zero-curvature equations. In this paper we obtain the original semidiscrete zero-curvature equations starting directly from the Toeplitz lattice and we generalize these computations to the matrix case. This generalization lead us to the semidiscrete zero-curvature equations for the non-abelian (or multicomponent) version of Ablowitz-Ladik equations \cite{GI}. In this way we extend the link between biorthogonal polynomials on the unit circle and Ablowitz-Ladik hierarchy to the matrix case.