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

Abstract

Adler and van Moerbeke (2001 Commun. Pure Appl. Math. 54 153–205) described a reduction of the 2D-Toda hierarchy called the Toeplitz lattice. This hierarchy turns out to be equivalent to the one originally described by Ablowitz and Ladik (1975 J. Math. Phys. 16 598–603) 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 leads us to the semidiscrete zero-curvature equations for the non-Abelian (or multicomponent) version of the Ablowitz–Ladik equations (Gerdzhikov and Ivanov 1982 Theor. Math. Phys. 52 676–85). In this way, we extend the link between biorthogonal polynomials on the unit circle and the Ablowitz–Ladik hierarchy to the matrix case.