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