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

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

2008-04-22

arxiv.org/abs/0804.3572v4

J. Phys. A: Math. Theor. 42 (2009) 365211

Mathematics

Classical Analysis and ODEs

23 pages, accepted on publication on J. Phys. A., electronic link: http://stacks.iop.org/1751-8121/42/365211

Scientific paper

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.

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