Matrix orthogonal polynomials, non-abelian Toda lattices, and Bäcklund transformations

A connection between matrix orthogonal polynomials and non-abelian integrable lattices is investigated in this paper. The normalization factors of matrix orthogonal polynomials expressed using quasi-determinants are shown to be the solutions to the non-abelian Toda lattice in semi-discrete and full-discrete cases. Moreover, with a moment modification method, we demonstrate that the Bäcklund transformation of the non-abelian Toda lattice given by Popowicz (1983) is equivalent to the non-abelian Volterra lattice, whose solutions can be expressed using quasi-determinants as well.