Toda lattice, special functions and their matrix analogues

The classical Toda lattice is a model for a one-dimensional crystal. After a transformation in Flaschka coordinates there exists a Lax pair, for which the operator acts as a three-term recurrence operator. This gives a link to orthogonal polynomials, special functions and Lie algebra representations. In the case of orthogonal polynomials, the time dependence in the Toda lattice corresponds to deformation of the orthogonality measure by an exponential. The Lax pair setting can be extended to include more generally special functions, or a multivariable setting. The nonabelian Toda lattice is a generalisation of the Toda lattice for which matrix valued orthogonal polynomials play a similar role. We discuss matrix polynomials, and we discuss an explicit example of such a nonabelian Toda lattice.