The algebra of differential operators associated to a family of matrix-valued orthogonal polynomials: Five instructive examples

Consider a fixed family of orthogonal matrix polynomials Pn that are common eigenfunctions of some differential operator L with matrix coefficients and a matrix-valued eigenvalue Λn, PnL = ΛnPn, n ≥ 0. We study the algebra of all such differential operators going along with the family Pn. The problem is explored through a detailed look at some explicit examples. Whereas in the scalar case (usually connected with the names of Hermite, Laguerre, and Jacobi) this algebra is commutative and has one generator, the examples given in this paper point to a very rich picture: each example discussed here behaves in a different fashion. In each example, we display the generators of the (generally noncommutative) algebra, and we give explicit polynomial relations satisfied by these generators. In the scalar case, the study of algebras like these ones has important connections with soliton-type equations. One can expect that these more complicated algebras would play a role in the study of nonabelian nonlinear evolution equations.