Algebraic interpretation of discrete families of matrix valued orthogonal polynomials

An algebraic interpretation of matrix-valued orthogonal polynomials (MVOPs) is provided. The construction is based on representations of a ($q$-deformed) Lie algebra $\mathfrak{g}$ into the algebra ${\mathrm{End}}_{M_n(\mathbb{C})}(M)$ of $M_n(\mathbb{C})$-linear maps over a $M_n(\mathbb{C})$-module $M$. Cases corresponding to the Lie algebras $\mathfrak{s}\mathfrak{u}(2)$ and $\mathfrak{s}\mathfrak{u}(1,1)$ as well as to the $q$-deformed algebra ${\mathfrak{s}\mathfrak{o}}_q(3)$ at $q$ a root of unity are presented; they lead to matrix analogs of the Krawtchouk, Meixner and discrete Chebyshev polynomials.