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We study algebras of differential and difference operators acting on matrixvalued orthogonal polynomials (MVOPs) with respect to a weight matrix of theform $W^{(\nu)}_{\phi}(x) = x^{\nu}e^{-\phi(x)} W^{(\nu)}_{pol}(x)$, where$\nu>0$, $W^{(\nu)}_{pol}(x)$ is certain matrix valued polynomial and $\phi$ anentire function. We introduce a pair differential operators $\mathcal{D}$,$\mathcal{D}^{\dagger}$ which are mutually adjoint with respect to the matrixinner product induced by $W^{(\nu)}_{\phi}(x)$. We prove that the Lie algebragenerated by $\mathcal{D}$ and $\mathcal{D}^{\dagger}$ is finite dimensional ifand only if $\phi$ is a polynomial, giving a partial answer to a problem by M.Ismail. In the case $\phi$ polynomial, we describe the structure of this Liealgebra. The case $\phi(x)=x$, is discussed in detail. We derive difference anddifferential relations for the MVOPs. We give explicit expressions for theentries of the MVOPs in terms of classical Laguerre and Dual Hahn polynomials.

Lie algebras of differential operators for Matrix valued Laguerre type polynomials