On Algebraic Classiication of Quasi-exactly Solvable Matrix Models

We suggest a generalization of the Lie algebraic approach for constructing quasi-exactly solvable one-dimensional Schrr odinger equations which is due to Shifman and Turbiner in order to include into consideration matrix models. This generalization is based on representations of Lie algebras by rst-order matrix diierential operators. We have classiied inequivalent representations of the Lie algebras of the dimension up to three by rst-order matrix diierential operators in one variable. Next we describe invariant nite-dimensional sub-spaces of the representation spaces of the one-, two-dimensional Lie algebras and of the algebra sl(2; R). These results enable constructing multi-parameter families of rst-and second-order quasi-exactly solvable models. In particular, we have obtained two classes of quasi-exactly solvable matrix Schrr odinger equations.