ON ALGEBRAIC CLASSIFICATION OF QUASI-EXACTLY SOLVABLE MATRIX MODELS

We suggest a generalization of the Lie algebraic approach for construc ting quasi-exactly solvable one-dimensional Schrodinger equations. Thi s generalization is based on representations of Lie algebras by first- order matrix differential operators. We have classified inequivalent r epresentations of the Lie algebras of dimensions up to three by first- order matrix differential operators in one variable. Next we describe invariant finite-dimensional subspaces of the representation spaces of the one-, two-dimensional Lie algebras and of the algebra sl(2, R). T hese results enable us to construct multiparameter families of first-a nd second-order quasi-exactly solvable models. in particular, we have obtained two classes of quasi-exactly solvable matrix Schrodinger equa tions.