The hidden symmetry algebras of a class of quasi - exactly solvable multi dimensional operators

Let P (N, V) denote the vector space of polynomials of maximal degree less than or equal to N in V independent variables. This space is preserved by the enveloping algebra generated by a set of linear, differential operators representing the Lie algebra gl(V + 1). We establish the counterpart of this property for the vector space P (M, V) ⊕ P (N, V) for any values of the integers M, N, V. We show that the operators preserving P (M, V) ⊕ P (N, V) generate an abstract superalgebra (non linear if ∆ =| M − N |≥ 2). A family of algebras is also constructed, extending this particular algebra by ∆ − 1 arbitrary complex parameters. de la Recherche Scientifique.