%0 Journal Article
%T The hidden symmetry algebras of a class of quasi-exactly solvable multi dimensional operators
%A Yves Brihaye
%A Jean Nuyts
%J Mathematics
%D 1997
%I arXiv
%R 10.1007/s002200050430
%X 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 $P(M,V) \oplus P(N,V)$ for any values of the integers $M,N,V$. We show that the operators preserving $P(M,V) \oplus P(N,V)$ generate an abstract superalgebra (non linear if $\Delta=\mid M-N\mid\geq 2$). A family of algebras is also constructed, extending this particular algebra by $\Delta -1$ arbitrary complex parameters.
%U http://arxiv.org/abs/q-alg/9701016v1