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arXiv · q-alg/9701016

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

Yves Brihaye·Jean Nuyts
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Abstract

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 $Δ=\mid M-N\mid\geq 2$). A family of algebras is also constructed, extending this particular algebra by $Δ-1$ arbitrary complex parameters.

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Yves Brihaye, Jean Nuyts. 1997-01-13. The hidden symmetry algebras of a class of quasi-exactly solvable multi dimensional operators. https://doi.org/10.1007/s002200050430

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