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

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

19 pages, latex

References in corpus (2)

• Remarks on Finite W Algebras
• Irreducible Representations of an Algebra underlying Hidden Symmetries of a class of Quasi Exactly Solvable Systems of Equations

Cited by in corpus (2)

• Quasi exactly solvable operators and Lie superalgebras
• spl(p,q) superalgebra and differential operators