The Lie Algebraic Structure of Differential Operators Admitting Invariant Spaces of Polynomials

Details The Lie Algebraic Structure of Differential Operators Admitting Invariant Spaces of Polynomials The Lie Algebraic Structure of Differential Operators Admitting Invariant Spaces of Polynomials

1996-12-18

arxiv.org/abs/q-alg/9612027v3

Adv. Appl. Math., 20 (1998) 300-322

Physics

Mathematical Physics

LaTeX, 26 pages. In final form, to appear in Adv. Appl. Math

Scientific paper

We prove that the scalar and $2\times 2$ matrix differential operators which preserve the simplest scalar and vector-valued polynomial modules in two variables have a fundamental Lie algebraic structure. Our approach is based on a general graphical method which does not require the modules to be irreducible under the action of the corresponding Lie (super)algebra. This method can be generalized to modules of polynomials in an arbitrary number of variables. We give generic examples of partially solvable differential operators which are not Lie algebraic. We show that certain vector-valued modules give rise to new realizations of finite-dimensional Lie superalgebras by first-order differential operators.

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