Abstract
Series of finite-dimensional representations of the superalgebrasspl(p,q) can be formulated in terms of linear differentialoperators acting on a suitable space of polynomials. We sketch the generalingredients necessary to construct these representations and presentexamples related to spl(2,1) and spl(2,2). By revisiting the products ofprojectivised representations of sl(2), we are able to construct new sets ofdifferential operators preserving some space of polynomials in two or morevariables. In particular, this allows us to express the representation ofspl(2,1) in terms of matrix differential operators in two variables. Thecorresponding operators provide the building blocks for the construction ofquasi-exactly solvable systems of two and four equations in two variables.We also present a quommutator deformation of spl(2,1) which, by constructionprovides an appropriate basis for analyzing the quasi exactly solvablesystems of finite difference equations.
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Brihaye, Y., Giller, S. & Kosinski, P. The Superalgebra spl(p,q) and Differential Operators. Acta Applicandae Mathematicae 54, 167–184 (1998). https://doi.org/10.1023/A:1006083505435
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DOI: https://doi.org/10.1023/A:1006083505435