2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/149130A general classification of linear differential and finite-difference operators possessing a finite-dimensional invariant subspace with a polynomial basis is given. The main result is that any operator with the above property must have a representation as a polynomial element of the universal enveloping algebra of the algebra of differential (difference) operators in finite-dimensional representation. In one-dimensional case a classification is given by algebras $sl_2({\bold R})$ (for differential operators in ${\bold R}$) and $sl_2({\bold R})_q$ (for finite-difference operators in ${\bold R}$), $osp(2,2)$ (operators in one real and one Grassmann variable, or equivalently, $2 \times 2$ matrix operators in ${\bold R}$) and $gl_2 ({\bold R})_K$ ( for the operators containing the differential operators and the parity operator). A classification of linear operators possessing infinitely many finite-dimensional invariant subspaces with a basis in polynomials is presented.33 pages, to appear as Chapter 12 in CRC Handbook of Lie Group Analysis of Differential Equations, Vol. 3 : New Trends in Theoretical Developments and Computational Methods, ed. N. H. Ibragimov, CRC Press, 1995High Energy Physics - TheoryFunctional Analysistext