Lie-algebras and linear operators with invariant subspaces. (English) Zbl 0809.17023

Turbiner, Alexander

Kamran, Niky (ed.) et al., Lie algebras, cohomology, and new applications to quantum mechanics. AMS special session on Lie algebras, cohomology, and new applications to quantum mechanics, March 20-21, 1992, Southwest Missouri State University, Springfield, MO, USA. Providence, RI: American Mathematical Society. Contemp. Math. 160, 263-310 (1994).

This paper is connected to the so called Bochner problem concerning the classification of linear differential operators with a finite-dimensional invariant subspace generated by polynomials. The results are presented in terms of representations of Lie algebras and enveloping algebras.
Main Theorem: Consider a Lie algebra \({\mathfrak g}\) of first order differential operators, which possesses a finite-dimensional irreducible representation \(P\). Any linear differential operator acting on \(P\) can be represented by a polynomial in generators of the algebra \(\mathfrak g\) plus an operator annihilating \(P\). Some lower-dimensional cases (operators in one or two variables, or in one real and one Grassmann variable) are discussed and classification results with respect to the Bochner problem are given. See also B. D. Lowe, M. Pilant, and W. Rundell [SIAM J. Math. Anal. 23, No. 2, 482-504 (1992; Zbl 0763.34005)].
For the entire collection see [Zbl 0793.00019].

Reviewer: G.Czichowski (Greifswald)

MSC:

81R05	Finite-dimensional groups and algebras motivated by physics and their representations
17B81	Applications of Lie (super)algebras to physics, etc.
22E70	Applications of Lie groups to the sciences; explicit representations
33C80	Connections of hypergeometric functions with groups and algebras, and related topics
34A05	Explicit solutions, first integrals of ordinary differential equations
39A70	Difference operators
81U05	\(2\)-body potential quantum scattering theory

Keywords:

Lie algebra of first order differential operators; Bochner problem; classification of linear differential operators; invariant subspace generated by polynomials; representation