Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Lie-algebras and linear operators with invariant subspaces   A general classification of linear differential and finite-difference
operators possessing a finite-dimensional invariant subspace with a polynomial
basis (the generalized Bochner problem) 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 some algebra of differential
(difference) operators in finite-dimensional representation plus an operator
annihilating the finite-dimensional invariant subspace. In low dimensions 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 ${\bold R}$),
$sl_3({\bold R})$, $sl_2({\bold R}) \oplus sl_2({\bold R})$ and $gl_2 ({\bold
R}) \ltimes {\bold R}^{r+1}\ , r$ a natural number (operators in ${\bold
R^2}$). A classification of linear operators possessing infinitely many
finite-dimensional invariant subspaces with a basis in polynomials is
presented. A connection to the recently-discovered quasi-exactly-solvable
spectral problems is discussed.

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The categories of this paper are: q-fin
12
The categories of this paper are: nucl-th
1
The categories of this paper are: cs
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The categories of this paper are: alg-geom
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The categories of this paper are: comp-gas
36
The categories of this paper are: cmp-lg
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The categories of this paper are: solv-int
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The categories of this paper are: math-ph
32
The categories of this paper are: atom-ph
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The categories of this paper are: nlin
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The categories of this paper are: patt-sol
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The categories of this paper are: econ
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The categories of this paper are: ao-sci
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The categories of this paper are: physics
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The categories of this paper are: bayes-an
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The categories of this paper are: chem-ph