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Real Lie algebras of differential operators and quasi-exactly solvable potentials. (English) Zbl 0872.17021

The primary objects of interest for the construction of real quasi-exactly solvable Schrödinger operators are finite-dimensional real Lie algebras of real-valued first-order differential operators on a manifold \(M\) which admit a finite-dimensional module of smooth complex-valued functions. They are represented by triples \(({\mathcal H, M},F)\), where \(\mathcal H\) is a finite-dimensional Lie algebra of vector fields on \(M\), \(\mathcal M\) is an \(\mathcal H\)-submodule of scalar-valued functions \({\mathcal F}(M)\), and \(F\in Z^1({\mathcal H,Q})\) is a \({\mathcal Q}={\mathcal F}/{\mathcal M}\)-valued 1-cocycle on \(\mathcal H\).
It is the (local) classification of such objects in the two-dimensional case which forms the primary focus of this paper.
The starting point is Lie’s complete classification of the finite-dimensional Lie algebras of vector fields in two complex variables. Such a classification in the real case was done by the authors in an earlier paper [Proc. Lond. Math. Soc., III. Ser. 64, No. 2, 339-368 (1992; Zbl 0872.17022)] as well as the classification of the finite-dimensional Lie algebras of first-order differential operators in two complex variables [Am. J. Math. 114, 1163-1185 (1992; Zbl 0781.17011)].
In particular, it is shown that the five additional Lie algebras of vector fields in the real plane not being simple restrictions of the complex ones in Lie’s list, the only one admitting an extension to a Lie algebra of first-order differential operators by a non-trivial real-valued cocycle is a family of central extensions of \(so(3,1)\). They are applied to find a few interesting new examples of real quasi-exactly solvable Schrödinger operators in two-dimensional space.

MSC:

17B66 Lie algebras of vector fields and related (super) algebras
17B81 Applications of Lie (super)algebras to physics, etc.
81R10 Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations
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