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Journal of Lie Theory 16 (2006), No. 3, 427--454
Copyright Heldermann Verlag 2006

Finite-dimensional Lie Subalgebras of the Weyl Algebra

Michel Rausch de Traubenberg
Lab. de Physique Théorique, CNRS UMR 7085, Université Louis Pasteur, 3 rue de l'Université, 67084 Strasbourg, France

Marcus J. Slupinski
Institut de Recherches en Mathématique Avancée, Université Louis Pasteur, 7 rue R. Descartes, 67084 Strasbourg, France

Adrian Tanasa
Lab. Mathématiques Informatique Applications, Université de Haute Alsace, Faculté de Sciences et Techniques, 4 rue des Frères Lumières, 68093 Mulhouse, France


We classify up to isomorphism all finite-dimensional Lie algebras that can be realised as Lie subalgebras of the complex Weyl algebra $A_1$. The list we obtain turns out to be countable and, for example, the only non-solvable Lie algebras with this property are: $\frak{sl}(2)$, $\frak{sl}(2)\times{\bf C}$ and $\frak{sl}(2)\ltimes{\cal H}_3$. We then give several different characterisations, normal forms and isotropy groups for the action of ${\rm Aut}(A_1)\times {\rm Aut}(\frak{sl}(2))$ on a class of realisations of $\frak{sl}(2)$ in $A_1$.

Keywords: Finite-dimensional Lie subalgebras, Weyl algebra, embeddings.

MSC: 16S32, 17B60

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