Journal of Lie Theory

no. 3
Finite-dimensional Lie Subalgebras of the Weyl Algebra
Journal of Lie Theory, Volume 16 (2006) no. 3, pp. 427-454
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 $A_1$.
DOI: 10.5802/jolt.418
Classification: 16S32, 17B60
Keywords: Finite-dimensional Lie subalgebras, Weyl algebra, embeddings 
@article{JOLT_2006_16_3_a1,
     author = {M. Rausch de Traubenberg and M. J. Slupinski and A. Tanasa},
     title = {Finite-dimensional {Lie} {Subalgebras} of the {Weyl} {Algebra}},
     journal = {Journal of Lie Theory},
     pages = {427--454},
     year = {2006},
     volume = {16},
     number = {3},
     doi = {10.5802/jolt.418},
     zbl = {1115.16011},
     url = {https://jolt.centre-mersenne.org/articles/10.5802/jolt.418/}
}
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M. Rausch de Traubenberg; M. J. Slupinski; A. Tanasa. Finite-dimensional Lie Subalgebras of the Weyl Algebra. Journal of Lie Theory, Volume 16 (2006) no. 3, pp. 427-454. doi: 10.5802/jolt.418

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