Finite-dimensional Lie subalgebras of the Weyl algebra

de Traubenberg, M. Rausch; Slupinski, M. J.; Tanasa, A.

Finite-dimensional Lie subalgebras of the Weyl algebra

Authors

de Traubenberg, M. Rausch

Slupinski, M. J.

Tanasa, A.

Description

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 discrete and for example, the only non-solvable Lie algebras with this property are: $sl(2)$, $sl(2)\times\mathbb C$ and $sl(2)\ltimes{\cal H}_3$. We then give several different characterisations, normal forms and isotropy groups for the action of $Aut (A_1)\times Aut (sl(2))$ on a particular class of realisations of $sl(2)$ in $A_1$.
Latex 27 pages; some proofs are given with more details