Finite-dimensional Lie subalgebras of the Weyl algebra

Abstract

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 $A_1$.