Derivations on differential operator algebra and Weyl algebra (Q1917813)

The author obtains the \(n\)-th Weyl algebra \(A_n(A)\) over an algebra \(A\) as an iterated smash product when an \(n\)-dimensional nilpotent Lie algebra of locally nilpotent derivations acts on \(A\). Specifically, if \(L\) is an \(n\)-dimensional nilpotent Lie algebra with standard basis \(\{x_1,\dots,x_n\}\) and universal enveloping algebra \(U(L)\), and if each \(x_i\) acts as a locally nilpotent derivation on the algebra \(A\) over the field \(K\), then \(A \otimes A_n(K)\) is isomorphic to \((A\# U(L))\# K[y_1,\dots,y_n]\) as a \(K\)-algebra in such a way that each \(y_i\) acts as a derivation on \(U(L)\) satisfying the conditions: \(y_i(x_j)=0\) when \(j<i\), \(y_i(x_i)=1\), \(y_i([x_j,x_s])=0\) when \(j\geq i\geq s\). The existence of such derivations of \(U(L)\), for any \(n\)-dimensional Lie algebra \(L\), is equivalent to \(L\) being nilpotent.