An algebra isomorphism on $U({\mathfrak{gl}}_n)$

Yang Li, Genqiang Liu

For each positive integer $n$, let ${\mathfrak{s}}_n={\mathfrak{gl}}_n⋉{\mathbb{C}}^n$. We show that $U(\mathfrak{s}_{n})_{X_{n}}\cong \mathcal{D}_{n}\otimes U(\mathfrak{s}_{n-1})…