of the quotient algebra $U({\mathfrak{n}})/J$, which is isomorphic to a Weyl algebra, see Theorem \[theo:Weyl\_quotient\]. Next, in Section \[sect:infinite\_dim\], we consider the infinite-dimensional case. In Subsection \[sst:centers\_ifd\], we briefly recall a description of $Z({\mathfrak{n}})$ from [@IgnatyevPenkov1 Theorem 2.6]. Finally, in Subsection \[sst:ideals\_ifd\] we restrict ourselves on the special choice of ${\mathfrak{n}}$ and, using Proposition \[prop:n\_annihilating\_n\_plus\_1\], prove our second main result, Theorem \[theo:ideal\_ifd\], describing centrally generated ideals of $U({\mathfrak{n}})$.

<span style="font-variant:small-caps;">Acknowledgments</span>. I express my gratitude to A. Melnikov, A. Panov, I. Penkov and A. Shevchenko for useful discussions. The work on Section \[sect:finite\_dim\] was done during my stay at the University of Haifa. I thank this institution for its hospitality.

**.. [The\[sst:center\_fd\] center of $U({\mathfrak{n}})$]{}.**

Let $n\in{\mathbb{Z}}_{>0}$. Throughout this subsection ${\mathfrak{g}}$ denotes one of the Lie algebras ${\mathfrak{sl}}_n({\mathbb{C}})$, ${\mathfrak{so}}_{2n}({\mathbb{C}})$, ${\mathfrak{so}}_{2n+1}({\mathbb{C}})$ or ${\mathfrak{sp}}_{2n}({\mathbb{C}})$. The algebra ${\mathfrak{so}}_{2n}({\mathbb{C}})$ (respectively, ${\mathfrak{so}}_{2n+1}({\mathbb{C}})$ and ${\mathfrak{sp}}_{2n}({\mathbb{C}})$) is realized as the subalgebra of ${\mathfrak{sl}}_{2n}({\mathbb{C}})$ (respectively, ${\mathfrak{sl}}_{2n+1}({\mathbb{C}})$ and ${\mathfrak{sl}}_{2n}({\mathbb{C}})$) consisting of all $x$ such that $\beta(u,xv)+\beta(xu,v)=0$ for all $u,v$ in ${\mathbb{C}}^{2n}$ (respectively, in ${\mathbb{C}}^{2n+1}$ and ${\mathbb{C}}^{2n}$), where $$\beta(u,v)=\begin{cases}
\sum\nolimits_{i=1}^n(u_iv_{-i}+u_{-i}v_i)&\text{for }{\mathfrak{so}}_{2n}({\mathbb{C}}),\\
u_0v_0+\sum\nolimits_{i=1}^n(u_iv_{-i}+u_{-i}v_i)&\text{for }{\mathfrak{so}}_{2n+1}({\mathbb{C}}),\\
\sum\nolimits_{i=1}^n(u_iv_{-i}-u_{-i}v_i)&\text{for }{\mathfrak{sp}}_{2n}({\mathbb{C}}).
\end{cases}$$ Here for ${\mathfrak{so}}_{2n}({\mathbb{C}})$ (respectively, for ${\mathfrak{so}}_{2n+1}$ and ${\mathfrak{sp}}_{2n}({\mathbb{C}}))$ we denote by $e_1,\ldots,e_n,e_{-n},\ldots,e_{-1}$ (respectively, by $e_1,\ldots,e_n,e_0,e_{-n},\ldots,e_{-1}$ and $e_1,\ldots,e_n,e_{-n},\ldots,e_{-1}$) the standard basis of ${\mathbb{C}}^{2n}$ (respectively, of ${\mathbb{C}}^{2n+1}$ 