Let $\mathfrak{g}$ be the Witt algebra or the positive Witt algebra. It is well known that the enveloping algebra $U(\mathfrak{g})$ has intermediate growth and thus infinite Gelfand–Kirillov (GK-) dimension. We prove that the GK-dimension of $U(\mathfrak{g})$ is just infinite in the sense that any proper quotient of $U(\mathfrak{g})$ has polynomial growth. This proves a conjecture of Petukhov and the second named author for the positive Witt algebra. We also establish the corresponding results for quotients of the symmetric algebra $S(\mathfrak{g})$ by proper Poisson ideals. In fact, we prove more generally that any central quotient of the universal enveloping algebra of the Virasoro algebra has just infinite GK-dimension. We give several applications. In particular, we easily compute the annihilators of Verma modules over the Virasoro algebra.

中文翻译：

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具有无限Gelfand–Kirillov维的包络代数

令\\ mathfrak {g} $为Witt代数或正Witt代数。众所周知，包络代数$ U（\ mathfrak {g}）$具有中间增长，因此具有无限的Gelfand–Kirillov（GK-）维。我们证明$ U（\ mathfrak {g}）$的GK维只是无限的，因为$ U（\ mathfrak {g}）$的任何适当商都具有多项式增长。这证明了Petukhov和正Witt代数的第二名作者的猜想。我们还通过正确的泊松理想建立了对称代数$ S（\ mathfrak {g}）$的商的相应结果。实际上，我们更笼统地证明，维拉索罗代数的通用包络代数的任何中心商都具有无限的GK维。我们给出了几个应用程序。特别是，我们可以轻松地在Virasoro代数上计算Verma模块的an灭子。