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Enveloping algebras with just infinite Gelfand-Kirillov dimension

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Published
<mark>Journal publication date</mark>4/03/2020
<mark>Journal</mark>Arkiv för Matematik
Issue number2
Volume58
Number of pages22
Pages (from-to)285-306
Publication StatusPublished
<mark>Original language</mark>English

Abstract

Let $\mf g$ be the Witt algebra or the positive Witt algebra. It is well known that the enveloping algebra $U(\mf g )$ has intermediate growth and thus infinite Gelfand-Kirillov (GK-) dimension. We prove that the GK-dimension of $U(\mf g)$ is {\em just infinite} in the sense that any proper quotient of $U(\mf g)$ 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 $S(\mf g)$ 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.