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# Arkiv för Matematik

## Volume 58 (2020)

### Number 2

### Enveloping algebras with just infinite Gelfand–Kirillov dimension

Pages: 285 – 306

DOI: https://dx.doi.org/10.4310/ARKIV.2020.v58.n2.a4

#### Authors

#### Abstract

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.

#### Keywords

Witt algebra, positive Witt algebra, Virasoro algebra, Gelfand–Kirillov dimension

#### 2010 Mathematics Subject Classification

Primary 16P90, 16S30, 17B68. Secondary 17B65.

This work is funded by the EPSRC grant EP/M008460/1/.

Received 17 October 2019

Received revised 21 February 2020

Accepted 1 March 2020

Published 3 November 2020