# Enveloping algebras with just infinite Gelfand-Kirillov dimension

## TL;DR

This paper proves that the enveloping algebra of the Witt algebra and related structures have just infinite Gelfand-Kirillov dimension, meaning all proper quotients have polynomial growth, confirming a conjecture and providing new insights into their structure.

## Contribution

It establishes that the enveloping algebra of the Witt algebra and Virasoro algebra have just infinite GK-dimension, confirming a conjecture and analyzing quotients and their growth.

## Key findings

- Enveloping algebra of Witt algebra has just infinite GK-dimension.
- Proper quotients of these algebras have polynomial growth.
- Results apply to quotients of symmetric and universal enveloping algebras.

## 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.