Enveloping algebras with just infinite Gelfand–Kirillov dimension

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.

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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