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October 2020 Enveloping algebras with just infinite Gelfand–Kirillov dimension
Natalia K. Iyudu, Susan J. Sierra
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Ark. Mat. 58(2): 285-306 (October 2020). DOI: 10.4310/ARKIV.2020.v58.n2.a4


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.

Funding Statement

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


Download Citation

Natalia K. Iyudu. Susan J. Sierra. "Enveloping algebras with just infinite Gelfand–Kirillov dimension." Ark. Mat. 58 (2) 285 - 306, October 2020.


Received: 17 October 2019; Revised: 21 February 2020; Published: October 2020
First available in Project Euclid: 16 January 2021

Digital Object Identifier: 10.4310/ARKIV.2020.v58.n2.a4

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

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

Rights: Copyright © 2020 Institut Mittag-Leffler


Vol.58 • No. 2 • October 2020
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