Enveloping algebras with just infinite Gelfand-Kirillov dimension

Iyudu, Natalia K .; Sierra, Susan J.

doi:10.4310/ARKIV.2020.v58.n2.a4

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Enveloping algebras with just infinite Gelfand-Kirillov dimension

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Iyudu, Natalia K .
Max Planck Institute for Mathematics, Max Planck Society;

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https://doi.org/10.4310/ARKIV.2020.v58.n2.a4
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arXiv:1905.07507.pdf
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Citation

Iyudu, N. K.., & Sierra, S. J. (2020). Enveloping algebras with just infinite Gelfand-Kirillov dimension. Arkiv för Matematik, 58(2), 285-306. doi:10.4310/ARKIV.2020.v58.n2.a4.

Cite as: https://hdl.handle.net/21.11116/0000-0007-8184-1

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.