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Enveloping algebras with just infinite Gelfand-Kirillov dimension

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Enveloping algebras with just infinite Gelfand-Kirillov dimension. / Iyudu, Natalia K.; Sierra, Susan J.
In: Arkiv för Matematik, Vol. 58, No. 2, 04.03.2020, p. 285-306.

Research output: Contribution to Journal/Magazine › Journal article › peer-review

Harvard

Iyudu, NK & Sierra, SJ 2020, 'Enveloping algebras with just infinite Gelfand-Kirillov dimension', Arkiv för Matematik, vol. 58, no. 2, pp. 285-306. https://doi.org/10.4310/ARKIV.2020.v58.n2.a4

APA

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

Vancouver

Iyudu NK, Sierra SJ. Enveloping algebras with just infinite Gelfand-Kirillov dimension. Arkiv för Matematik. 2020 Mar 4;58(2):285-306. doi: 10.4310/ARKIV.2020.v58.n2.a4

Author

Iyudu, Natalia K. ; Sierra, Susan J. / Enveloping algebras with just infinite Gelfand-Kirillov dimension. In: Arkiv för Matematik. 2020 ; Vol. 58, No. 2. pp. 285-306.

Bibtex

@article{1d2d815f02f249b8a948059be4a2c737,
title = "Enveloping algebras with just infinite Gelfand-Kirillov dimension",
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.",
keywords = "Witt algebra, positive Witt algebra, Virasoro algebra, Gelfand-Kirillov dimension",
author = "Iyudu, {Natalia K.} and Sierra, {Susan J.}",
year = "2020",
month = mar,
day = "4",
doi = "10.4310/ARKIV.2020.v58.n2.a4",
language = "English",
volume = "58",
pages = "285--306",
journal = "Arkiv f{\"o}r Matematik",
issn = "0004-2080",
publisher = "Springer Netherlands",
number = "2",

}

RIS

TY - JOUR

T1 - Enveloping algebras with just infinite Gelfand-Kirillov dimension

AU - Iyudu, Natalia K.

AU - Sierra, Susan J.

PY - 2020/3/4

Y1 - 2020/3/4

N2 - 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.

AB - 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.

KW - Witt algebra

KW - positive Witt algebra

KW - Virasoro algebra

KW - Gelfand-Kirillov dimension

U2 - 10.4310/ARKIV.2020.v58.n2.a4

DO - 10.4310/ARKIV.2020.v58.n2.a4

M3 - Journal article

VL - 58

SP - 285

EP - 306

JO - Arkiv för Matematik

JF - Arkiv för Matematik

SN - 0004-2080

IS - 2

ER -