Enveloping algebras with just infinite Gelfand-Kirillov dimension

Iyudu, Natalia K. and Sierra, Susan J. (2020) Enveloping algebras with just infinite Gelfand-Kirillov dimension. Arkiv för Matematik, 58 (2). pp. 285-306. ISSN 0004-2080

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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.

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Arkiv för Matematik
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09 Jun 2021 19:10
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22 Nov 2022 10:19