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