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PreprintCC BY 4.0arXiv

The ozone group of Uq+(B2)

  • James Gómez
  • Helbert Venegas
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Abstract

We determine the ozone group of Uq+(B2). More precisely, if ℓ=ord(q2) when q is a root of unity, we obtain Oz(Uq+(B2)) ≅ μ2 , & if (q) is not a root of unity, [2mm] μgcd(ℓ,2), & if (q) is a primitive (m)−th root of unity, (m≥5). We also study several homological properties of Uq+(B2), showing that it is Artin--Schelter regular of global dimension 4, Auslander-regular, Cohen--Macaulay, strongly Noetherian, and skew Calabi--Yau. Using two different iterated Ore presentations, we show that its Nakayama automorphism is trivial for every q ∈ K*, so Uq+(B2) is Calabi--Yau of dimension $4$; for q not a root of unity this also follows from work of Goodearl and Yakimov. When q is a primitive m-th root of unity with m≥5, we deduce that the center is Gorenstein, we check directly that the ozone group acts with trivial homological determinant, and we describe the nonzero normal elements in terms of the ozone group.

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Cite

BibTeXRIS
BibTeX
@misc{gomez2026ozone,
  title = {{The ozone group of $U_q^+(B_2)$}},
  author = {James Gómez and Helbert Venegas},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2609.04919},
  eprint = {2609.04919},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.04919},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - The ozone group of U_q^+(B_2)
AU  - James Gómez
AU  - Helbert Venegas
PY  - 2026
DA  - 2026/10/14
PB  - arXiv
DO  - 10.48550/arxiv.2609.04919
UR  - https://arxiv.org/abs/2609.04919
AB  - We determine the ozone group of U_q^+(B_2). More precisely, if ℓ=ord(q^2) when q is a root of unity, we obtain Oz(U_q^+(B_2)) ≅ μ_2 , & if (q) is not a root of unity, [2mm] μ_gcd(ℓ,2), & if (q) is a primitive (m)−th root of unity, (m≥5). We also study several homological properties of U_q^+(B_2), showing that it is Artin--Schelter regular of global dimension 4, Auslander-regular, Cohen--Macaulay, strongly Noetherian, and skew Calabi--Yau. Using two different iterated Ore presentations, we show that its Nakayama automorphism is trivial for every q ∈ K^*, so U_q^+(B_2) is Calabi--Yau of dimension $4$; for q not a root of unity this also follows from work of Goodearl and Yakimov. When q is a primitive m-th root of unity with m≥5, we deduce that the center is Gorenstein, we check directly that the ozone group acts with trivial homological determinant, and we describe the nonzero normal elements in terms of the ozone group.
ER  -