The ozone group of $U_q^{+}(B_2)$

Abstract

We determine the ozone group of \(U_q^{+}(B_2)\). More precisely, if \(\ell=\operatorname{ord}(q^2)\) when \(q\) is a root of unity, we obtain $$\mathrm{Oz}(U_q^{+}(B_2))\cong \{\begin{array}{ll}{\mu }_2,& \text{if q is not a root of unity},\\ {\mu }_{\gcd (\ell ,2)},& \text{if q is a primitive m-th root of unity, m≥5.}\end{array}$$ 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 $q \in \K^*$, 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 $m\ge 5$, 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.

arXiv commentThe Calabi-Yau property is now proved for every q (previously only when ell is odd); added results on the homological determinant of the ozone group action and minor corrections