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

RESUMO

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 $$ \operatorname{Oz}(U_q^{+}(B_2)) \cong \begin{cases} \mu_2 , & \text{if \(q\) is not a root of unity},\\[2mm] \mu_{\gcd(\ell,2)}, & \text{if \(q\) is a primitive \(m\)-th root of unity, \(m\geq5\).} \end{cases} $$ 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. Finally, in the root-of-unity case, we relate the ozone group to the normal elements of \(U_q^{+}(B_2)\) and show that, when \(\ell\) is odd, \(U_q^{+}(B_2)\) is Calabi--Yau and its center is Gorenstein.