arXiv subjects

James Gómez

Publications and source records attributed to James Gómez.

4 recordsLinked to original sources

arXiv2026

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

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 $$ \operatorname{Oz}(U_q^{+}(B_2)) \cong \begin{cases} μ_2 , & \text{if $q$ is not a root of unity},\\[2mm] μ_{\gcd(\ell,2)}, & \text{if $q$ is a primitive $m$-th root of unity, $m\geq5$.} \end{cases} $$ 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. 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 $m\geq5$, 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.

math.RA↗

arXiv2026

The ozone groups of the algebras $B_q(f)$

Let $q$ be a primitive $n$-th root of unity, $n>1$, and let $f$ be a nonzero polynomial such that $n\nmid(j+1)$ for every $j\in\supp(f)$. Set $e=\gcd(n,\{j+1:j\in\supp(f)\})$. We show that $\Oz(B_q(f))\congμ_e\timesμ_e$: the defining relations are homogeneous for a $\mathbb{Z}/e\times\mathbb{Z}/e$ grading, the center sits in degree zero, and the ozone group is the character group of that grading. The determination of the ozone group only requires the central elements $u^n$, $v^n$, and $Ω$. The regular normal elements modulo the center form the same group, generated by $u^{n/e}$ and $v^{n/e}$, so every normal element is central exactly when $e=1$. For $f=t^2$ we recover a computation of Chan, Gaddis, Won and Zhang, and for $e>1$ we obtain an infinite family of Calabi--Yau algebras with nontrivial ozone group.

math.RA↗

arXiv2026

The PI Property in Algebras of Polynomial Type

In this article, we study the PI property for several families of noncommutative algebras of polynomial type. Specifically, we review criteria for the PI property in double Ore extensions, two-parameter quantum Heisenberg algebras, two-parameter quantum matrix algebras, the algebra $U_q^+(B_2)$, multiparametric quantum Weyl algebras, biquadratic algebras with three generators, Noetherian Down--Up algebras, and the recently introduced algebras $B_q(f)$. In several cases, we include detailed proofs of known results, provide proofs of some identities used in the literature, and present alternative proofs of results characterizing the PI property for some of these algebras. For Noetherian Down--Up algebras, we highlight the relationship between the PI property, finiteness over the center, and the FBN property. Finally, for the algebras $B_q(f)$, we prove that they admit a PBW basis and show that the PI property can be controlled in terms of the support of the polynomial $f$.

math.RA↗

arXiv2026

The PI property of skew PBW extensions

In this article we study the polynomial identity (PI) property of skew PBW extensions. We show that every bijective skew PBW extension over a prime PI-algebra has nontrivial center. This fact allows us to determine, from the known description of the center in several classes of examples, whether such extensions satisfy a polynomial identity. Furthermore, building on results of Brown and Zhang \cite{BrownZhang2022}, we investigate the PI property of certain $\K$-algebras over fields of positive characteristic.

math.RA↗