arXiv subjects

Helbert Venegas

Publications and source records attributed to Helbert Venegas.

6 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 $$ \operatorname{Oz}(U_q^{+}(B_2)) \cong \begin{cases} \mu_2 , & \text{if $q$ is not a root of unity},\\[2mm] \mu_{\gcd(\ell,2)}, & \text{if $q$ is a primitive $m$-th root of unity, $m\geq5$.} \end{cases} $$ 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.

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 $\Oz(B_q(f))\cong\mu_e\times\mu_e$: 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 $\Omega$. 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.

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arXiv2019

The center of the total ring of fractions

Let $A$ be a right Ore domain, $Z(A)$ be the center of $A$ and $Q_r(A)$ be the right total ring of fractions of $A$. If $K$ is a field and $A$ is a $K$-algebra, in this short paper we prove that if $A$ is finitely generated and ${\rm GKdim}(A)<{\rm GKdim}(Z(A))+1$, then $Z(Q_r(A))\cong Q(Z(A))$. Many examples that illustrate the theorem are included, most of them within the skew $PBW$ extensions.

math.RA↗

arXiv2019

Gelfand-Kirillov dimension for rings

The classical Gelfand-Kirillov dimension for algebras over fields has been extended recently by J. Bell and J.J Zhang to algebras over commutative domains. However, the behavior of this new notion has not been enough investigated for the principal algebraic constructions as polynomial rings, matrix rings, localizations, filtered-graded rings, skew $PBW$ extensions, etc. In this paper, we present complete proofs of the computation of this more general dimension for the mentioned algebraic constructions for algebras over commutative domains. The Gelfand-Kirillov dimension for modules and the Gelfand-Kirillov transcendence degree will be also considered. The obtained results can be applied in particular to algebras over the ring of integers, i.e, to arbitrary rings.

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arXiv2019

Noncommutative analogues of a cancellation theorem of Abhyankar, Eakin, and Heinzer

Let $k$ be a field and let $A$ be a finitely generated $k$-algebra. The algebra $A$ is said to be cancellative if whenever $B$ is another $k$-algebra with the property that $A[x]\cong B[x]$ then we necessarily have $A\cong B$. An important result of Abhyankar, Eakin, and Heinzer shows that if $A$ is a finitely generated commutative integral domain of Krull dimension one then it is cancellative. We consider the question of cancellation for finitely generated not-necessarily-commutative domains of Gelfand-Kirillov dimension one, and show that such algebras are necessarily cancellative when the characteristic of the base field is zero. In particular, this recovers the cancellation result of Abhyankar, Eakin, and Heinzer in characteristic zero when one restricts to the commutative case. We also provide examples that show affine domains of Gelfand-Kirillov dimension one need not be cancellative when the base field has positive characteristic, giving a counterexample to a conjecture of Tang, the fourth-named author, and Zhang. In addition, we prove a skew analogue of the result of Abhyankar-Eakin-Heinzer, in which one works with skew polynomial extensions as opposed to ordinary polynomial rings.

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arXiv2016

Some homological properties of skew PBW extensions arising in non-commutative algebraic geometry

In this short paper we study for the skew PBW (Poincaré-Birkhoff-Witt) extensions some homological properties arising in non-commutative algebraic geometry, namely, Auslander-Gorenstein regularity, Cohen-Macaulayness and strongly noetherianity. Skew PBW extensions include a considerable number of non-commutative rings of polynomial type such that classical PBW extensions, quantum polynomial rings, multiplicative analogue of the Weyl algebra, some Sklyanin algebras, operator algebras, diffusion algebras, quadratic algebras in 3 variables, among many others. For some key examples we present the parametrization of its point modules.

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