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Héctor Suárez

Publications and source records attributed to Héctor Suárez.

16 recordsLinked to original sources
arXiv2025

Homological properties of some quantum Heisenberg algebras

In this paper we study the properties Koszul, Artin-Schelter regular and (skew) Calabi-Yau of some special types of quantum and generalized Heisenberg algebras and also analyze relations between these algebras, (graded) iterated Ore extensions and (graded) skew PBW extensions. The first-named author and Razavinia introduced the quantum generalized Heisenberg algebras, which depend on a parameter $q$ and two polynomials $f, g\in K[t]$. We prove that under certain conditions for $f$, $g$ these algebras are Koszul, Artin-Shelter regular, Calabi-Yau and graded Calabi-Yau.

math.RA↗
arXiv2022

$Σ$-semicommutative rings and their skew PBW extensions

In this paper, we introduce the concept of $Σ$-semicommutative ring, for $Σ$ a finite family of endomorphisms of a ring $R$. We relate this class of rings with other classes of rings such that Abelian, reduced, $Σ$-rigid, nil-reversible and rings satisfying the $Σ$-skew reflexive nilpotent property. Also, we study some ring-theoretical properties of skew PBW extensions over $Σ$-semicommutative rings. We prove that if a ring $R$ is $Σ$-semicommutative with certain conditions of compatibility on derivations, then for every skew PBW extension $A$ over $R$, $R$ is Baer if and only if $R$ is quasi-Baer, and equivalently, $A$ is quasi-Baer if and only if $A$ is Baer. Finally, we consider some topological conditions for skew PBW extensions over $Σ$-semicommutative rings.

math.RA↗
arXiv2021

Homogenized skew PBW extensions

In this paper, we provide a new and more general filtration to the family of noncommutative rings known as skew PBW extensions. We introduce the notion of $σ$-filtered skew PBW extension and study some homological properties of these algebras. We show that the homogenization of a $σ$-filtered skew PBW extension $A$ over a ring $R$ is a graded skew PBW extension over the homogenization of $R$. Using this fact, we prove that if the homogenization of $R$ is Auslander-regular, then the homogenization of $A$ is a domain Noetherian, Artin-Schelter regular, and $A$ is Noetherian, Zariski and (ungraded) skew Calabi-Yau.

math.QA↗
arXiv2021

On $Σ$-skew reflexive-nilpotents-property for rings

In this paper, we study the reflexive-nilpotents-property (briefly, RNP) for Ore extensions of injective type, and more generally, skew PBW extensions. With this aim, we introduce the notions of $Σ$-skew CN rings and $Σ$-skew reflexive (RNP) rings, for $Σ$ a finite family of ring endomorphisms of a ring $R$. Under certain conditions of compatibility, we study the transfer of the $Σ$-skew RNP property from a ring of coefficients to an Ore extension or skew PBW extension over this ring. We also consider this property for localizations of these noncommutative rings. Our results extend those corresponding presented by Bhattacharjee \cite{Bhattacharjee2020}.

math.RA↗
arXiv2021

On NI and NJ skew PBW extensions

We establish necessary or sufficient conditions to guarantee that skew Poincaré-Birkhoff-Witt extensions are NI or NJ rings. Our results extend those corresponding for skew polynomial rings and establish similar properties for other families of noncommutative rings such as universal enveloping algebras and examples of differential operators.

math.RA↗
arXiv2018

Double Ore extensions versus graded skew PBW extensions

In this paper we present necessary and sufficient conditions for a graded (trimmed) double Ore extension to be a graded (quasi-commutative) skew PBW extension. Using this fact, we prove that a graded skew PBW extension $A = σ(R)\langle x_1,x_2 \rangle$ of an Artin-Schelter regular algebra $R$ is Artin-Schelter regular. As a consequence, every graded skew PBW extension $A = σ(R)\langle x_1,x_2 \rangle$ of a connected skew Calabi-Yau algebra $R$ of dimension $d$ is skew Calabi-Yau of dimension $d+2$.

math.RA↗
arXiv2018

Skew PBW extensions over weak symmetric and $(Σ,Δ)$-weak symmetric rings

In this paper we study skew Poincaré-Birkhoff-Witt extensions over weak symmetric and $(Σ,Δ)$-weak symmetry rings. Since these extensions generalize Ore extensions of injective type and another noncommutative rings of polynomial type, we unify and extend several results in the literature concerning the property of being symmetry. Under adequate conditions, we transfer the property of being weak symmetric or $(Σ,Δ)$-weak symmetric from a ring of coefficients to a skew PBW extension over this ring. We illustrate our results with remarkable examples of algebras appearing in noncommutative algebraic geometry and theoretical physics.

math.QA↗
arXiv2018

Skew Poincaré-Birkhoff-Witt extensions over weak $Σ$-rigid rings

In this paper we introduce the notion of weak $Σ$-rigid ring which extends $α$-rigid rings and $Σ$-rigid rings defined for Ore extensions and skew PBW extensions, respectively. We also present the notion of weak $Σ$-skew Armendariz ring which extends that of $α$-skew Armendariz ring. In this way we generalize several results in the literature, from Ore extensions of injective type to the more general setting of skew PBW extensions.

math.QA↗
arXiv2018

Burchnall-Chaundy theory for skew Poincaré-Birkhoff-Witt extensions

In this paper we review some classical results on the algebraic dependence of commuting elements in several noncommutative algebras as differential operator rings and Ore extensions. Then we extend all these results to a more general setting, the family of noncommutative rings known as skew Poincaré-Birkhoff-Witt extensions.

math.QA↗
arXiv2018

A generalized Koszul property for skew PBW extensions

Let R be a commutative algebra. In this paper we show that constant skew PBW extensions of a generalized Koszul algebra R are also generalized Koszul. Let A be a semi-commutative skew PBW extension of R such that A is R-augmented. We show also that if R is augmented Koszul then A is R-augmented Koszul.

math.AG↗
arXiv2017

Nakayama automorphism of quasi-commutative skew PBW extensions over AS-regular algebras

Graded quasi-commutative skew PBW extensions are isomorphic to graded iterated Ore extensions of endomorphism type, whence graded quasi-commutative skew PBW extensions with coefficients in AS-regular algebras are skew Calabi-Yau and the Nakayama automorphism exists for these extensions. With this in mind, in this paper we give a description of Nakayama automorphism for these non-commutative algebras using the Nakayama automorphism of the ring of the coefficients.

math.QA↗
arXiv2016

Koszulity for skew PBW extensions over fields

Koszul and homogeneous Koszul algebras were defined by Priddy in \cite{Priddy1970}. There exist some relations between these algebras and the skew PBW extensions introduced in \cite{LezamaGallego}. In this paper we give conditions to guarantee that skew PBW extensions over fields are Koszul or homogeneous Koszul. We also show that a constant skew PBW extension of a field is a PBW deformation of its homogeneous version.

math.RA↗
arXiv2016

$σ$-PBW Extensions of Skew Armendariz Rings

The aim of this paper is to investigate a general notion of $σ$-PBW extensions over Armendariz rings. As an application, the properties of being Baer, quasi-Baer, p.p. and p.q.-Baer are established for these extensions. We generalize several results in the literature for Ore extensions of injective type and skew PBW extensions.

math.RA↗
arXiv2016

Koszulity for graded skew PBW extensions

Pre-Koszul and Koszul algebras were defined by Priddy. There exist some relations between these algebras and the skew PBW extensions defined. We have established conditions to guarantee that skew PBW extensions over fields it turns out homogeneous pre-Koszul or Koszul algebra. In this paper we complement these results defining graded skew PBW extensions and showing that if R is a finite presented Koszul K-algebra then every graded skew PBW extension of R is Koszul.

math.RA↗