PBW Extension Research Articles

On Σ-skew reflexive-nilpotents-property for rings

In this paper, we study the reflexive-nilpotents-property (briefly, RNP) for skew PBW extensions. With this aim, we introduce the Σ-skew CN and Σ-skew reflexive (RNP) rings. Under conditions of compatibility, we investigate the transfer of there flexive-nilpotents-property from a ring of coefficients to a skew PBW extension. We also consider this property for localizations on these families of noncommutative rings. Our results extend those corresponding presented by Bhattacharjee [9].

On the diameter of the zero-divisor graph over skew PBW extensions

The aim of this paper is to investigate the interplay between the algebraic properties of a skew Poincaré–Birkhoff–Witt extesion ring [Formula: see text] and the graph-theoretic properties of its zero-divisor graph. We are interested in studying the diameter of the zero-divisor graph of skew PBW extension rings. Among other results, we give a complete characterization of the possible diameters of [Formula: see text] in terms of the diameter of [Formula: see text].

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 sigma -filtered skew PBW extension and study some homological properties of these algebras. We show that the homogenization of a sigma -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.

Some special types of determinants in graded skew PBW extensions

In this paper, we prove that the Nakayama automorphism of a graded skew PBW extension over a finitely presented Koszul Auslander-regular algebra has trivial homological determinant. For A = σ(R)<x1, x2> a graded skew PBW extension over a connected algebra R, we compute its P-determinant and the inverse of σ. In the particular case of quasi-commutative skew PBW extensions over Koszul Artin-Schelter regular algebras, we show explicitly the connection between the Nakayama automorphism of the ring of coefficients and the extension. Finally, we give conditions to guarantee that A is Calabi-Yau. We provide illustrative examples of the theory concerning algebras of interest in noncommutative algebraic geometry and noncommutative differential geometry.

Associated prime ideals over skew PBW extensions

In this article, we continue the study of ideals of the noncommutative rings of polynomial type known as skew Poincaré-Birkhoff-Witt extensions. More exactly, we focus on the associated prime ideals of these extensions. Our purpose here is to describe how the associated primes of a ring of coefficients behave under passage to a skew PBW extension over this ring. As we show, all results established here coincide and generalize those corresponding in the commutative case and the Ore extensions.

Poisson brackets on some skew PBW extensions

In [1] the author gives a description of Poisson brackets on some algebras of quantum polynomials \(\mathcal{O}_q\), which is called the general algebra of quantum polynomials. The main of this paper is to present a generalization of [1] through a description of Poisson brackets on some skew PBW extensions of a ring \(A\) by the extensions \(\mathcal{O}_{q,\delta}^{r,n}\), which are generalization of \(\mathcal{O}_q\), and show some examples of skew PBW extension where we can apply this description.

Radicals and Köthe’s Conjecture for Skew PBW Extensions

The aim of this paper is to investigate different radicals (Wedderburn radical, lower nil radical, Levitzky radical, upper nil radical, the set of all nilpotent elements, the sum of all nil left ideals) of the noncommutative rings known as skew Poincare–Birkhoff–Witt extensions. We characterize minimal prime ideals of these rings and prove that the Kothe’s conjecture holds for these extensions. Finally, we establish the transfer of several ring-theoretical properties (reduced, symmetric, reversible, 2-primal) from the coefficients ring of a skew PBW extension to the extension itself.

Double Ore extensions versus graded skew PBW extensions

In this article, 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 of an Artin–Schelter regular algebra R is Artin–Schelter regular. As a consequence, every graded skew PBW extension of a connected skew Calabi–Yau algebra R of dimension d is skew Calabi–Yau of dimension d + 2.

Nakayama Automorphism of Some Skew PBW Extensions

Let R be an Artin-Schelter regular algebra and A=σ(R)〈x1,...,xn〉be agraded quasi-commutative skew PBW extension overR. In this paper wedescribe the Nakayama automorphism ofAusing the Nakayama automor-phism of the ring of coefficients R. We calculate explicitly the Nakayamaautomorphism of some skew PBW extensions.

Algunas propiedades homológicas del plano de Jordan

El plano de Jordan puede ser visto como un álgebra cociente, como una extensión de Ore graduada y como unaextensión PBW torcida graduada. Usando estas interpretaciones, se muestra de forma explícita que el plano de Jordan es un álgebra Artin-Schelter regular y Calabi-Yau torcida, además se calcula de forma explícita su automorfismo de Nakayama.

Symmetry and reversibility properties for quantum algebras and skew Poincaré-Birkhoff-Witt extensions

Our aim in this paper is to investigate symmetry and reversibility pro-perties for quantum algebras and skew PBW extensions. Under certainconditions we prove that these properties transfer from a ring of coeffi-cients to a quantum algebra or skew PBW extension over this ring. In thisway we generalize several results established in the literature and consideralgebras which have not been studied before. We illustrate our results withremarkable examples of theoretical physics

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.

BASES FOR QUANTUM ALGEBRAS AND SKEW POINCARÉ-BIRKHOFF-WITT EXTENSIONS

Para las álgebras cuánticas y las extensiones torcidas de Poincaré-Birkhoff-Witt definidas por un anillo y un conjunto de variables con relaciones entre ellas, estamos interesados en establecer un criterio y algunos algoritmos que nos permitan decidir si una estructura algebraica, definida en términos de generadores y relaciones, puede expresarse como una extensión torcida de Poincaré-Birkhoff-Witt, de manera que se determine la base de la misma. Ilustramos nuestro tratamiento con diversas álgebras de la física cuántica.

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.

Koszulity for graded skew PBW extensions

ABSTRACTPre-Koszul and Koszul algebras were defined by Priddy [15]. There exist some relations between these algebras and the skew PBW extensions defined in [8]. In [24] we gave 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.

Filtered-graded transfer of noncommutative Gröbner bases

As the case of free k -algebras and PBW algebras, given a bijective skew PBW extension A , we will show that it is possible transfer Grobner bases between A and its associated graded ring.

Automorphisms for Skew PBW Extensions and Skew Quantum Polynomial Rings

In this work, I study the automorphisms of skew PBW extensions and skew quantum polynomials. I use Artamonov's works as reference for getting the principal results about automorphisms for generic skew PBW extensions and generic skew quantum polynomials. In general, if I have an endomorphism on a generic skew PBW extension and there are some x i , x j , x u such that the endomorphism is not zero on these elements and the principal coefficients are invertible, then endomorphisms act over x i as a i x i for some a i in the ring of coefficients. Of course, this is valid for quantum polynomial rings, with r = 0, as such Artamonov shows in his work. We use this result for giving some more general results for skew PBW extensions, using filtred-graded techniques. Finally, I use localization to characterize some class the endomorphisms and automorphisms for skew PBW extensions and skew quantum polynomials over Ore domains.

Some Homological Properties of Skew PBW Extensions

We prove that if R is a left Noetherian and left regular ring then the same is true for any bijective skew PBW extension A of R. From this we get Serre's Theorem for such extensions. We show that skew PBW extensions and its localizations include a wide variety of rings and algebras of interest for modern mathematical physics such as PBW extensions, well-known classes of Ore algebras, operator algebras, diffusion algebras, quantum algebras, quadratic algebras in 3-variables, skew quantum polynomials, among many others. We estimate the global, Krull and Goldie dimensions, and also Quillen's K-groups.

Maximality of pbw extensions of orders

It is shown that if R is a maximal order in a simple Artinian ring Q, then any PBW extension 5 of R is also a maximal order in a simple Artinian ring. Furthermore, if R is Noetherian, the maximal v-ideals of S are either induced from R or cut down from QS, and a representation of 5 using localizations at these maximal v-ideals is given.