A Generalized Koszul Property for Skew PBW Extensions

Suárez, Héctor; Reyes, Armando

doi:10.17654/ms101020301

Cited by 13 publications

(21 citation statements)

References 17 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Then we proceed to review results obtained by other authors for particular examples of skew PBW extensions mentioned above (differential operator rings, q-Weyl algebras, and Ore extensions), and finally we describe the Burchnall-Chaundy theory for these noncommutative rings generalizing the results presented in [21]. We consider that this study enriches the investigation of ring and homological properties of these extensions in the sense of [14], [12], [17], [18], [19], [23], [24], [15] and [20].…”

Section: Introductionmentioning

confidence: 81%

“…. , n (these extensions were called constant by the authors in [23]), then Theorem 4.3 can be illustrated. Some examples of these extensions are the following: PBW extensions defined by Bell and Goodearl (which include the classical commutative polynomial rings, universal enveloping algebra of a Lie algebra, and others); some operator algebras (for example, the algebra of linear partial differential operators, the algebra of linear partial shift operators, the algebra of linear partial difference operators, the algebra of linear partial q-dilation operators, and the algebra of linear partial q-differential operators); the class of diffusion algebras; Weyl algebras; additive analogue of the Weyl algebra; multiplicative analogue of the Weyl algebra; some quantum Weyl algebras as A 2 (J a,b ); the quantum algebra U ′ (so(3, k)); the family of 3-dimensional skew polynomial algebras (there are exactly fifteen of these algebras, see [17]); Dispin algebra U (osp(1, 2)); Woronowicz algebra W v (sl(2, k)); the complex algebra V q (sl 3 (C)); q-Heisenberg algebra H n (q); the Hayashi algebra W q (J), and several algebras of quantum physics (for instance, Weyl algebras, additive and multiplicative analogue of the Weyl algebra, quantum Weyl algebras, q-Heisenberg algebra, and others).…”

Section: Burchnall-chaundy Theory For Skew Poincaré-birkhoff-witt Extmentioning

confidence: 98%

See 1 more Smart Citation

Burchnall-Chaundy Theory for Skew Poincaré-Birkhoff-Witt Extensions

Reyes¹,

Suárez²

2018

FJMS

Self Cite

View full text Add to dashboard Cite

show abstract

Section: Introductionmentioning

confidence: 81%

Section: Burchnall-chaundy Theory For Skew Poincaré-birkhoff-witt Extmentioning

confidence: 98%

Burchnall-Chaundy Theory for Skew Poincaré-Birkhoff-Witt Extensions

Reyes¹,

Suárez²

2018

FJMS

Self Cite

View full text Add to dashboard Cite

show abstract

“…The following examples are graded skew PBW extensions of the classical polynomial ring R with coefficients in a field K, which are not quasicommutative and where R has the usual graduation (see [36], Example 2.9). In [8,19] and [38] we can find further details of these algebras. By Theorem 4.5-(ii), these extensions are skew Calabi-Yau algebras, since R is a connected Auslander-regular algebra.…”

Section: Calabi-yau Algebrasmentioning

confidence: 99%

“…In this section we present some definitions, properties and examples related with skew PBW extensions. For more details and to check other recent properties related to skew PBW extensions, see [1,2,9,7,17,18,19,23,24,26,25,27,37,16], [28,29,30,31,36,38], [39,40], and [32].…”

Section: Graded Skew Pbw Extensionsmentioning

confidence: 99%

Calabi-Yau property for graded skew PBW extensions

2017

Self Cite

View full text Add to dashboard Cite

Las extensiones PBW torcidas graduadas fueron definidas por el primer autor como una generalización de las extensiones de Ore iteradas graduadas [36]. El propósito de este artículo es estudiar las condiciones Artin-Schelter regular y Calabi-Yau (torcida) para esta clase de extensiones. Demostramos que cada extensión PBW torcida cuasi-conmutativa graduada de un álgebra Artin-Schelter regular también es Artin-Schelter regular, y, como consecuencia, que cada extensión PBW torcida cuasi-conmutativa graduada de un álgebra conexa Calabi-Yau torcida es Calabi-Yau torcida. Finalmente, mostramos que las extensiones PBW torcidas graduadas de álgebras Auslander-regular finitamente presentadas y conexas son Calabi-Yau torcidas.

show abstract

“…. , n, or equivalently, σ i = id R and δ i = 0, for every i (these extensions were called constant by the authors in [44]), then it is clear that R is (Σ, ∆)-compatible. Some examples of constant σ-PBW extensions are the following: PBW extensions defined by Bell and Goodearl (which include the classical commutative polynomial rings, universal enveloping algebra of a Lie algebra, and others); some operator algebras (for example, the algebra of linear partial differential operators, the algebra of linear partial shift operators, the algebra of linear partial difference operators, the algebra of linear partial q-dilation operators, and the algebra of linear partial q-differential operators); the class of diffusion algebras; Weyl algebras; additive analogue of the Weyl algebra; multiplicative analogue of the Weyl algebra; some quantum Weyl algebras as A 2 (J a,b ); the quantum algebra U (so(3, k)); the family of 3-dimensional skew polynomial algebras (there are exactly fifteen of these algebras, see [39]); Dispin algebra U(osp(1, 2)); Woronowicz algebra W v (sl(2, k)); the complex algebra V q (sl 3 (C)); q-Heisenberg algebra H n (q); the Hayashi algebra W q (J), and more.…”

Section: Definition 32 Consider a Ring R With A Family Of Endomorphmentioning

confidence: 99%

A notion of compatibility for Armendariz and Baer properties over skew PBW extensions

Reyes¹,

Suárez²

2017

Rev. Un. Mat. Argentina

Self Cite

View full text Add to dashboard Cite

In this paper we are interested in studying the properties of Armendariz, Baer, quasi-Baer, p.p. and p.q.-Baer over skew PBW extensions. Using a notion of compatibility, we generalize several propositions established for Ore extensions and present new results for several noncommutative rings which can not be expressed as Ore extensions (universal enveloping algebras, diffusion algebras, and others). 2010 Mathematics Subject Classification. Primary: 16S36; 16E50. Secondary: 16W60. 157 158 ARMANDO REYES AND HÉCTOR SUÁREZ in [6], they proved that a ring B is right p.q.-Baer if and only if B[x] is right p.q.-Baer. In the context of noncommutative rings, more exactly polynomial extensions known as Ore extensions B [x; σ, δ] of injective type, i.e., when σ is injective, we found several works (cf. [10,5,6,7,13,19,8,9,17,18,15,16]). Some of these works consider the case δ = 0 and σ an automorphism, or the case where σ is the identity. It is important to say that the Baerness and quasi-Baerness of a ring B and an Ore extension B[x; σ, δ] of B do not depend on each other. More exactly, there are examples which show that there exists a Baer ring B but the Ore extension B[x; σ, δ] is not right p.q.-Baer; similarly, there exist Ore extensions B[x; σ, δ] which are quasi-Baer, but B is not quasi-Baer (see [19] for more details).One of the most important kinds of rings for which all the above properties have been studied are the σ-rigid rings (cf. [24,19,18]). Following Krempa [24], an endomorphism σ of a ring B is called rigid if aσ(a) = 0 implies a = 0, for a ∈ B, and a ring B is said to be σ-rigid, if there exists a rigid endomorphism σ of B. One can see that any rigid endomorphism of a ring is a monomorphism, and σ-rigid rings are reduced rings ([19, p. 218]). With the aim of generalizing the σ-rigid rings in the context of Ore extensions, in [2] Annin introduced the notion of compatibility: a ring B is called σ-compatible if for every a, b ∈ B, we have ab = 0 if and only if aσ(b) = 0 (necessarily, the endomorphism σ is injective); B is called δ-compatible if for each a, b ∈ B, ab = 0 ⇒ aδ(b) = 0. If B is both σ-compatible and δ-compatible, B is called (σ, δ)-compatible. Note that σ-rigid rings are (σ, δ)compatible rings ([15, Lemma 3.3]), but the converse is false ([15, Examples 2.1, 2.2 and 2.3]). Nevertheless, Hashemi et al., in [16, Lemma 2.2], showed that a ring B is (σ, δ)-compatible and reduced if and only if B is σ-rigid. Hence σ-compatible rings generalize σ-rigid rings for the case where B is not assumed to be reduced. All the above properties have been also studied for (σ, δ)-compatible rings: in [16] it was imposed the (σ, δ)-compatibility on the ring B and it was proved that (i) the ring B is quasi-Baer if and only if B[x; σ, δ] is quasi-Baer; (ii) the ring B is left p.q.-Baer if and only if B[x; σ, δ] is left p.q.-Baer. In this way, the treatment in [16] is a generalization of [6, Theorem 1.8] and [9, Theorem 3.1].With all the above facts in mind, a natural question for a given class of Baer, quasi-Baer, p....

show abstract

A Generalized Koszul Property for Skew PBW Extensions

Cited by 13 publications

References 17 publications

Burchnall-Chaundy Theory for Skew Poincaré-Birkhoff-Witt Extensions

Burchnall-Chaundy Theory for Skew Poincaré-Birkhoff-Witt Extensions

Calabi-Yau property for graded skew PBW extensions

A notion of compatibility for Armendariz and Baer properties over skew PBW extensions

Contact Info

Product

Resources

About