A class of AS-regular algebras of dimension five

Wang, S.-Q.; Wu, Q.-S.

doi:10.1016/j.jalgebra.2012.04.012

Cited by 18 publications

(26 citation statements)

References 22 publications

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“…Theorem. [19,Theorem 5] The connected graded algebra C is Artin-Schelter regular of global dimension five. Moreover, C is an Auslander regular, Cohen-Macaulay, and strongly noetherian domain.…”

Section: 4mentioning

confidence: 99%

A Note on Generic Clifford Algebras of Binary Cubic Forms

Wang

2019

Algebr Represent Theor

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We study the representation theoretic results of the binary cubic generic Clifford algebra C, which is an Artin-Schelter regular algebra of global dimension five. In particular, we show that C is a PI algebra of PI degree three and compute its point variety and discriminant ideals. As a consequence, we give a necessary and sufficient condition on a binary cubic form f for the associated Clifford algebra C f to be an Azumaya algebra.2010 Mathematics Subject Classification. 16G30, 16R99.

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Section: 4mentioning

confidence: 99%

A Note on Generic Clifford Algebras of Binary Cubic Forms

Wang

2019

Algebr Represent Theor

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“…9. In [44], AS -regular algebras of dimension 5 generated by two generators of degree 1 with three generating relations of degree 4 are classified under some generic condition. There are nine types such AS -regular algebras in this classification list.…”

Section: Letmentioning

confidence: 99%

“…There are nine types such AS -regular algebras in this classification list. Among them, the algebras D and G are given by iterated Ore extensions (see [44], Section 5.2).…”

Section: Letmentioning

confidence: 99%

Algunas relaciones entre álgebras N-Koszul, Artin-Schelter regular y Calabi-Yau con extensiones PBW torcidas

Suárez

Lezama

Reyes

2015

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Some authors have studied relations between Artin-Schelter regular algebras, N-Koszul algebras and CalabiYau algebras (resp. skew Calabi-Yau) of dimension d. In this paper we want to show through examples and counterexamples some relations between these classes of algebras with skew PBW extensions. In addition, we also exhibit some examples of the preservation of these properties by Ore extensions.Key words: Skew PBW extensions, Calabi-Yau algebras, N-Koszul algebras, AS -regular algebras, Ore extensions. ResumenAlgunos autores han estudiado las relaciones entre las álgebras Artin-Schelter regular, las álgebras N -Koszul y las álgebras Calabi-Yau (resp. skew Calabi-Yau) de dimensión d . En este artículo queremos mostrar a través de ejemplos y contraejemplos algunos relaciones entre estas clases de álgebras y las extensiones PBW torcidas. Además, mostraremos algunos ejemplos de preservación de estas propiedades en las extensiones de Ore.

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“…Artin-Schelter regular algebras of global dimension two and global dimension three were classified in [1], but there are also many open questions about these algebras; for example, the classification of Artin-Schelter regular algebras of global dimension greater than three. Different authors have focused on studying Artin-Schelter regular algebras, especially those of global dimension four and global dimension five (see for example [14,15,16,17]). One of the main objectives in noncommutative algebraic geometry is the classification of noncommutative projective spaces.…”

Section: Introductionmentioning

confidence: 99%

Double Ore extensions versus graded skew PBW extensions

Gómez

Suárez

2019

Communications in Algebra

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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) x 1 , x 2 of an Artin-Schelter regular algebra R is Artin-Schelter regular. As a consequence, every graded skew PBW extension A = σ(R) x 1 , x 2 of a connected skew Calabi-Yau algebra R of dimension d is skew Calabi-Yau of dimension d + 2.Calabi-Yau property is equivalent to the Artin-Schelter regular property. Calabi-Yau algebras are skew Calabi-Yau, but some skew Calabi-Yau algebras are not Calabi-Yau (for example the Jordan plane).Zhang and Zhang in [15] introduced a special class of algebras called double Ore extensions. They proved that a connected graded double Ore extension of an Artin-Schelter regular algebra is Artin-Schelter regular. The same authors in [16] constructed 26 families of Artin-Schelter regular algebras of global dimension four, using double Ore extensions. Carvalho and Matczuk in [2] described double Ore extensions that can be presented as iterated Ore extensions. Other properties of double Ore extensions have been studied (see for example [17]).Skew PBW extensions were defined in [3], and recently studied in many papers (see for example [5,8,9,10]). The second author in [12] defined the graded skew PBW extensions for an algebra R. This definition generalizes graded iterated Ore extensions. Some properties of graded skew PBW extensions have been studied in [12,13]. The Artin-Schelter regular property and the skew Calabi-Yau condition for graded skew PBW extensions were studied in [13]; they proved that every graded quasi-commutative skew PBW extension of an Artin-Schelter regular algebra is Artin-Schelter regular algebra, every graded quasi-commutative skew PBW extension of a connected skew Calabi-Yau algebra is skew Calabi-Yau, and graded skew PBW extensions of connected Auslander regular algebras are Artin-Schelter regular and skew Calabi-Yau.In the current literature, as far as we are aware, there is no study of the relationships between skew PBW extensions and double Ore extensions. In this paper we study some relations between (graded) right double Ore extensions and (graded) skew PBW extensions, and between (graded) trimmed right double Ore extensions and (graded) quasi-commutative skew PBW extensions. We conclude that a graded skew PBW extension A = σ(R) x 1 , x 2 of an Artin-Schelter regular algebra R is Artin-Schelter regular, and a graded skew PBW extension A = σ(R) x 1 , x 2 of a connected skew Calabi-Yau algebra R of dimension d is skew Calabi-Yau of dimension d + 2.The main results of this paper are found in Section 3. In Theorem 3.1 we give conditions for a double Ore extension to be a skew PBW extension. In Theorem 3.2 we show that a connected graded skew PBW extension A = σ(R) x 1 , x 2 is a connected graded double Ore extension of R. As a consequence of this theorem we show in Corollary 3.3 that if A = σ(R) x 1 , x 2 is a connected graded quasi-comm...

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A class of AS-regular algebras of dimension five

Cited by 18 publications

References 22 publications

A Note on Generic Clifford Algebras of Binary Cubic Forms

A Note on Generic Clifford Algebras of Binary Cubic Forms

Algunas relaciones entre álgebras N-Koszul, Artin-Schelter regular y Calabi-Yau con extensiones PBW torcidas

Double Ore extensions versus graded skew PBW extensions

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