Double Ore extensions versus graded skew PBW extensions

Abstract

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...

Algunos tipos especiales de determinantes en extensiones PBW torcidas graduadas

“…In this way, for all r ∈ R, σ 1 (r) = c 1,r , σ 2 (r) = c 2,r , δ 1 (r) = c 1,0 , δ 2 (r) = c 2,0 , where σ 1 , σ 2 , δ 1 and δ 2 are as in Proposition 2.5. From [9], Theorem 3.2, we can assert that A is a connected graded double Ore extension R P [x 1 , x 2 ; σ, δ, τ ]. (ii) Comparing relations ( 5) and ( 8), we have that…”

“…As we can expect from these facts, graded iterated Ore extensions are strictly contained in graded skew PBW extensions, see Suárez [35], Remark 2.11. Examples of graded skew PBW extensions can be found in Suárez et al, [9,35,36].…”

“…, x n by using the Nakayama automorphism of an Artin-Schelter regular algebra R, and also they calculated explicitly the Nakayama automorphism of some skew PBW extensions. About doble Ore extensions, Gómez and Suárez [9] gave necessary and sufficient conditions for a graded (trimmed) double Ore extension to be a graded (quasi-commutative) skew PBW extension. They proved that graded skew PBW extensions A = σ(R) x 1 , x 2 over Artin-Schelter regular algebras R are also Artin-Schelter regular, and graded skew PBW extensions…”

Propiedad χ en extensiones PBW torcidas graduadas

“…Gallego y Lezama en [4] definieron una clase especial de anillos de tipo polinomial, los cuales son llamados extensiones PBW torcidas. Varias propiedades de estas extensiones han sido ampliamente estudiadas (véase por ejemplo [5,6,7,9,11,12,13,14,15,16,17,19,21]). Gran parte de los ejemplos, las propiedades y otros aspectos importantes de las extensiones PBW torcidas se encuentran compiladas en [3].…”

“…El primer autor en [18] definió las extensiones PBW torcidas graduadas como una generalización de las extensiones de Ore iteradas graduadas. Algunas propiedades de estas extensiones graduadas han sido estudiadas recientemente (véase por ejemplo [6,20]).…”