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

Reyes, Armando; Suárez, Héctor

doi:10.17654/ms106010237

FJMS

2018

DOI: 10.17654/ms106010237

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

Armando Reyes¹,

Héctor Suárez²

Abstract: 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.

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“…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.…”

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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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confidence: 99%

Double Ore extensions versus graded skew PBW extensions

Gómez

Suárez

2019

Communications in Algebra

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

Cited by 1 publication

References 22 publications

Double Ore extensions versus graded skew PBW extensions

Double Ore extensions versus graded skew PBW extensions

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