Skew Poincaré–Birkhoff–Witt extensions over weak compatible rings

2019 ◽  
Vol 19 (12) ◽  
pp. 2050225 ◽  
Author(s):  
Armando Reyes ◽  
Héctor Suárez
Keyword(s):  
Algebraic Geometry ◽  
Theoretical Physics ◽  
Quantum Algebras ◽  
Noncommutative Rings ◽  
Noncommutative Algebraic Geometry ◽  
Ore Extensions ◽  
Skew Pbw Extensions ◽  
Weak Notion

In this paper, we introduce weak [Formula: see text]-compatible rings and study skew Poincaré–Birkhoff–Witt extensions over these rings. We characterize the weak notion of compatibility for several noncommutative rings appearing in noncommutative algebraic geometry and some quantum algebras of theoretical physics. As a consequence of our treatment, we unify and extend results in the literature about Ore extensions and skew PBW extensions over compatible rings.

scholarly journals Algunos tipos especiales de determinantes en extensiones PBW torcidas graduadas

2021 ◽  
Vol 39 (1) ◽  
Author(s):  
Héctor Suárez ◽  
Duban Cáceres ◽  
Armando Reyes
Keyword(s):  
Differential Geometry ◽  
Algebraic Geometry ◽  
Regular Algebra ◽  
Noncommutative Algebraic Geometry ◽  
Noncommutative Differential Geometry ◽  
Regular Algebras ◽  
Skew Pbw Extensions ◽  
Finitely Presented

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.

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

2017 ◽  
pp. 157-178 ◽  
Author(s):  
Armando Reyes ◽  
Héctor Suárez
Keyword(s):  
Universal Enveloping Algebras ◽  
Noncommutative Rings ◽  
Enveloping Algebras ◽  
Ore Extensions ◽  
Skew Pbw Extensions

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).

scholarly journals NAKAYAMA AUTOMORPHISMS OF ORE EXTENSIONS OVER POLYNOMIAL ALGEBRAS

2019 ◽  
Vol 62 (3) ◽  
pp. 518-530 ◽  
Author(s):  
LIYU LIU ◽  
WEN MA
Keyword(s):  
Algebraic Geometry ◽  
Cyclic Group ◽  
Invariant Theory ◽  
Polynomial Algebra ◽  
Ore Extension ◽  
Polynomial Algebras ◽  
Noncommutative Algebraic Geometry ◽  
Ore Extensions ◽  
Group Sharing ◽  
Noncommutative Invariant Theory

AbstractNakayama automorphisms play an important role in the fields of noncommutative algebraic geometry and noncommutative invariant theory. However, their computations are not easy in general. We compute the Nakayama automorphism ν of an Ore extension R[x; σ, δ] over a polynomial algebra R in n variables for an arbitrary n. The formula of ν is obtained explicitly. When σ is not the identity map, the invariant EG is also investigated in terms of Zhang’s twist, where G is a cyclic group sharing the same order with σ.

scholarly journals Some remarks about minimal prime ideals of skew Poincaré-Birkhoff-Witt extensions

2020 ◽  
Vol 30 (2) ◽  
pp. 207-229
Author(s):  
A. Niño ◽  
◽  
A. Reyes ◽  
Keyword(s):  
Prime Ideals ◽  
Noncommutative Rings ◽  
Polynomial Type ◽  
Ore Extensions ◽  
Skew Pbw Extensions ◽  
Minimal Prime

In this paper, we characterize the minimal prime ideals of skew PBW extensions over several classes of rings. We unify different results established in the literature for Ore extensions, and extend all of them to a several families of noncommutative rings of polynomial type which cannot be expressed as these extensions.

scholarly journals Minimal prime ideals of skew PBW extensions over 2-primal compatible rings

2020 ◽  
Vol 54 (1) ◽  
pp. 39-63
Author(s):  
Mohamed Louzari ◽  
Armando Reyes
Keyword(s):  
Commutative Rings ◽  
Prime Ideals ◽  
Noncommutative Rings ◽  
Polynomial Type ◽  
Ore Extensions ◽  
Skew Pbw Extensions ◽  
Minimal Prime

In this paper, we characterize the units of skew PBW extensions over compatible rings. With this aim, we recall the transfer of the property of being 2-primal for these extensions. As a consequence of our treatment, the results established here generalize those corresponding for commutative rings and Ore extensions of injective type. In this way, we present new results for several noncommutative rings of polynomial type not considered before in the literature.

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

2018 ◽  
Vol 14 (27) ◽  
pp. 29-52 ◽  
Author(s):  
Armando Reyes ◽  
Julio Jaramillo
Keyword(s):  
Quantum Algebra ◽  
Theoretical Physics ◽  
Quantum Algebras ◽  
Coeffi Cients ◽  
Skew Pbw 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

scholarly journals Hopf algebras and skew PBW extensions

2019 ◽  
Vol 10 (2) ◽  
Author(s):  
Luis Alfonso Salcedo Plazas
Keyword(s):  
Hopf Algebra ◽  
Hopf Algebras ◽  
Ore Extensions ◽  
Skew Pbw Extensions

In this article we relate some Hopf algebra structures over Ore extensions and over skew PBW extensions ofa Hopf algebra. These relations are illustrated with examples. We also show that Hopf Ore extensions andgeneralized Hopf Ore extensions are Hopf skew PBW extensions.

APPLICATIONS OF COMPUTER ALGEBRA IN QUANTUM GROUPS

1994 ◽  
Vol 05 (04) ◽  
pp. 701-706
Author(s):  
W.-H. STEEB
Keyword(s):  
Quantum Groups ◽  
Computer Algebra ◽  
Kronecker Product ◽  
Quantum Algebra ◽  
Theoretical Physics ◽  
Baxter Equation ◽  
Quantum Algebras ◽  
Helpful Tool

Quantum groups and quantum algebras play a central role in theoretical physics. We show that computer algebra is a helpful tool in the investigations of quantum groups. We give an implementation of the Kronecker product together with the Yang-Baxter equation. Furthermore the quantum algebra obtained from the Yang-Baxter equation is implemented. We apply the computer algebra package REDUCE.

scholarly journals VANISHING OF THE NEGATIVE HOMOTOPY -THEORY OF QUOTIENT SINGULARITIES

2017 ◽  
Vol 18 (3) ◽  
pp. 619-627
Author(s):  
Gonçalo Tabuada
Keyword(s):  
Algebraic Geometry ◽  
Upper Bound ◽  
Homotopy Theory ◽  
Krull Dimension ◽  
Noncommutative Algebraic Geometry ◽  
Noetherian Scheme ◽  
Quotient Singularities ◽  
Cyclic Quotient Singularities

Making use of Gruson–Raynaud’s technique of ‘platification par éclatement’, Kerz and Strunk proved that the negative homotopy$K$-theory groups of a Noetherian scheme$X$of Krull dimension$d$vanish below$-d$. In this article, making use of noncommutative algebraic geometry, we improve this result in the case of quotient singularities by proving that the negative homotopy$K$-theory groups vanish below$-1$. Furthermore, in the case of cyclic quotient singularities, we provide an explicit ‘upper bound’ for the first negative homotopy$K$-theory group.

scholarly journals A Complete Classification of 3-dimensional Quadratic AS-regular Algebras of Type EC

2020 ◽  
pp. 1-19
Author(s):  
Masaki Matsuno
Keyword(s):  
Algebraic Geometry ◽  
Complete Classification ◽  
Complete List ◽  
Graded Algebra ◽  
Regular Algebra ◽  
3 Dimensional ◽  
Noncommutative Algebraic Geometry ◽  
Regular Algebras ◽  
Point Scheme

Abstract Classification of AS-regular algebras is one of the main interests in noncommutative algebraic geometry. We say that a $3$ -dimensional quadratic AS-regular algebra is of Type EC if its point scheme is an elliptic curve in $\mathbb {P}^{2}$ . In this paper, we give a complete list of geometric pairs and a complete list of twisted superpotentials corresponding to such algebras. As an application, we show that there are only two exceptions up to isomorphism among all $3$ -dimensional quadratic AS-regular algebras that cannot be written as a twist of a Calabi–Yau AS-regular algebra by a graded algebra automorphism.

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