Artin Schelter Regular Algebras Research Articles

Growth of graded twisted Calabi-Yau algebras

We initiate a study of the growth and matrix-valued Hilbert series of N-graded twisted Calabi-Yau algebras that are homomorphic images of path algebras of weighted quivers, generalizing techniques previously used to investigate Artin-Schelter regular algebras and graded Calabi-Yau algebras. Several results are proved without imposing any assumptions on the degrees of generators or relations of the algebras. We give particular attention to twisted Calabi-Yau algebras of dimension d≤3, giving precise descriptions of their matrix-valued Hilbert series and partial results describing which underlying quivers yield algebras of finite GK-dimension. For d=2, we show that these are algebras with mesh relations. For d=3, we show that the resulting algebras are a kind of derivation-quotient algebra arising from an element that is similar to a twisted superpotential.

A Note on Generic Clifford Algebras of Binary Cubic Forms

We study the representation theoretic results of the binary cubic generic Clifford algebra $\mathcal C$, which is an Artin-Schelter regular algebra of global dimension five. In particular, we show that $\mathcal C$ 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 $\mathcal C_f$ to be an Azumaya algebra.

Pertinency of Hopf actions and quotient categories of Cohen–Macaulay algebras

We study invariants and quotient categories of fixed subrings of Artin–Schelter regular algebras under Hopf algebra actions.

Double Ore extensions versus graded skew PBW extensions

In this article, 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 of an Artin–Schelter regular algebra R is Artin–Schelter regular. As a consequence, every graded skew PBW extension of a connected skew Calabi–Yau algebra R of dimension d is skew Calabi–Yau of dimension d + 2.

Nakayama Automorphism of Some Skew PBW Extensions

Let R be an Artin-Schelter regular algebra and A=σ(R)〈x1,...,xn〉be agraded quasi-commutative skew PBW extension overR. In this paper wedescribe the Nakayama automorphism ofAusing the Nakayama automor-phism of the ring of coefficients R. We calculate explicitly the Nakayamaautomorphism of some skew PBW extensions.

New Artin-Schelter regular and Calabi-Yau algebras via normal extensions

We introduce a new method to construct 4-dimensional Artin-Schelter regular algebras as normal extensions of (not necessarily noetherian) 3-dimensional ones. The method produces large classes of new 4-dimensional Artin-Schelter regular algebras. When applied to a 3-Calabi-Yau algebra our method produces a flat family of central extensions of it that are 4-Calabi-Yau, and all 4-Calabi-Yau central extensions having the same generating set as the original 3-Calabi-Yau algebra arise in this way. Each normal extension has the same generators as the original 3-dimensional algebra, and its relations consist of all but one of the relations for the original algebra and an equal number of new relations determined by “the missing one” and a tuple of scalars satisfying some numerical conditions. We determine the Nakayama automorphisms of the 4-dimensional algebras obtained by our method and as a consequence show that their homological determinant is 1. This supports the conjecture in [J. Algebra 446 (2016), pp. 373–399] that the homological determinant of the Nakayama automorphism is 1 for all Artin-Schelter regular connected graded algebras. Reyes-Rogalski-Zhang proved this is true in the noetherian case [Trans. Amer. Math. Soc. 369 (2017), pp. 309–340, Cor. 5.4].

Graded Lie Algebras of Dimension Five and Some Artin–Schelter Regular Algebras

In this paper, we show that there are only seven graded Lie algebras of dimension 5 generated in degree 1 up to isomorphism. By parameterizing the relations of the universal enveloping algebras of three of those graded Lie algebras, we construct some new Artin–Schelter regular algebras of global dimension 5. We prove that those algebras are all strongly noetherian, Auslander regular and Cohen–Macaulay, and describe their Nakayama automorphisms.

Skew Clifford algebras

We introduce a generalization, called a skew Clifford algebra, of a Clifford algebra, and relate these new algebras to the notion of graded skew Clifford algebra that was defined in 2010. In particular, we examine homogenizations of skew Clifford algebras, and determine which skew Clifford algebras can be homogenized to create Artin-Schelter regular algebras. Just as (classical) Clifford algebras are the Poincaré-Birkhoff-Witt (PBW) deformations of exterior algebras, skew Clifford algebras are the Z2-graded PBW deformations of quantum exterior algebras. We also determine the possible dimensions of skew Clifford algebras and provide several examples.

McKay correspondence for semisimple Hopf actions on regular graded algebras. II

We continue our study of the McKay correspondence for grading preserving actions of semisimple Hopf algebras H on (noncommutative) Artin–Schelter regular algebras A . Here, we establish correspondences between module categories over A^H , over A\#H , and over End _{A^H}(A) . We also study homological properties of (endomorphism rings of) maximal Cohen–Macaulay modules over A^H .

On quantum groups associated to a pair of preregular forms

We define the universal quantum group \mathcal{H} that preserves a pair of Hopf comodule maps, whose underlying vector space maps are preregular forms defined on dual vector spaces. This generalizes the construction of Bichon and Dubois-Violette (2013), where the target of these comodule maps are the ground field. We also recover the quantum groups introduced by Dubois-Violette and Launer (1990), by Takeuchi (1990), by Artin, Schelter, and Tate (1991), and by Mrozinski (2014), via our construction. As a consequence, we obtain an explicit presentation of a universal quantum group that coacts simultaneously on a pair of N-Koszul Artin–Schelter regular algebras with arbitrary quantum determinant.

The quadratic algebras in Artin-Schelter regular algebras of dimension five

We study the quadratic algebras in Artin–Schelter regular algebras of dimension 5 generated in degree 1 under the hypothesis that . All the algebras obtained are proved to be Koszul algebras or piecewise-Koszul algebras. In addition, we find that there don’t exist d-Koszul (d > 2) Artin-Schelter regular algebras of dimension 5.

Properties of the fixed ring of a preprojective algebra

For a finite group acting on a polynomial ring, the Chevalley–Shephard–Todd Theorem proves that the fixed subring is isomorphic to a polynomial ring if and only if the group is generated by pseudo-reflections. In recent years, progress was made in work of Kirkman, Kuzmanovich, Zhang, and others to extend this result to Artin–Schelter regular algebras by expanding pseudo-reflections to quasi-reflections. Naturally, the question arises if the theory generalizes further to non-connected noncommutative algebras. Our objects of study will be preprojective algebras which are certain factor algebras of path algebras corresponding to extended Dynkin diagrams of type A, D or E. This work answers the question what conditions need to be satisfied by the fixed ring in order to make a rich theory possible. On our way, we will point out additional difficulties in establishing quasi-reflections using the trace and reveal situations which do not occur for Artin–Schelter regular algebras.

Artin-Schelter regularity of twisted tensor products

We prove an Artin-Schelter regularity result for the method of twisted tensor products under a certain form. Such twisted tensor products, whose twisting maps are determined by the action on the generators, include Ore extensions and double Ore extensions. It is helpful to construct high-dimensional Artin-Schelter regular algebras.

Derived Categories of Noncommutative Quadrics and Hilbert Squares

AbstractA noncommutative deformation of a quadric surface is usually described by a three-dimensional cubic Artin–Schelter regular algebra. In this paper we show that for such an algebra its bounded derived category embeds into the bounded derived category of a commutative deformation of the Hilbert scheme of two points on the quadric. This is the second example in support of a conjecture by Orlov. Based on this example we formulate an infinitesimal version of the conjecture and provide some evidence in the case of smooth projective surfaces.

Algunas propiedades homológicas del plano de Jordan

El plano de Jordan puede ser visto como un álgebra cociente, como una extensión de Ore graduada y como unaextensión PBW torcida graduada. Usando estas interpretaciones, se muestra de forma explícita que el plano de Jordan es un álgebra Artin-Schelter regular y Calabi-Yau torcida, además se calcula de forma explícita su automorfismo de Nakayama.

McKay correspondence for semisimple Hopf actions on regular graded algebras, I

In establishing a more general version of the McKay correspondence, we prove Auslander's theorem for actions of semisimple Hopf algebras H on noncommutative Artin–Schelter regular algebras A of global dimension two, where A is a graded H-module algebra, and the Hopf action on A is inner faithful with trivial homological determinant. We also show that each fixed ring AH under such an action arises as an analogue of a coordinate ring of a Kleinian singularity.

Nakayama automorphisms of twisted tensor products

In this paper, we study homological properties of twisted tensor products of connected graded algebras. We focus first on the Ext-algebras of twisted tensor products with a certain form of twisting maps. We show those Ext-algebras are also twisted tensor products, and describe the twisting maps for such Ext-algebras explicitly. With those preparations, we describe Nakayama automorphisms of twisted tensor products of noetherian Artin–Schelter regular algebras.

The representation theory of non-commutative $\mathcal O$(GL$_2)$

In our companion paper "The Manin Hopf algebra of a Koszul Artin–Schelter regular algebra is quasi-hereditary" we used the Tannaka–Krein formalism to study the universal coacting Hopf algebra \underline {\mathrm {aut}}(A) for a Koszul Artin–Schelter regular algebra A . In this paper we study in detail the case A=k[x,y] . In particular we give a more precise description of the standard and costandard representations of \underline {\mathrm {aut}}(A) as a coalgebra and we show that the latter can be obtained by induction from a Borel quotient algebra. Finally we give a combinatorial characterization of the simple \underline {\mathrm {aut}}(A) -representations as tensor products of \underline {\mathrm {end}}(A) -representations and their duals.

A NEW FAMILY OF POISSON ALGEBRAS AND THEIR DEFORMATIONS

Let $\Bbbk$ be a field of characteristic zero. For any positive integer $n$ and any scalar $a\in \Bbbk$, we construct a family of Artin–Schelter regular algebras $R(n,a)$, which are quantizations of Poisson structures on $\Bbbk [x_{0},\ldots ,x_{n}]$. This generalizes an example given by Pym when $n=3$. For a particular choice of the parameter $a$ we obtain new examples of Calabi–Yau algebras when $n\geqslant 4$. We also study the ring theoretic properties of the algebras $R(n,a)$. We show that the point modules of $R(n,a)$ are parameterized by a bouquet of rational normal curves in $\mathbb{P}^{n}$, and that the prime spectrum of $R(n,a)$ is homeomorphic to the Poisson spectrum of its semiclassical limit. Moreover, we explicitly describe $\operatorname{Spec}R(n,a)$ as a union of commutative strata.

Graded Clifford Algebras and Graded Skew Clifford Algebras and Their Role in the Classification of Artin–Schelter Regular Algebras

This paper is a survey of work done on $$\mathbb {Z}$$ -graded Clifford algebras (GCAs) and $$\mathbb {Z}$$ -graded skew Clifford algebras (GSCAs) by Vancliff et al. (Commun Algebra 26(4): 1193–1208, 1998), Stephenson and Vancliff (J Algebra 312(1): 86–110, 2007), Cassidy and Vancliff (J Lond Math Soc 81:91–112, 2010), Nafari et al. (J Algebra 346(1): 152–164, 2011), Vancliff and Veerapen (Contemp Math 592: 241–250, 2013), (J Algebra 420:54–64, 2014). In particular, we discuss the hypotheses necessary for these algebras to be Artin Schelter-regular (Adv Math 66:171–216, 1987), (The Grothendieck Festschrift. Birkhauser, Boston, 1990) and show how certain ‘points’ called, point modules, can be associated to them. We may view an AS-regular algebra as a noncommutative analog of the polynomial ring. We begin our survey with a fundamental result in Vancliff et al. (Commun Algebra 26(4): 1193–1208, 1998) that is essential to subsequent results discussed here: the connection between point modules and rank-two quadrics. Using, in part, this connection the authors in Stephenson and Vancliff (J Algebra 312(1): 86–110, 2007) provide a method to construct GCAs with finitely many distinct isomorphism classes of point modules. In Cassidy and Vancliff (J Lond Math Soc 81:91–112, 2010), Cassidy and Vancliff introduce a quantized analog of a GCA, called a graded skew Clifford algebra and Nafari et al. (J Algebra 346(1): 152–164, 2011) show that most Artin Schelter-regular algebras of global dimension three are either twists of graded skew Clifford algebras of global dimension three or Ore extensions of graded Clifford algebras of global dimension two. Vancliff and Veerapen (Contemp Math 592: 241–250, 2013), (J Algebra 420:54–64, 2014) go a step further and generalize the result of Vancliff et al. (Commun Algebra 26(4): 1193–1208, 1998), between point modules and rank-two quadrics, by showing that point modules over GSCAs are determined by (noncommutative) quadrics of $$\mu $$ -rank at most two.