Artin Schelter Regular Algebras Research Articles

Group coactions on two-dimensional Artin-Schelter regular algebras

We describe all possible coactions of finite groups (equivalently, all group gradings) on two-dimensional Artin-Schelter regular algebras. We give necessary and sufficient conditions for the associated Auslander map to be an isomorphism, and determine precisely when the invariant ring for the coaction is Artin-Schelter regular. The proofs of our results are combinatorial and exploit the structure of the McKay quiver associated to the coaction.

A cogroupoid associated to preregular forms

We construct a family of cogroupoids associated to preregular forms and recover the Morita–Takeuchi equivalence for Artin–Schelter regular algebras of dimension two, observed by Raedschelders and Van den Bergh. Moreover, we study the 2-cocycle twists of pivotal analogues of these cogroupoids, by developing a categorical description of preregularity in any tensor category that has a pivotal structure.

Hecke Symmetries, Associated with Artin–Schelter Regular Algebras of Type $$\boldsymbol{E}$$ and $$\boldsymbol{H}$$

Hecke Symmetries, Associated with Artin–Schelter Regular Algebras of Type $$\boldsymbol{E}$$ and $$\boldsymbol{H}$$

Auslander theorem for PI Artin-Schelter regular algebras

We prove a version of a theorem of Auslander for finite group actions or coactions on noetherian polynomial identity Artin-Schelter regular algebra.

Superpotentials and Quiver Algebras for Semisimple Hopf Actions

We consider the action of a semisimple Hopf algebra H on an m-Koszul Artin–Schelter regular algebra A. Such an algebra A is a derivation-quotient algebra for some twisted superpotential w, and we show that the homological determinant of the action of H on A can be easily calculated using w. Using this, we show that the smash product A#H is also a derivation-quotient algebra, and use this to explicitly determine a quiver algebra Λ to which A#H is Morita equivalent, generalising a result of Bocklandt–Schedler–Wemyss. We also show how Λ can be used to determine whether the Auslander map is an isomorphism. We compute a number of examples, and show how several results for the quantum Kleinian singularities studied by Chan–Kirkman–Walton–Zhang follow using our techniques.

Twisted algebras of geometric algebras

AbstractA twisting system is one of the major tools to study graded algebras; however, it is often difficult to construct a (nonalgebraic) twisting system if a graded algebra is given by generators and relations. In this paper, we show that a twisted algebra of a geometric algebra is determined by a certain automorphism of its point variety. As an application, we classify twisted algebras of three-dimensional geometric Artin–Schelter regular algebras up to graded algebra isomorphism.

Twisted Segre products

We introduce the notion of the twisted Segre product A∘ψB of Z-graded algebras A and B with respect to a twisting map ψ. It is proved that if A and B are noetherian Koszul Artin-Schelter regular algebras and ψ is a twisting map such that the twisted Segre product A∘ψB is noetherian, then A∘ψB is a noncommutative graded isolated singularity. To prove this result, the notion of densely (bi-)graded algebras is introduced. Moreover, we show that the twisted Segre product A∘ψB of A=k[u,v] and B=k[x,y] with respect to a diagonal twisting map ψ is a noncommutative quadric surface (so in particular it is noetherian), and we compute the stable category of graded maximal Cohen-Macaulay modules over it.

The Calabi–Yau property of Ore extensions of two-dimensional Artin–Schelter regular algebras and their PBW deformations

Let A be a noncommutative Artin–Schelter regular algebra of dimension 2 with the Nakayama automorphism μ A and U a PBW deformation of A with the Nakayama automorphism μ U . We prove that any graded Ore extension A[z;μ A ,δ] and any filtered Ore extension U[z;μ U ,δ ˜] are 3-Calabi–Yau.

Algebras defined by Lyndon words and Artin-Schelter regularity

LetX={x1,x2,⋯,xn}X= \{x_1, x_2, \cdots , x_n\}be a finite alphabet, and letKKbe a field. We study classesC(X,W)\mathfrak {C}(X, W)of gradedKK-algebrasA=K⟨X⟩/IA = K\langle X\rangle / I, generated byXXand witha fixed set of obstructionsWW. Initially we do not impose restrictions onWWand investigate the case when the algebras inC(X,W)\mathfrak {C} (X, W)have polynomial growth and finite global dimensiondd. Next we consider classesC(X,W)\mathfrak {C} (X, W)of algebras whose sets of obstructionsWWare antichains of Lyndon words. The central question is “when a classC(X,W)\mathfrak {C} (X, W)contains Artin-Schelter regular algebras?” Each classC(X,W)\mathfrak {C} (X, W)defines a Lyndon pair(N,W)(N,W), which, ifNNis finite, determines uniquely the global dimension,gldimAgl\,dimA, and the Gelfand-Kirillov dimension,GKdimAGK dimA, for everyA∈C(X,W)A \in \mathfrak {C}(X, W). We find a combinatorial condition in terms of(N,W)(N,W), so that the classC(X,W)\mathfrak {C}(X, W)contains the enveloping algebraUgU\mathfrak {g}, of a Lie algebrag\mathfrak {g}. We introduce monomial Lie algebras defined by Lyndon words, and prove results on Gröbner-Shirshov bases of Lie ideals generated by Lyndon-Lie monomials. Finally we classify all two-generated Artin-Schelter regular algebras of global dimension66and77occurring as envelopingU=UgU = U\mathfrak {g}ofstandard monomial Lie algebras. The classification is made in terms of their Lyndon pairs(N,W)(N, W), each of which determines also the explicit relations ofUU.

Auslander’s theorem for dihedral actions on preprojective algebras of type A

AbstractGiven an algebra R and G a finite group of automorphisms of R, there is a natural map $\eta _{R,G}:R\#G \to \mathrm {End}_{R^G} R$ , called the Auslander map. A theorem of Auslander shows that $\eta _{R,G}$ is an isomorphism when $R=\mathbb {C}[V]$ and G is a finite group acting linearly and without reflections on the finite-dimensional vector space V. The work of Mori–Ueyama and Bao–He–Zhang has encouraged the study of this theorem in the context of Artin–Schelter regular algebras. We initiate a study of Auslander’s result in the setting of nonconnected graded Calabi–Yau algebras. When R is a preprojective algebra of type A and G is a finite subgroup of $D_n$ acting on R by automorphism, our main result shows that $\eta _{R,G}$ is an isomorphism if and only if G does not contain all of the reflections through a vertex.

Degree bounds for Hopf actions on Artin–Schelter regular algebras

We study semisimple Hopf algebra actions on Artin–Schelter regular algebras and prove several upper bounds on the degrees of the minimal generators of the invariant subring, and on the degrees of syzygies of modules over the invariant subring. These results are analogues of results for group actions on commutative polynomial rings proved by Noether, Fogarty, Fleischmann, Derksen, Sidman, Chardin, and Symonds.

Examples of Non-Semisimple Hopf Algebra Actions on Artin-Schelter Regular Algebras

Let \(\Bbbk \) be a base field of characteristic p > 0 and let U be the restricted enveloping algebra of a 2-dimensional nonabelian restricted Lie algebra. We classify all inner-faithful U-actions on noetherian Koszul Artin-Schelter regular algebras of global dimension up to three.

Artin–Schelter regular algebras of dimension five with three generators

In this paper, we investigate Artin–Schelter regular algebras of dimension 5 with three generators in degree 1 under the hypothesis that [Formula: see text], in which the degree types of the relations for the number of the generating relations less than five can be determined. We prove that the only possible degree type of three generating relations is (2, 2, 3) and the only possible degree type of four generating relations is (2, 2, 3, 4).

Some Open Problems in the Context of Skew PBW Extensions and Semi-graded Rings

In this paper, we discuss some open problems of non-commutative algebra and non-commutative algebraic geometry from the approach of skew PBW extensions and semi-graded rings. More exactly, we will analyze the isomorphism arising in the investigation of the Gelfand–Kirillov conjecture about the commutation between the center and the total ring of fractions of an Ore domain. The Serre’s conjecture will be discussed for a particular class of skew PBW extensions. The questions about the Noetherianity and the Zariski cancellation property of Artin–Schelter regular algebras will be reformulated for semi-graded rings. Advances for the solution of some of the problems are included.

Some special types of determinants in graded skew PBW extensions

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.

Nakayama automorphisms of graded Ore extensions of Koszul Artin-Schelter regular algebras

Let A be a Koszul Artin-Schelter regular algebra, σ a graded automorphism of A and δ a degree-one σ-derivation of A. We introduce an invariant for δ called the σ-divergence of δ. We describe the Nakayama automorphism of the graded Ore extension B=A[z;σ,δ] explicitly using the σ-divergence of δ, and construct a twisted superpotential ωˆ for B so that it is a derivation quotient algebra defined by ωˆ. We also determine all graded Ore extensions of noetherian Artin-Schelter regular algebras of dimension 2 and compute their Nakayama automorphisms.

Actions of small groups on two-dimensional Artin-Schelter regular algebras

In commutative invariant theory, a classical result due to Auslander says that if R=k[x1,…,xn] and G is a finite subgroup of Autgr(R)≅GL(n,k) which contains no reflections, then there is a natural graded isomorphism R#G≅EndRG(R). In this paper, we show that a version of Auslander's Theorem holds if we replace R by an Artin-Schelter regular algebra A of global dimension 2, and G by a finite subgroup of Autgr(A) which contains no quasi-reflections. This extends work of Chan–Kirkman–Walton–Zhang. As part of the proof, we classify all such pairs (A,G), up to conjugation of G by an element of Autgr(A). In all but one case, we also write down explicit presentations for the invariant rings AG, and show that they are isomorphic to factors of AS regular algebras.

Stably Noetherian Algebras of Polynomial Growth

Let A be a right noetherian algebra over a field k. If the base field extension A ⊗kK remains right noetherian for all extension fields K of k, then A is called stably right noetherian over k. We develop an inductive method to show that certain algebras of finite Gelfand-Kirillov dimension are stably noetherian, using critical composition series. We use this to characterize which algebras satisfying a polynomial identity are stably noetherian. The method also applies to many $\mathbb {N}$ -graded rings of finite global dimension; in particular, we see that a noetherian Artin-Schelter regular algebra must be stably noetherian. In addition, we study more general variations of the stably noetherian property where the field extensions are restricted to those of a certain type, for instance purely transcendental extensions.

The Jacobian, Reflection Arrangement and Discriminant for Reflection Hopf Algebras

AbstractWe study finite-dimensional semisimple Hopf algebra actions on noetherian connected graded Artin–Schelter regular algebras and introduce definitions of the Jacobian, the reflection arrangement, and the discriminant in a noncommutative setting.

Three infinite families of reflection Hopf algebras

Let H be a semisimple Hopf algebra acting on an Artin-Schelter regular algebra A, homogeneously, inner-faithfully, preserving the grading on A, and so that A is an H-module algebra. When the fixed subring AH is also AS regular, thus providing a generalization of the Chevalley-Shephard-Todd Theorem, we say that H is a reflection Hopf algebra for A. We show that each of the semisimple Hopf algebras H2n2 of Pansera, and A4m and B4m of Masuoka is a reflection Hopf algebra for an AS regular algebra of dimension 2 or 3.