Artin Schelter Regular Algebras Research Articles

INVARIANT THEORY OF FINITE GROUP ACTIONS ON DOWN-UP ALGEBRAS

We study Artin-Schelter Gorenstein fixed subrings of some Artin-Schelter regular algebras of dimension 2 and 3 under a finite group action and prove a noncommutative version of the Kac-Watanabe and Gordeev theorem for these algebras.

Hopf actions and Nakayama automorphisms

Let H be a Hopf algebra with antipode S, and let A be an N-Koszul Artin–Schelter regular algebra. We study connections between the Nakayama automorphism of A and S2 of H when H coacts on A inner-faithfully. Several applications pertaining to Hopf actions on Artin–Schelter regular algebras are given.

Monomial algebras defined by Lyndon words

Assume that X={x1,…,xg} is a finite alphabet and K is a field. We study monomial algebras A=K〈X〉/(W), where W is an antichain of Lyndon words in X of arbitrary cardinality. We find a Poincaré–Birkhoff–Witt type basis of A in terms of its Lyndon atoms N, but, in general, N may be infinite. We prove that if A has polynomial growth of degree d then A has global dimension d and is standard finitely presented, with d−1⩽|W|⩽d(d−1)/2. Furthermore, A has polynomial growth iff the set of Lyndon atoms N is finite. In this case A has a K-basis N={l1α1l2α2⋯ldαd|αi⩾0,1⩽i⩽d}, where N={l1,…,ld}. We give an extremal class of monomial algebras, the Fibonacci–Lyndon algebras, Fn, with global dimension n and polynomial growth, and show that the algebra F6 of global dimension 6 cannot be deformed, keeping the multigrading, to an Artin–Schelter regular algebra.

Artin–Schelter regular algebras of dimension five with two generators

We study and classify Artin–Schelter regular algebras of dimension five with two generators under an additional Z2-grading by Hilbert driven Gröbner basis computations. All the algebras we obtained are strongly noetherian, Auslander regular, and Cohen–Macaulay. One of the results provides an answer to Fløystad–Vatneʼs question in the context of Z2-grading. Our results also achieve a connection between Lyndon words and Artin–Schelter regular algebras.

Returning arrows for self-injective algebras and Artin–Schelter regular algebras

We show that when constructing twisted trivial extensions for a graded self-injective algebra, the returning arrows appear in the quiver, and in the meantime the complexity increases by one in Koszul cases, and the representation dimension also increases by one under certain additional conditions. We apply our results to Koszul Artin–Schelter regular algebras and obtain a family of extensions of a Koszul Artin–Schelter regular algebra, among them one is central extension and one is Calabi–Yau.

Hopf actions on filtered regular algebras

We study finite dimensional Hopf algebra actions on so-called filtered Artin–Schelter regular algebras of dimension n, particularly on those of dimension 2. The first Weyl algebra is an example of such an algebra with n=2, for instance. Results on the Gorenstein condition and on the global dimension of the corresponding fixed subrings are also provided.

Skew polynomial algebras with coefficients in Koszul Artin–Schelter regular algebras

Let A be a Koszul Artin–Schelter regular algebra with Nakayama automorphism ξ. We show that the Yoneda Ext-algebra of the skew polynomial algebra A[z;ξ] is a trivial extension of a Frobenius algebra. Then we prove that A[z;ξ] is Calabi–Yau; and hence each Koszul Artin–Schelter regular algebra is a subalgebra of a Koszul Calabi–Yau algebra. A superpotential wˆ is also constructed so that the Calabi–Yau algebra A[z;ξ] is isomorphic to the derivation quotient of wˆ. The Calabi–Yau property of a skew polynomial algebra with coefficients in a PBW-deformation of a Koszul Artin–Schelter regular algebra is also discussed.

Koszul-like algebras and modules

In this paper, the notion of Koszul-like algebra is introduced; this notion generalizes the notion of Koszul algebra and includes some Artin-Schelter regular algebras of global dimension 5 as special examples. Basic properties of Koszul-like modules are discussed. In particular, some necessary and sufficient conditions for KL(A) = L(A) are provided, where KL(A) and L(A) denote the categories of Koszul-like modules and modules with linear presentations (see [1]–[3], etc.) respectively, and A is a Koszul-like algebra. We construct new Koszul-like algebras from the known ones by the “one-point extension.” Some criteria for a graded algebra to be Koszul-like are provided. Finally, we construct many classical Koszul objects from the given Koszul-like objects.

Some Constructions of Bi-Koszul Algebras

Bi-Koszul algebras, including two classes of non-Koszul Artin-Schelter regular algebras of global dimension 4, were a class of graded algebras with non-pure resolutions, introduced in [8]. There is a natural question: can we construct bi-Koszul algebras from algebras with pure resolutions? In this paper, we study this question in terms of normal extensions and Ore extensions. More precisely, we attempt to obtain bi-Koszul algebras from algebras with pure resolutions by these two kinds of extensions. Furthermore, some homological properties of bi-Koszul algebras obtained in such ways are discussed.

Deformations of Koszul Artin–Schelter Gorenstein algebras

We compute the Nakayama automorphism of a Poincaré–Birkhoff–Witt (PBW)-deformation of a Koszul Artin–Schelter (AS) Gorenstein algebra of finite global dimension, and give a criterion for an augmented PBW-deformation of a Koszul Calabi–Yau algebra to be Calabi–Yau. The relations between the Calabi–Yau property of augmented PBW-deformations and that of non-augmented cases are discussed. The Nakayama automorphisms of PBW-deformations of Koszul AS–Gorenstein algebras of global dimensions 2 and 3 are given explicitly. We show that if a PBW-deformation of a graded Calabi–Yau algebra is still Calabi–Yau, then it is defined by a potential under some mild conditions. Some classical results are also recovered. Our main method used in this article is elementary and based on linear algebra. The results obtained in this article will be applied in a subsequent paper (He et al., Skew polynomial algebras with coefficients in AS regular algebras, preprint, 2011).

A class of non-Hopf bi-Frobenius algebras

We construct a class of bi-Frobenius algebras, which are not Hopf algebras. In some special cases, the constructed algebras are related to the Yoneda algebras of Artin-Schelter regular algebras of dimension three.

A class of AS-regular algebras of dimension five

We classify 5-dimensional Artin–Schelter regular algebras generated by two generators of degree 1 with three generating relations of degree 4 under a generic condition. All the algebras obtained are proved to be strongly noetherian, Auslander regular and Cohen–Macaulay with respect to the Gelfand–Kirillov dimension.

Double Ore Extensions Versus Iterated Ore Extensions

Motivated by the construction of new examples of Artin–Schelter regular algebras of global dimension four, Zhang and Zhang [6] introduced an algebra extension A P [y 1, y 2; σ, δ, τ] of A, which they called a double Ore extension. This construction seems to be similar to that of a two-step iterated Ore extension over A. The aim of this article is to describe those double Ore extensions which can be presented as iterated Ore extensions of the form A[y 1; σ1, δ1][y 2; σ2, δ2]. We also give partial answers to some questions posed in Zhang and Zhang [6].

DG algebra structures on AS-regular algebras of dimension 2

Let A be a connected cochain DG algebra, whose underlying graded algebra is an Artin-Schelter regular algebra of global dimension 2 generated in degree 1. We give a description of all possible differential of A and compute H(A). Such kind of DG algebras are proved to be strongly Gorenstein. Some of them serve as examples to indicate that a connected DG algebra with Koszul underlying graded algebra may not be a Koszul DG algebra defined in He and We (J Algebra, 2008, 320: 2934–2962). Unlike positively graded chain DG algebras, we give counterexamples to show that a bounded below DG A-module with a free underlying graded A#-module may not be semi-projective.

Graded Morita equivalences for generic Artin-Schelter regular algebras

Let A=A(E,σ),A'=A(E',σ') be Noetherian Artin-Schelter regular geometric algebras with dimkA1=dimkA1'=n, and let ν,ν' be generalized Nakayama automorphisms of A,A'. In this paper, we study relationships between the conditions (A) A is graded Morita equivalent to A', and (B) A(E,ν∗σn) is isomorphic to A(E',(ν')∗(σ')n) as graded algebras. It is proved that if A,A' are “generic” 3-dimensional quadratic Artin-Schelter regular algebras, then (A) is equivalent to (B), and if A,A' are n-dimensional skew polynomial algebras, then (A) implies (B).

Artin-Schelter regular algebras of dimension five

We show that there are exactly three types of Hilbert series of Artin-Schelter regular algebras of dimension five with two generators. One of these cases (the most extreme) may not be realized by an enveloping algebra of a graded Lie algebra. This is a n

Artin–Schelter regular algebras and categories

Motivated by constructions in the representation theory of finite dimensional algebras we generalize the notion of Artin–Schelter regular algebras of dimension n to algebras and categories to include Auslander algebras and a graded analogue for infinite representation type. A generalized Artin–Schelter regular algebra or a category of dimension n is shown to have common properties with the classical Artin–Schelter regular algebras. In particular, when they admit a duality, then they satisfy Serre duality formulas and the Ext -category of nice sets of simple objects of maximal projective dimension n is a finite length Frobenius category.

Koszulity of Algebras with Nonpure Resolutions

We discuss certain homological properties of graded algebras whose trivial modules admit nonpure resolutions. Such algebras include both of Artin–Schelter regular algebras of types (12221) and (13431). Under certain conditions, a module with nonpure resolution is decomposed to form an extension by two modules with pure resolutions.

Gorenstein subrings of invariants under Hopf algebra actions

This paper concerns conditions on the action of a finite dimensional semisimple Hopf algebra on an Artin–Schelter regular algebra that force the subring of invariants to satisfy the Artin–Schelter Gorenstein condition. Classical results such as Watanabe's Theorem and Stanley's Theorem are extended from the case of a group action to the context of a Hopf algebra action. A Hopf algebra version of the homological determinant is introduced, and it becomes an important tool in the generalization from group actions to Hopf algebra actions.

Double extension regular algebras of type (14641)

We construct several families of Artin–Schelter regular algebras of global dimension four using double Ore extension and then prove that all these algebras are strongly noetherian, Auslander regular, Koszul and Cohen–Macaulay domains. Many regular algebras constructed in the paper are new and are not isomorphic to either a normal extension or an Ore extension of an Artin–Schelter regular algebra of global dimension three.