Hui-Xiang Chen, Ding-Guo Wang, James J. Zhang
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.
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We show certain invariants of graded algebras of which all obstructions are Lyndon words and provide some methods to construct Artin-Schelter regular algebras from a closed set of Lyndon words.
Auslander theorem for PI Artin-Schelter regular algebras
本文证明了在 noetherian 多项式恒等 Artin-Schelter 正则代数上，有限群作用或余作用下的 Auslander 定理的一个版本。
We study and classify Artin-Schelter regular algebras of dimension five with two generators under an additional $\mathbb Z^2$-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-Vatne's question in the context of $\mathbb Z^2$-grading. Our results also achieve a connection between Lyndon words and Artin-Schelter regular algebras.
We identify two Batalin-Vilkovisky algebra structures, one obtained by Kowalzig and Krahmer on the Hochschild cohomology of an Artin-Schelter regular algebra with semisimple Nakayama automorphism and the other obtained by Lambre, Zhou and Zimmermann on the Hochschild cohomology of a Frobenius algebra also with semisimple Nakayama automorphism, provided that these two algebras are Koszul dual to each other.
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 $\hatω$ for $B$ so that it is a derivation quotient algebra defined by $\hatω$. We also determine all graded Ore extensions of noetherian Artin-Schelter regular algebras of dimension 2 and compute their Nakayama automorphisms.
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.
This is a survey of results that extend notions of the classical invariant theory of linear actions by finite groups on $k[x_1, \dots, x_n]$ to the setting of finite group or Hopf algebra $H$ actions on an Artin-Schelter regular algebra $A$. We investigate when $A^H$ is AS regular, or AS Gorenstein, or a "complete intersection" in a sense that is defined. Directions of related research are explored briefly.
The ozone groups of PI Artin-Schelter regular algebras are abelian
本文证明了 PI Artin-Schelter 正则代数的 ozone 组是交换群，并展示了其同调确定的平凡性。
PI Artin--Schelter regular algebras of dimension 3 are unique factorization rings
本文证明了三维PI Artin-Schelter正则代数是唯一分解环，非交换代数中对应正则局部环三维情况的类比，为Auslander-Buchsbaum定理的证明提供了关键支撑。
This paper studies the homological determinants and Nakayama automorphisms of not-necessarily-noetherian $m$-Koszul twisted Calabi-Yau or, equivalently, $m$-Koszul Artin-Schelter regular, algebras. Dubois-Violette showed that such an algebra is isomorphic to a derivation quotient algebra D(w,i) for a unique-up-to-scalar-multiples twisted superpotential w in a tensor power of some vector space V. By definition, D(w,i) is the quotient of the tensor algebra TV by the ideal generated by all i-th order left partial derivatives of w. We identify the group of graded algebra automorphisms of D(w,i) wi…
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 new phenomenon compared to lower dimensions, where all resolution types may be realized by such enveloping algebras.
We compute the Nakayama automorphism of a PBW-deformation of a Koszul Artin-Schelter 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 Artin-Schelter 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 de…
We consider algebras that can be realized as PBW deformations of (Artin-Schelter) regular algebras. This is equivalent to the homogenization of the algebra being regular. It is shown that the homogenization, when it is a geometric algebra, contains a component whose points are in 1-1 correspondence with the simple modules of the deformation. We classify all PBW deformations of 2-dimensional regular algebras and give examples of 3-dimensional deformations. Other properties, such as the skew Calabi-Yau property and closure under tensor products, are considered.
Ozone groups of Artin--Schelter regular algebras satisfying a polynomial identity
该研究针对满足多项式恒等式的诺特Artin–Schelter正则代数的ozone群展开，刻画了对应极大秩ozone群的斜多项式环，分类了平凡ozone群的Calabi–Yau代数，并给出相关应用。
Take $A$ to be a regular quadratic algebra of global dimension three. We observe that there are examples of $A$ containing a dimension three regular cubic algebra $C$. If $B$ is another dimension three regular quadratic algebra, also containing $C$ as a subalgebra, then we can form the pushout algebra $D$ of the inclusions $i_1:C\hookrightarrow A$ and $i_2:C\hookrightarrow B$. We show that for a certain class of regular algebras $C\hookrightarrow A,B$, their pushouts $D$ are regular quadratic algebras of global dimension four. Furthermore, some of the point module structures of the dimension t…
Group coactions on two-dimensional Artin-Schelter regular algebras
本文完全分类了有限群在二维Artin-Schelter正则代数上的余作用，给出了Auslander映射为同构的充要条件，并确定了不变环保持Artin-Schelter正则性的精确判据。
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 …
A double Ore extension is a natural generalization of the Ore extension. We prove that a connected graded double Ore extension of an Artin-Schelter regular algebra is Artin-Schelter regular. Some other basic properties such as the determinant of the DE-data are studied. Using the double Ore extension, we construct 26 families of Artin-Schelter regular algebras of global dimension four in a sequel paper.
Some Artin-Schelter Regular Algebras From Dual Reflection Groups and their Geometry
本文提出对偶反射群概念作为Shephard-Todd-Chevalley定理的推广，给出阶为16的两个例子，并研究了三个相关四维Artin-Schelter正则代数的代数与几何性质。
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 $\hat{w}$ is also constructed so that the Calabi-Yau algebra $A[z;ξ]$ is isomorphic to the derivation quotient of $\hat{w}$. The Calabi-Yau property of a skew polynomial algebra with coefficients in a PBW-deformation of a Koszul Arti…
Artin-Schelter Gorenstein property of Hopf Galois extensions
本文探讨了Hopf Galois扩展的AS Gorenstein性质，证明了在特定条件下该性质可被继承，并揭示了AS Gorenstein Hopf代数的投射维数单体不变量特性。
In commutative invariant theory, a classical result due to Auslander says that if $R = \Bbbk[x_1, \dots, x_n]$ and $G$ is a finite subgroup of $\text{Aut}_{\text{gr}}(R) \cong \text{GL}(n,\Bbbk)$ which contains no reflections, then there is a natural graded isomorphism $R \hspace{1pt} \# \hspace{1pt} G \cong \text{End}_{R^G}(R)$. 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 $\text{Aut}_{\text{gr}}(A)$ which contains no quasi-reflections. This extends work of C…
In this paper, quadratic algebras on which $\tilde{H}_4$, the Heisenberg group of order 64, acts as degree-preserving algebra automorphisms are studied. In particular, we show that if $\mathcal{A}$ is a four-dimensional Artin-Schelter regular quadratic $\tilde{H}_4$-algebra with the degree one part isomorphic to the Schrödinger representation of $\tilde{H}_4$, then $\mathcal{A}$ is (a twist of) a four-dimensional Sklyanin algebra or (a twist of) a quantum Clifford algebra of global dimension 4.
On differential smoothness of certain Artin-Schelter regular algebras of dimension 5
本文研究了五维Artin-Schelter正则代数的微分光滑性，发现部分代数无法接受微分学，而五生成格雷格 Clifford 代数则为正例。
Artin-Schelter Regular Algebras
本文综述了作为非交换射影几何核心对象的 Artin-Schelter 正则代数，回顾其早期基础成果并梳理近30年研究的主要主题。
Nakayama automorphisms of graded double Ore extensions of Koszul Artin-Schelter regular algebras with nontrivial skew derivations
本文研究Koszul Artin-Schelter正则代数的分次双重Ore扩张，通过构造极小自由预解证明其保持Koszul性质，并引入$ς$-散度同调不变量精确刻画中山自同构，进而构造扭曲超势实现导子商代数同构。
We determine the possible Hilbert functions of graded rank one torsion free modules over three dimensional Artin-Schelter regular algebras. It turns out that, as in the commutative case, they are related to Castelnuovo functions. From this we obtain an intrinsic proof that the space of torsion free rank one modules on a non-commutative projective plane is connected. A different proof of this fact, based on deformation theoretic methods and the known commutative case has recently been given by Nevins and Stafford. For the Weyl algebra it was proved by Wilson.
In this paper, we discuss returning arrows with respect to the Nakayama translation appearing in the quivers of some important algebras when we construct extensions. When constructing twisted trivial extensions for a graded self-injective algebra, we show that the returning arrows appear in the quiver, that the complexity increases by 1 in Koszul cases, and the representation dimension also increases by 1 under certain additional conditions. By applying Koszul duality, for each Koszul Artin-Schelter regular algebra of global dimension l and Gelfand-Kirilov dimension $c$, we construct a family …
We study Artin-Schelter Gorenstein fixed subrings of some Artin-Schelter regular algebras of dimension 2 and 3 under finite group actions, and prove a noncommutative version of the Kac-Watanabe and Gordeev theorem for these algebras.
Actions of Drinfeld doubles of finite groups on Artin-Schelter regular algebras
本文研究有限群的Drinfeld双群在Artin-Schelter正则代数上的内忠实作用及其不变量代数，并对8阶和16阶四元数群给出显式计算结果。
Actions of Taft Algebras on Noetherian Down-Up Algebras
本文分类了Taft代数在Noetherian分次down-up代数上的作用，确定了不变环的交换性、Artin-Schelter正则与Gorenstein性质，并指出省略半单假设会使相关文献结论与猜想失效。
Discrete Invariants of Koszul Artin-Schelter Regular Algebras of Dimension four
研究四维Koszul Artin-Schelter正则代数的离散不变量，通过Schur-Weyl对偶性和Borel-Weil定理，将其与K3表面关联，用于区分不同代数。
Trivial extensions of Koszul Artin-Schelter regular algebras
研究Koszul Artin-Schelter正则代数的平凡扩展的稳定范畴与其Koszul对偶代数的导出范畴的三角等价性，并给出连通分量情况下的等价条件。
For any Koszul Artin-Schelter regular algebra A, we consider a version of the universal Hopf algebra aut(A) coacting on A, introduced by Manin. To study the representations (i.e. finite dimensional comodules) of this Hopf algebra, we use the Tannaka-Krein formalism. Specifically, we construct an explicit combinatorial rigid monoidal category U, equipped with a functor M to finite dimensional vector spaces such that aut(A)= coend_U(M). Using this pair (U,M) we show that aut(A) is quasi-hereditary as a coalgebra and in addition is derived equivalent to the representation category of U.
Some components of the moduli space of Koszul Artin-Schelter regular algebras of dimension four
通过计算已知四维Koszul Artin-Schelter正则代数族的霍奇上同调和柯达依-斯宾塞映射，结合映射满射、双射的性质分析，确定了该类代数模栈的若干分支。
This is a general study of twisted Calabi-Yau algebras that are $\mathbb{N}$-graded and locally finite-dimensional, with the following major results. We prove that a locally finite graded algebra is twisted Calabi-Yau if and only if it is separable modulo its graded radical and satisfies one of several suitable generalizations of the Artin-Schelter regularity property, adapted from the work of Martinez-Villa as well as Minamoto and Mori. We characterize twisted Calabi-Yau algebras of dimension 0 as separable $k$-algebras, and we similarly characterize graded twisted Calabi-Yau algebras of dime…
Let $k$ be a field and $X$ be a set of $n$ elements. We introduce and study a class of quadratic $k$-algebras called \emph{quantum binomial algebras}. Our main result shows that such an algebra $A$ defines a solution of the classical Yang-Baxter equation (YBE), if and only if its Koszul dual $A^{!}$ is Frobenius of dimension $n,$ with a \emph{regular socle} and for each $x,y \in X $ an equality of the type $xyy=αzzt,$ where $α\in k \setminus\{0\},$ and $z,t \in X$ is satisfied in $A$. We prove the equivalence of the notions \emph{a binomial skew polynomial ring} and \emph{a binomial solution o…
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.
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 …
Rank two Artin-Schelter regular algebras and non commuting derivations
本文研究由满足特定交换关系的导子 $Δ$ 和 $Γ$ 构造的代数 $(A,\ast)$，证明其过滤半经典极限为秩二泊松结构，并在每个至少四维的维度上给出 Artin-Schelter 正则代数和 Calabi-Yau 代数的多参数族新例子。
We show that Artin-Schelter regularity of a $\mathbb{Z}$-graded algebra can be examined by its associated $\mathbb{Z}^r$-graded algebra. We prove that there is exactly one class of four-dimensional Artin-Schelter regular algebras with two generators of degree one in the Jordan case. This class is strongly noetherian, Auslander regular, and Cohen-Macaulay. Their automorphisms and point modules are described.
Dual reflection groups for three-dimensional Artin--Schelter regular algebras
本文完全分类了三维 Artin--Schelter 正则代数的非交换对偶反射群，发现三次代数无此类群，二次代数中来自 Ore 扩张或 Sklyanin 代数，并构造了新无限族，回答了相关猜想。
Homological regularities and concavities
本文研究非交换代数的同调正则性，引入凹性和Artin-Schelter正则性两个新不变量，并建立判定诺特连通分次代数是否为Artin-Schelter正则的准则。
We study quadratic algebras over a field $\textbf{k}$. We show that an $n$-generated PBW algebra $A$ has finite global dimension and polynomial growth \emph{iff} its Hilbert series is $H_A(z)= 1 /(1-z)^n$. Surprising amount can be said when the algebra $A$ has \emph{quantum binomial relations}, that is the defining relations are nondegenerate square-free binomials $xy-c_{xy}zt$ with non-zero coefficients $c_{xy}\in \textbf{k}$. In this case various good algebraic and homological properties are closely related. The main result shows that for an $n$-generated quantum binomial algebra $A$ the fol…
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 $A = σ(R)\langle x_1,x_2 \rangle$ of an Artin-Schelter regular algebra $R$ is Artin-Schelter regular. As a consequence, every graded skew PBW extension $A = σ(R)\langle x_1,x_2 \rangle$ of a connected skew Calabi-Yau algebra $R$ of dimension $d$ is skew Calabi-Yau of dimension $d+2$.
A family of algebras with trivial ozone group
研究包含与结点三次曲线相关的二次Artin-Schelter正则代数的一族Calabi-Yau代数，证明其臭氧群平凡，且该族中分次代数在非平凡分次自同构群作用下不变环非Artin-Schelter正则。
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 S^2 of H when H coacts on A inner-faithfully. Several applications pertaining to Hopf actions on Artin-Schelter regular algebras are given.
We study homological properties and rigidity of group coactions on Artin-Schelter regular algebras.
On GK Dimension and Generator Bounds for a Class of Graded Algebras
本文引入单调代数概念，证明其GK维数以上整体维数为上界、最小生成元数有类似界，并证Artin-Schelter正则代数的整体维数与GK维数差为偶数。
By any measure, semisimple modules form one of the most important classes of modules and play a distinguished role in the module theory and its applications. One of the most fundamental results in this area is the Wedderburn-Artin theorem. In this paper, we establish natural generalizations of semisimple modules and give a generalization of the Wedderburn-Artin theorem. We study modules in which every submodule is isomorphic to a direct summand and name them {\it virtually semisimple modules}. A module $_RM$ is called {\it completely virtually semisimple} if each submodules of $M$ is a virtual…
The Nakayama automorphism of a class of connected graded Artin-Schelter regular algebras is calculated explicitly.
We construct four families of Artin-Schelter regular algebras of global dimension four. Under some generic conditions, this is a complete list of Artin-Schelter regular algebras of global dimension four that are generated by two elements of degree 1. These algebras are also strongly noetherian, Auslander regular and Cohen-Macaulay. One of the main tools is Keller's higher-multiplication theorem on A-infinity Ext-algebras.
We study invariants and quotient categories of fixed subrings of Artin-Schelter regular algebras under Hopf algebra actions.
We classify reflexive graded right ideals, up to isomorphism and shift, of generic cubic three dimensional Artin-Schelter regular algebras. We also determine the possible Hilbert functions of these ideals. These results are obtained by using similar methods as for quadratic Artin-Schelter algebras. In particular our results apply to the enveloping algebra of the Heisenberg-Lie algebra from which we deduce a classification of right ideals of an invariant ring of the first Weyl algebra.
We study a noncommutative version of the Zariski cancellation problem for some classes of connected graded Artin-Schelter regular algebras of global dimension three.
Nakayama automorphism is used to study group actions and Hopf algebra actions on Artin-Schelter regular algebras of global dimension three.
Cohen-Macaulay representations of Artin-Schelter Gorenstein algebras of dimension one
本文研究了一维Artin-Schelter Gorenstein代数的Cohen-Macaulay表示，证明了稳定范畴存在silting对象的条件，并探讨了tilting对象的存在性及非交换射影二次超曲面的导出范畴结构。
Algebras defined by Lyndon words and Artin-Schelter regularity
研究由 Lyndon 字定义的分级代数类的 Artin-Schelter 正则性，通过 Lyndon 对给出组合条件以包含 Lie 代数包络代数，并分类全局维数 6 和 7 的二生成 Artin-Schelter 正则代数。
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.
Homological Integrals for Weak Hopf Algebras
本文引入了无限维弱Hopf代数的同调积分概念，并借此证明了若干结构定理，包括Artin-Schelter性质与Van den Bergh条件的等价性、对极映射的自动可逆性，以及关于仿射中心上有限弱Hopf代数的分解定理。
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