Johan Richter, Sergei Silvestrov, Alex Tumwesigye
In this article we consider the Ore extension Algebra for the algebra $\mathcal{A}$ of functions with finite support on a countable set. We derive explicit formulas for twisted derivations on $\mathcal{A}.$ We give a description for the centralizer of $\mathcal{A},$ and the center of the Ore extension algebra when the derivation is zero.
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Motivated by the construction of new examples of Artin-Schelter regular algebras of global dimension four, J.J. Zhang and J. Zhang (2008) 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 paper is to describe those double Ore extensions which can be presented as iterated Ore extensions of the form $A[y_1;σ_1, δ_1][y_2;σ_2, δ_2]$. We also give partial answers to some questions posed in Zhang and Zhang (2008).
It is proved that the Poisson enveloping algebra of a double Poisson-Ore extension is an iterated double Ore extension. As an application, properties that are preserved under iterated double Ore extensions are invariants of the Poisson enveloping algebra of a double Poisson-Ore extension.
We derive necessary and sufficient conditions for an Ore extension of a Hopf algebra to have a Hopf algebra structure of a certain type. This construction generalizes the notion of Hopf-Ore extension, called a generalized Hopf-Ore extension. We describe the generalized Hopf-Ore extensions of the enveloping algebras of Lie algebras. For some Lie algebras g, the generalized Hopf-Ore extensions of U(g) are classified.
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.
We show that there exist noncommutative Ore extensions in which every right ideal is two-sided. This answers a problem posed by Marks in Duo Rings and Ore extensions, J.Algebra 280(2), (2004). We also provide an easy construction of one sided duo rings.
Ore extensions of multiplier Hopf coquasigroups
本文研究乘子Hopf余拟群的Ore扩张，给出其成为乘子Hopf余拟群的充要条件，并讨论两个此类扩张的同构。
We present Buchberger Theory and Algorithm of Gröbner bases for multivariate Ore extensions of rings presented as modules over a principal ideal domain. The algorithms are based on Möller Lifting Theorem.
Ore Extension of Group-cograded Hopf Coquasigroups
本文研究群余分次Hopf拟群的Ore扩张，给出其范畴解释、实例及Ore扩张保持群余分次Hopf拟群结构的充要条件，并讨论了Ore扩张间的同构问题。
We extend recent results in order to construct projective resolutions for modules over twisted tensor products of truncated polynomial rings. We begin by taking note of the conditions necessary to think of these algebras as a type of Ore extension. We then use this parallel with Ore extensions to develop a method for constructing projective resolutions. Finally we use the new construction to compute a resolution for a family of examples.
We prove that the universal enveloping algebra of a Poisson-Ore extension is a length two iterated Ore extension of the original universal enveloping algebra. As consequences, we observe certain ring-theoretic invariants of the universal enveloping algebras that are preserved under iterated Poisson-Ore extensions. We apply our results to iterated quadratic Poisson algebras arising from semiclassical limits of quantized coordinate rings and a family of graded Poisson algebras of Poisson structures of rank at most two.
Let $R$ be a ring and $S=R[x;σ,δ]$ its Ore extension. We prove under some conditions that $R$ is a quasi-Baer ring if and only if the Ore extension $R[x;σ,δ]$ is a quasi-Baer ring. Examples are provided to illustrate and delimit our results.
In order to study AS-regular algebras of dimension 5, we consider dimension 5 graded iterated Ore extensions generated in degree one. We classify the possible degrees of relations and structure of the free resolution for extensions with 3 and 4 generators. We show that every known type of algebra of dimension 5 can be realized by an Ore extension and we consider which of these types cannot be realized by an enveloping algebra.
Let $R$ be a ring, $σ$ an injective endomorphism of $R$ and $δ$ a $σ$-derivation of $R$. We prove that if $R$ is semiprime left Goldie then the same holds for the Ore extension $R[x;σ,δ]$ and both rings have the same left uniform dimension.
We begin by reviewing a classical result on the algebraic dependence of commuting elements in Weyl algebras. We proceed by describing generalizations of this result to various classes of Ore extensions, both results that have already been published and a new result.
In this short note we present an elementary matrix-constructive proof of Quillen-Suslin theorem for Ore extensions: If $K$ is a division ring and $A:=K[x;σ,δ]$ is an Ore extension, with $σ$ bijective, then every finitely generated projective $A$-module is free. We will show an algorithm that computes the basis of a given finitely generated projective module. The algorithm has been implemented in a computational package, and some illustrative examples are included.
Necessary and sufficient conditions for an Ore extension $S=R[x;\si,\de]$ to be a {\rm PI} ring are given in the case $\si$ is an injective endomorphism of a semiprime ring $R$ satisfying the {\rm ACC} on annihilators. Also, for an arbitrary endomorphism $τ$ of $R$, a characterization of Ore extensions $R[x;τ]$ which are {\rm PI} rings is given, provided the coefficient ring $R$ is noetherian.
In this paper we consider centralizers of single elements in Ore extensions of the ring of polynomials in one variable over a field. We show that they are commutative and finitely generated as an algebra. We also show that for certain classes of elements their centralizer is singly generated as an algebra.
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 $d+2$.
In this paper we investigate extended modules for a special class of Ore extensions. We will assume that $R$ is a ring and $A$ will denote the Ore extension $A:=R[x_1,\dots,x_n;σ]$ for which $σ$ is an automorphism of $R$, $x_ix_j=x_jx_i$ and $x_ir=σ(r)x_i$, for every $1\leq i,j\leq n$. With some extra conditions over the ring $R$, we will prove Vaserstein's, Quillen's patching, Horrocks' and Quillen-Suslin's theorems for this type of non-commutative rings.
A view toward homomorphisms and cv-polynomials between double Ore extensions
本文针对双 Ore 扩张建立同态与 cv-多项式的初步理论，刻画 cv-多项式及其与系数环内导子的关系，并通过 Nakayama 自同构等例子说明同态计算的非平凡性。
In this article, we study Ore extensions of non-unital associative rings. We provide a characterization of simple non-unital differential polynomial rings $R[x;δ]$, under the hypothesis that $R$ is $s$-unital and $\ker(δ)$ contains a nonzero idempotent. This result generalizes a result by Öinert, Richter and Silvestrov from the unital setting. We also present a family of examples of simple non-unital differential polynomial rings.
Let R = D[x;σ;δ] be an Ore extension over a commutative Dedekind domain D, where σis an automorphism on D. In the case δ= 0 Marubayashi et. al. already investigated the class of minimal prime ideals in term of their contraction on the coefficient ring D. In this note we extend this result to a general case δnot 0.
For iterated Ore extensions satisfying a polynomial identity we present an elementary way of erasing derivations. As a consequence we recover some results obtained by Haynal in "PI degree parity in q-skew polynomial rings" (J. Algebra 319, 2008, 4199-4221). We also prove, under mild assumptions on $R_n=R[x_1;\si_1,\de_1]...[x_n;\si_n;\de_n]$ that the Ore extension $R[x_1;\si_1]...[x_n;\si_n]$ exists and is PI if $R_n$ is PI.
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.
Let A be a connected-graded algebra with trivial module k, and let B be a graded Ore extension of A. We relate the structure of the Yoneda algebra E(A) := Ext_A(k,k) to E(B). Cassidy and Shelton have shown that when A satisfies their K_2 property, B will also be K_2. We prove the converse of this result.
Let $R$ be a ring and $(σ,δ)$ a quasi-derivation of $R$. In this paper, we show that if $R$ is an $(σ,δ)$-skew Armendariz ring and satisfies the condition $(\mathcal{C_σ})$, then $R$ is right p.q.-Baer if and only if the Ore extension $R[x;σ,δ]$ is right p.q.-Baer. As a consequence we obtain a generalization of \cite{hong/2000}.
In this paper, we study the representations of the Hopf-Ore extensions $kG(χ^{-1}, a, 0)$ of group algebra $kG$, where $k$ is an algebraically closed field. We classify all finite dimensional simple $kG(χ^{-1}, a, 0)$-modules under the assumption $|χ|=\infty$ and $|χ|=|χ(a)|<\infty$ respectively, and all finite dimensional indecomposable $kG(χ^{-1}, a, 0)$-modules under the assumption that $kG$ is finite dimensional and semisimple, and $|χ|=|χ(a)|$. Moreover, we investigate the decomposition rules for the tensor product modules over $kG(χ^{-1}, a, 0)$ when char$(k)$=0. Finally, we consider the…
Ore meets homothetic extensions
本文推导了将斜导出扩展到同核扩展的条件，并证明了Ore扩展的唯一嵌入性。
We observe \cite[Proposition 4.1]{LaLe} that Poisson polynomial extensions appear as semiclassical limits of a class of Ore extensions. As an application, a Poisson generalized Weyl algebra $A_1$ considered as a Poisson version of the quantum generalized Weyl algebra is constructed and its Poisson structures are studied. In particular, it is obtained a necessary and sufficient condition such that $A_1$ is Poisson simple and established that the Poisson endomorphisms of $A_1$ are Poisson analogues of the endomorphisms of the quantum generalized Weyl algebra.
We introduce hom-associative Ore extensions as non-unital, non-associative Ore extensions with a hom-associative multiplication, and give some necessary and sufficient conditions when such exist. Within this framework, we construct families of hom-associative quantum planes, universal enveloping algebras of a Lie algebra, and Weyl algebras, all being hom-associative generalizations of their classical counterparts, as well as prove that the latter are simple. We also provide a way of embedding any multiplicative hom-associative algebra into a multiplicative, weakly unital hom-associative algebr…
Hilbert's Basis Theorem for generalized nonassociative Ore extensions
本文提出广义非结合Ore扩张，并证明其扩展的希尔伯特基定理，通过欧几里得除法算法建立非交换多项式环的左右诺特性与算法之间的直接联系。
Many rings and algebras arising in quantum mechanics can be interpreted as skew PBW (Poincaré-Birkhoff-Witt) extensions. Indeed, Weyl algebras, enveloping algebras of finite-dimensional Lie algebras (and its quantization), Artamonov quantum polynomials, diffusion algebras, Manin algebra of quantum matrices, among many others, are examples of skew PBW extensions. In this paper we extend the classical Ore and Goldie theorems, known for skew polynomial rings, to this wide class of non-commutative rings. As application, we prove the quantum version of the Gelfand-Kirillov conjecture for the skew q…
Connected Hopf Algebras that are Not Hopf Ore Extensions of Enveloping Algebras
本文构造了一族连通Hopf代数，证明其并非原始部分的通用 enveloping 代数的迭代Hopf Ore扩展，并展示了这些代数可表示为 enveloping 代数的迭代交叉积。
Nakayama automorphisms play an important role in several mathematical branches, which are known to be tough to compute in general. We compute the Nakayama automorphism $ν$ of any 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 $E^G$ is also investigated in term of Zhang's twist, where $G$ is a cyclic group sharing the same order with $σ$.
An algebra $A$ satisfies the Dixmier-Moeglin equivalence if we have the equivalences: $$P~{\rm primitive}\iff P~{\rm rational}\iff P ~{\rm locally~closed~}\qquad~{\rm for}~P\in {\rm Spec}(A).$$ We study the robustness of the Dixmier-Moeglin equivalence under extension of scalars and under the formation of Ore extensions. In particular, we show that the Dixmier-Moeglin equivalence is preserved under base change for finitely generated complex noetherian algebras. We also study Ore extensions of finitely generated complex noetherian algebras $A$. If $T:A\to A$ is either a $\mathbb{C}$-algebra aut…
Hilbert's basis theorem for non-associative and hom-associative Ore extensions
本文证明了Hilbert基定理的同结合版本，涵盖非结合与经典结合情形，发展了同模理论，并给出了非结合与同结合Ore扩张的Noether性实例。
We consider Ore extensions of the form $T:=R[x;σ,δ]$ with $R$ a commutative integral domain that is finitely generated over a field $k$. We show that if $T$ has Gelfand-Kirillov dimension less than four then a prime ideal $P\in {\rm Spec}(T)$ is primitive if and only if $\{P\}$ is locally closed in ${\rm Spec}(T)$, if and only if the Goldie ring of quotients of $T/P$ has centre that is an algebraic extension of $k$. We also show that there are examples for which these equivalences do not all hold for $T$ of integer Gelfand-Kirillov dimension greater than or equal to $4$.
We investigate when a skew polynomial extension T = R[x; σ, δ] of a Hopf algebra R admits a Hopf algebra structure, substantially generalising a theorem of Panov. When this construction is applied iteratively in characteristic 0 one obtains a large family of connected noetherian Hopf algebras of finite Gelfand-Kirillov dimension, including for example all enveloping algebras of finite dimensional solvable Lie algebras and all coordinate rings of unipotent groups. The properties of these Hopf algebras are investigated.
Let $A$ be a Koszul Artin-Schelter regular algebra and $σ$ an algebra homomorphism from $A$ to $M_{2\times 2}(A)$. We compute the Nakayama automorphisms of a trimmed double Ore extension $A_P[y_1, y_2; σ]$ (introduced in \cite{ZZ08}). Using a similar method, we also obtain the Nakayama automorphism of a skew polynomial extension $A[t; θ]$, where $θ$ is a graded algebra automorphism of $A$. These lead to a characterization of the Calabi-Yau property of $A_P[y_1, y_2; σ]$, the skew Laurent extension $A[t^{\pm 1}; θ]$ and $A[y_1^{\pm 1}, y_2^{\pm 1}; σ]$ with $σ$ a diagonal type.
Let $K$ be a field, let $σ$ be an automorphism of $K$, and let $δ$ be a derivation of $K$. We show that if $D$ is one of $K(x;σ)$ or $K(x;δ)$, then $D$ either contains a free algebra over its center on two generators, or every finitely generated subalgebra of $D$ satisfies a polynomial identity. As a corollary, we are able to show that the quotient division ring of any iterated Ore extension of an affine domain satisfying a polynomial identity either again satisfies a polynomial identity or it contains a free algebra over its center on two variables.
Let $B$ be a Poisson algebra $\Bbb C[x_1,\ldots, x_k]$ with Poisson bracket such that $$\{x_j,x_i\}=c_{ji}x_ix_j+p_{ji}$$ for all $j>i$, where $c_{ji}\in\Bbb C$ and $p_{ji}\in\Bbb C[x_1,\ldots,x_i]$. Here we obtain an iterated skew polynomial algebra such that its semiclassical limit is equal to $B$ and the results are illustrated by examples.
Let $σ$ be an endomorphism and $δ$ an $σ$-derivation of a ring $R$. In this paper, we show that if $R$ is $(σ,δ)$-skew Armendariz and $aσ(b)=0$ implies $ab=0$ for $a,b\in R$. Then $R$ is symmetric (respectively, reversible) if and only if $R$ is $σ$-symmetric (respectively, $σ$-reversible) if and only if $R[x;σ,δ]$ is symmetric (respectively, reversible). Moreover, we study on the relationship between the Baerness, quasi-Baerness and p.q.-Baerness of a ring $R$ and these of the Ore extension $R[x;σ,δ]$. As a consequence we obtain a partial generalization of \cite{hong/2000}.
Structure and arithmetic of multivariate Ore extensions
本文研究了多变量 Ore 扩张的结构与算术，引入了伪多线性变换并探讨了其在多项式根集结构中的作用。
A double Ore extension was introduced by James Zhang and Jun Zhang [26] to study a class of Artin-Schelter regular algebras. Here we give a definition of Poisson double extension which may be considered as an analogue of double Ore extension and show that algebras in a class of double Ore extensions are deformation quantizations of Poisson double extensions. We also investigate the modular derivations of Poisson double extensions and the relationship between Poisson double extensions and iterated Poisson polynomial extensions. Results are illustrated by examples.
We show that if $A$ is a finite dimensional associative $H$-module algebra for an arbitrary Hopf algebra $H$, then the proof of the analog of Amitsur's conjecture for $H$-codimensions of $A$ can be reduced to the case when $A$ is $H$-simple. (Here we do not require that the Jacobson radical of $A$ is an $H$-submodule.) As an application, we prove that if $A$ is a finite dimensional associative $H$-module algebra where $H$ is a Hopf algebra $H$ over a field of characteristic $0$ such that $H$ is constructed by an iterated Ore extension of a finite dimensional semisimple Hopf algebra by skew-pri…
On Locally Finite Derivations in Ore Extensions
将van den Essen的多项式代数局部有限导子分类推广到非交换Ore扩张情形，完成三类Ore扩张的局部有限导子分类并证明相关子代数性质。
The aim of this article is to describe necessary and sufficient conditions for simplicity of Ore extension rings, with an emphasis on differential polynomial rings. We show that a differential polynomial ring, R[x;id,δ], is simple if and only if its center is a field and R is δ-simple. When R is commutative we note that the centralizer of R in R[x;σ,δ] is a maximal commutative subring containing R and, in the case when σ=id, we show that it intersects every non-zero ideal of R[x;id,δ] non-trivially. Using this we show that if R is δ-simple and maximal commutative in R[x;id,δ], then R[x;id,δ] i…
We study a class of graded algebras obtained from Ore extensions of graded Calabi-Yau algebras of dimension 2. It is proved that these algebras are graded Calabi-Yau and graded coherent. The superpotentials associated to these graded Calabi-Yau algebras are also constructed.
Panov proved necessary and sufficient conditions to extend the Hopf algebra structure of an algebra $R$ to an Ore extension $R[x;σ,δ]$ with $x$ being a skew-primitive element. In this paper we extend Panov's result to Ore extensions over weak Hopf algebras. As an application we study Ore extensions of connected groupoid algebras.
Let $Λ(f) = K[x][y; f\frac{d}{dx} ]$ be an Ore extension of a polynomial algebra $K[x]$ over an arbitrary field $K$ of characteristic $p>0$ where $f\in K[x]$. For each polynomial $f$, the automorphism group of the algebras $Λ(f)$ is explicitly described. The automorphism group ${\rm Aut}_K(Ł(f))=S\rtimes G_f$ is a semidirect product of two explicit groups where $G_f$ is the {\em eigengroup} of the polynomial $f$ (the set of all automorphisms of $K[x]$ such that $f$ is their common eigenvector). For each polynomial $f$, the eigengroup $G_f$ is explicitly described. It is proven that every subgr…
For a derivation d of a commutative Noetherian complex algebra A, a homeomorphism is established between the prime spectrum of the Ore extension A[z;d] and the Poisson prime spectrum of the polynomial algebra A[z] endowed with the Poisson bracket such that {A,A}=0 and {z,a}=d(a) for all a in A. Several illustrative examples are discussed including some where A is known to be d-simple.
Let $Ł(f) = K[x][y; f\frac{d}{dx} ]$ be an Ore extension of a polynomial algebra $K[x]$ over a field $K$ of characteristic zero where $f\in K[x]$. For a given polynomial $f$, the automorphism group of the algebra $Ł(f) $ is explicitly described. The polynomial case $Ł(0) = K[x,y]$ and the case of the Weyl algebra $A_1= K[x][y; \frac{d}{dx} ]$ were done done by Jung (1942) and van der Kulk (1953), and Dixmier (1968), respectively. In 1997, Alev and Dumas proved that the algebras $Ł(f)$ and $Ł(g)$ are isomorphic iff $g(x) = łf(αx+β)$ for some $ł, α\in K\backslash \{ 0\}$ and $β\in K$. In 2015, B…
Suppose that $E=A[x;σ,δ]$ is an Ore extension with $σ$ an automorphism. It is proved that if $A$ is twisted Calabi-Yau of dimension $d$, then $E$ is twisted Calabi-Yau of dimension $d+1$. The relation between their Nakayama automorphisms is also studied. As an application, the Nakayama automorphisms of a class of 5-dimensional Artin-Schelter regular algebras are given explicitly.
Hilbert's basis theorem for Poisson Ore extensions
本文证明了Poisson Ore扩张和Poisson Laurent Ore扩张的Hilbert基定理，并推广到迭代情形，给出非Noetherian但Poisson-Noetherian的例子。
The rank condition and strong rank conditions for Ore extensions
本文研究Ore扩张的秩条件及强秩条件，证明环R满足秩条件当且仅当Ore扩张R[x;σ,δ]满足，同时探讨左右强秩条件的性质，并提供Susan Montgomery关于斜幂级数环有限性的新证明。
A note on explicit homological invariants of graded double Ore extensions
本文计算了量子平面的格雷德双奥尔扩展的显式同调不变量，确定了平凡右模的最小格雷德自由解析和格雷德贝蒂数，并计算了两个自然循环右模的线性解析。
On homomorphisms and cv-polynomials between iterated Ore extensions
本文研究迭代 Ore 扩张中的同态与 cv-多项式，证明已有结果的推广具有非平凡性，并以多个非交换代数加以说明。
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.
Noncommutative coding theory and algebraic sets for skew PBW extensions
本文综述了单个Ore扩张在非交换编码理论中的代数工具与性质，并创新性地将代数集研究拓展到斜PBW扩张这一更广泛的非交换多项式环类。
In this paper, we study pointed rank one Hopf algebras and Hopf-Ore extensions of group algebras, over an arbitrary field $k$. It is proved that the rank of a Hopf-Ore extension of a group algebra is one or two or infinite. It is also shown that an arbitrary (finite or infinite dimensional) pointed Hopf algebra of rank one is isomorphic to a quotient of a Hopf-Ore extension of its coradical, a group algebra. We classify the finite dimensional simple modules and describe a family of indecomposable modules over a Hopf-Ore extension $H=kG(χ, a,δ)$ and its quotient $H'$ of rank one, where $χ(a)\ne…
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