双重Ore扩张与迭代Ore扩张

迭代Ore多项式与斜Reed-Muller码的评估

Evaluation of iterated Ore polynomials and skew Reed-Muller codes

本文通过比较两种评估方式，利用其构造Reed-Muller码并计算相关数据。

Double Ore extensions versus graded skew PBW extensions

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 $d+2$.

迭代 Ore 扩张之间的同态与 cv-多项式

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。

Iterated Hopf Ore extensions in positive characteristic

Iterated Hopf Ore extensions (IHOEs) over an algebraically closed base field k of positive characteristic p are studied. We show that every IHOE over k satisfies a polynomial identity, with PI-degree a power of p, and that it is a filtered deformation of a commutative polynomial ring. We classify all 2-step IHOEs over k, thus generalising the classification of 2-dimensional connected unipotent algebraic groups over k. Further properties of 2-step IHOEs are described: for example their simple modules are classified, and every 2-step IHOE is shown to possess a large Hopf center and hence an anal…

A classification of relation types of Ore extensions of dimension 5

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.

群环上的迭代Hopf Ore扩张

Iterated Hopf Ore Extensions over Group Rings

引入并研究群代数上的两步Ore扩张构造的一类Hopf代数，统一推广了广义Taft代数和与sl2相关的Hopf代数，分析了环论性质并分类了有限维单模。

Poisson Ore扩张的Hilbert基定理

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。

Ore扩张的商环的自由子代数

Free subalgebras of quotient rings of Ore extensions

令$K$为字段，令$σ$为$K$的自同构，令$δ$为$K$的导出。证明了如果$D$是$K(x;σ)$或$K(x;δ)$中的一个，则$D$要么在其中心上包含一个关于两个生成元的自由代数，要么$D$的每个生成子代数都满足多项式恒等式.作为推论，我们能够证明，商除环的任何迭代的Ore扩张的仿射域满足一个多项式单位再次满足一个多项式单位或它包含一个自由代数在其中心的两个变量。

Double Poisson extensions

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.

双 Ore 扩张之间同态与 cv-多项式的展望

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。

Actions of Ore extensions and growth of polynomial $H$-identities

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…

Ore Extensions and Infinite Triangularization

We give infinite triangularization and strict triangularization results for algebras of operators on infinite dimensional vector spaces. We introduce a class of algebras we call Ore-solvable algebras: these are similar to iterated Ore extensions but need not be free as modules over the intermediate subrings. Ore-solvable algebras include many examples as particular cases, such as group algebras of polycyclic groups or finite solvable groups, enveloping algebras of solvable Lie algebras, quantum planes and quantum matrices. We prove both triangularization and strict triangularization results fo…

一般李色代数

Generic Lie Color Algebras

本文描述了一类李色代数，我们称之为泛色李色代数，它的泛包络代数是一个具有有限整体维数的整环。此外，它是一个迭代的Ore扩展。给出了Grobner基方法的一个应用，并证明了该方法可用于研究一般李色代数的因子的泛包络代数.

From quantum Ore extensions to quantum tori via noncommutative UFDs

All iterated skew polynomial extensions arising from quantized universal enveloping algebras of Kac-Moody algebras are special examples of a very large, axiomatically defined class of algebras, called CGL extensions. For the purposes of constructing initial clusters for quantum cluster algebra structures on an algebra R, and classification of the automorphisms of R, one needs embeddings of R into quantum tori T which have the property that R contains the corresponding quantum affine space algebra A. We explicitly construct such an embedding A \subseteq R \subset T for each CGL extension R usin…