Armando Reyes, Héctor Suárez
The aim of this paper is to establish necessary and sufficient algorithmic conditions to guarantee that an algebra is actually a 3-dimensional skew polynomial algebra in the sense of Bell and Smith.
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In this paper, we provide a new and more general filtration to the family of noncommutative rings known as skew PBW extensions. We introduce the notion of $σ$-filtered skew PBW extension and study some homological properties of these algebras. We show that the homogenization of a $σ$-filtered skew PBW extension $A$ over a ring $R$ is a graded skew PBW extension over the homogenization of $R$. Using this fact, we prove that if the homogenization of $R$ is Auslander-regular, then the homogenization of $A$ is a domain Noetherian, Artin-Schelter regular, and $A$ is Noetherian, Zariski and (ungrade…
The aim of this paper is to develop the theory of skew Armendariz and quasi-Armendariz modules over skew PBW extensions. We generalize the results of several works in the literature concerning Ore extensions to another non-commutative rings which can not be expressed as iterated Ore extensions. As a consequence of our treatment, we extend and unify different results about the Armendariz, Baer, p.p., and p.q.-Baer properties for Ore extensions and skew PBW extensions.
Graded quasi-commutative skew PBW extensions are isomorphic to graded iterated Ore extensions of endomorphism type, whence graded quasi-commutative skew PBW extensions with coefficients in AS-regular algebras are skew Calabi-Yau and the Nakayama automorphism exists for these extensions. With this in mind, in this paper we give a description of Nakayama automorphism for these non-commutative algebras using the Nakayama automorphism of the ring of the coefficients.
In this paper we present the notion of skew $Π$-Armendariz for the non-commutative rings known as $σ$-PBW extensions. This concept generalizes several definitions of Armendariz rings presented in the literature for these extensions, and in particular, for Ore extensions of injective type. We investigate the relations between skew $Π$-Armendariz, $Σ$-rigid, $(Σ,Δ)$-skew Armendariz and $Σ$-skew Armendariz rings rings.
In this paper we study skew Poincaré-Birkhoff-Witt extensions over weak symmetric and $(Σ,Δ)$-weak symmetry rings. Since these extensions generalize Ore extensions of injective type and another noncommutative rings of polynomial type, we unify and extend several results in the literature concerning the property of being symmetry. Under adequate conditions, we transfer the property of being weak symmetric or $(Σ,Δ)$-weak symmetric from a ring of coefficients to a skew PBW extension over this ring. We illustrate our results with remarkable examples of algebras appearing in noncommutative algebra…
Koszul and homogeneous Koszul algebras were defined by Priddy in \cite{Priddy1970}. There exist some relations between these algebras and the skew PBW extensions introduced in \cite{LezamaGallego}. In this paper we give conditions to guarantee that skew PBW extensions over fields are Koszul or homogeneous Koszul. We also show that a constant skew PBW extension of a field is a PBW deformation of its homogeneous version.
On the uniform dimension and the associated primes of skew PBW extensions
本文研究了斜 PBW 扩张上诱导模的一致维数与结合素理想。
We establish necessary or sufficient conditions to guarantee that skew Poincaré-Birkhoff-Witt extensions are NI or NJ rings. Our results extend those corresponding for skew polynomial rings and establish similar properties for other families of noncommutative rings such as universal enveloping algebras and examples of differential operators.
In this paper, we introduce the concept of $Σ$-semicommutative ring, for $Σ$ a finite family of endomorphisms of a ring $R$. We relate this class of rings with other classes of rings such that Abelian, reduced, $Σ$-rigid, nil-reversible and rings satisfying the $Σ$-skew reflexive nilpotent property. Also, we study some ring-theoretical properties of skew PBW extensions over $Σ$-semicommutative rings. We prove that if a ring $R$ is $Σ$-semicommutative with certain conditions of compatibility on derivations, then for every skew PBW extension $A$ over $R$, $R$ is Baer if and only if $R$ is quasi-…
On the annihilator condition for skew PBW extensions
本文研究了斜 PBW 扩张的零化子条件，证明了斜多项式环的相关结果同样适用，并指出零化子行为由扩张的定义关系决定，推广了已有结论。
We examine PBW deformations of finite group extensions of skew polynomial rings, in particular the quantum Drinfeld orbifold algebras defined by the first author. We give a homological interpretation, in terms of Gerstenhaber brackets, of the necessary and sufficient conditions on parameter functions to define a quantum Drinfeld orbifold algebra, thus clarifying the conditions. In case the acting group is trivial, we determine conditions under which such a PBW deformation is a generalized enveloping algebra of a color Lie algebra; our PBW deformations include these algebras as a special case.
Smooth geometry of skew PBW extensions over commutative polynomial rings I
研究交换多项式环上斜PBW扩张的微分光滑性，针对一个和两个不定元情形。
We investigate the notions of weak annihilator and nilpotent associated prime defined by Ouyang and Birkenmeier \cite{OuyangBirkenmeier2012} in the setting of noncommutative rings having PBW bases. We extend several results formulated in the literature concerning annihilators and associated primes of commutative rings and skew polynomial rings to a more general setting of algebras not considered before. We exemplify our results with families of algebras appearing in the theory of enveloping algebras, differential operators on noncommutative spaces, noncommutative algebraic geometry, and theore…
We give a necessary and sufficient PBW basis criterion for Hopf algebras generated by skew-primitive elements and abelian group of group-like elements with action given via characters. This class of pointed Hopf algebras has shown great importance in the classification theory and can be seen as generalized quantum groups. We apply the criterion to classical examples and liftings of Nichols algebras which were determined in arxiv.org:1003.5882.
A first approach to the Burchnall-Chaundy theory for quadratic algebras having PBW bases
该研究面向具有 PBW 基的二次代数，初步探讨由二次关系生成的高阶斜 Ore 多项式的 Burchnall-Chaundy 理论。
Combinatorial aspects of normal ordering of 3-dimensional skew polynomial rings
本文研究了三维非交换多项式环的正常排列组合特性，利用SageMath软件优化了PBW形式和交换规则中的正常排列计算。
We construct finite-dimensional pointed Hopf algebras \mathfrak u_{r,s}(G_2) (i.e. restricted 2-parameter quantum groups) from the 2-parameter quantum group U_{r,s}(G_2) defined in \cite{HS}, which turn out to be of Drinfel'd doubles, where a crucial point is to give a detailed combinatorial construction of the convex PBW-type Lyndon basis for type G_2 in 2-parameter quantum version. After furnishing possible commutation relations among quantum root vectors, we show that the restricted quantum groups are ribbon Hopf algebras under certain conditions through determining their left and right int…
Construction of PBW and canonical bases in the folding extension algebras for symmetrizable types
研究源于Lusztig构造的扩张代数，扩展Varagnolo和Vasserot方法构造斜群代数，证明相关过渡矩阵性质，为对称型情形下PBW基与典范基及其关系提供几何范畴化。
In this paper, our objects of interest are Hopf Galois extensions (e.g., Hopf algebras, Galois field extensions, strongly graded algebras, crossed products, principal bundles, etc.) and families of noncommutative rings (e.g., skew polynomial rings, PBW extensions and skew PBW extensions, etc.). We collect and systematize questions, problems, properties and recent advances in both theories by explicitly developing examples and doing calculations that are usually omitted in the literature. In particular, for Hopf Galois extensions we consider approaches from the point of view of quantum torsors …
The crystals for finite dimensional representations of sl(n+1) can be realized using Young tableaux. The infinity crystal on the other hand is naturally realized using multisegments, and there is a simple description of the embedding of each finite crystal into the infinity crystal in terms of these realizations. The infinity crystal is also parameterized by Lusztig's PBW basis with respect to any reduced expression for the longest word in the Weyl group. We give an explicit description of the unique crystal isomorphism from PBW bases to multisegments for one standard choice of reduced express…
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…
We show explicitly a generalised Lie algebra embedded in the positive and negative parts of the Drinfeld-Jimbo quantum groups of type A_n. Such a generalised Lie algebra satisfy axioms closely related to the ones found by S.L. Woronowicz. For the universal enveloping algebra of such generalised Lie algebras we establish several conditions in order to obtain bases of type Poincaré-Birkhoff-Witt. Besides a graded algebra is proposed and some relations with the quantum Yang-Baxter equation are studied.
We determine convex PBW-type Lyndon bases for two-parameter quantum groups $U_{r,s}(F_4)$ with detailed commutation relations. We construct a finite-dimensional Hopf algebra $\mathfrak u_{r,s}(F_4)$, as a quotient of $U_{r,s}(F_4)$ by a Hopf ideal generated by certain central elements, which is pointed, and of a Drinfel'd double structure under a certain condition. All of Hopf isomorphisms of $\mathfrak u_{r,s}(F_4)$ are determined which are important for seeking the possible new pointed objects in low order with $(\ell, 210)\ne 1$. Finally, necessary and sufficient conditions for $\mathfrak u…
PBW bases for $\imath$quantum groups
本文利用相对辫子群对称性为任意有限型 $\imath$ 量子群建立 PBW 型基，提供秩 1 类型的根向量公式，并证明其生成修正 $\imath$ 量子群的积分基，首项与经典 PBW 基一致。
PBW bases for modified quantum groups
本文构造了有限型修正量子群的PBW基，证明其关于自然双线性型正交且典范基的PBW展开矩阵为单位三角矩阵，并基于最低权与最高权模张量积中元素序列的极限给出了任意型修正量子群的新构造。
PBW theory for Bosonic extensions of quantum groups
本文为任意有限型量子群的玻色子扩张建立PBW理论，引入不变双线性型与伴随算子，构造具有PBW根向量和正交PBW基的子代数，推广量子幺正坐标环。
PBW bases and marginally large tableaux in type D
本文刻画D型$B(\infty)$的边缘大表实现与基于最长词良好约化表达式的PBW单项式实现之间的唯一晶体同构。
We reduce the basis construction problem for Hopf algebras generated by skew-primitive semi-invariants to a study of special elements, called ``super-letters,'' which are defined by Shirshov standard words. In this way we show that above Hopf algebras always have sets of PBW-generators (``hard'' super-letters). It is shown also that these Hopf algebras having not more than finitely many ``hard'' super-letters share some of the properties of universal enveloping algebras of finite-dimensional Lie algebras. The background for the proofs is the construction of a filtration such that the associate…
We obtain a presentation by generators and relations of any Nichols algebra of diagonal type with finite root system. We prove that the defining ideal is finitely generated. The proof is based in Kharchenko's theory of PBW basis of Lyndon words. We prove that the lexicographic order on Lyndon words is convex for such PBW generators and so the PBW basis is orthogonal with respect to the canonical non-degenerate form associated to the Nichols algebra.
PBW bases and marginally large tableaux in types B and C
本文研究B型和C型中Kashiwara无穷晶体的两种组合实现，明确给出边缘大表与PBW基导出的Kostant分拆之间的同构，并引入栈记号简化Kostant分拆表示。
Recently, Kuniba, Okado and Yamada proved that the transition matrix of PBW-type bases of the positive-half of a quantized universal enveloping algebra $U_q(\mathfrak{g})$ coincides with a matrix coefficients of the intertwiner between certain irreducible modules over the corresponding quantized coordinate ring $A_q(\mathfrak{g})$, introduced by Soibelman. In the present article, we give a new proof of their result, by using representation theory of the $q$-boson algebra, and the Drinfeld paring of $U_q(\mathfrak{g})$.
Recently Pascal Baseilhac and Stefan Kolb obtained a PBW basis for the $q$-Onsager algebra $\mathcal O_q$. They defined the PBW basis elements recursively, and it is obscure how to express them in closed form. To mitigate the difficulty, we bring in the universal Askey-Wilson algebra $Δ_q$. There is a natural algebra homomorphism $\natural \colon \mathcal O_q \to Δ_q$. We apply $\natural $ to the above PBW basis, and express the images in closed form. Our results make heavy use of the Chebyshev polynomials of the second kind.
In this paper, we give a recursive formula for the interesting PBW basis $E_{A}$ of composition subalgebras of Ringel-Hall algebras $\fkH_\vartri(n)$ of cyclic quivers after \cite{DengDuXiao2007generic}, and another construction of canonical bases of $\U_v^+(\wih{\fks\fkl}_n)$ from the monomial bases $m^{(A)}$ follow \cite{DuZhaomultiplication}. As an application, we will determined all the canonical basis of $\U_v^+(\wih{\fks\fkl}_2)$ associated with modules of Lowery length $\leqs3$. Finally, we will discuss the relation of canonical bases of Ringel-Hall algebras and those of affine quantum …
This note has two purposes. First we establish that the map defined in [L, $§40.2.5$ (a)] is an isomorphism for certain admissible sequences. Second we show the map gives rise to a convex basis of Poincaré--Birkhoff--Witt (PBW) type for $\bup$, an affine untwisted quantized enveloping algebra of Drinfel$'$d and Jimbo. The computations in this paper are made possible by extending the usual braid group action by certain outer automorphisms of the algebra.
We prove the existence of a basis of Poincare-Birkhoff-Witt type for braided Hopf algebras R generated by a braided subspace V of P(R) if the braiding on V fulfils a triangularity condition. We apply our result to pointed Hopf algebras with abelian coradical and to Nichols algebras of low dimensional simple U_q(sl_2) modules.
Around 1990 Soibelman constructed certain irreducible modules over the quantized coordinate algebra. A. Kuniba, M. Okado, Y. Yamada recently found that the relation among natural bases of Soibelman's irreducible module can be described using the relation among the PBW-type bases of the positive part of the quantized enveloping algebra, and proved this fact using case-by-case analysis in rank two cases. In this paper we will give a realization of Soibelman's module as an induced module, and give a unified proof of the above result of Kuniba, Okado, Yamada. We also verify Conjecture 1 of Kuniba,…
A uniform approach to the Damiani, Beck, and alternating PBW bases for the positive part of $U_q(\widehat{\mathfrak{sl}}_2)$
本文研究 $U_q(\widehat{\mathfrak{sl}}_2)$ 正部分的三种 PBW 基，引入含整数参数 $m$ 的指数生成函数统一其指数公式，并给出该函数的分解与系数的封闭形式表达。
Monomial bases and canonical bases for quantum affine algebras
本文研究了量子仿射代数中单项基与规范基的构造方法，提出了一种基于PBW基的简单算法来计算规范基，并探讨了由此得到的规范基与Lusztig利用有向图几何构造的规范基之间的关系。
Mirkovic-Vilonen (MV) polytopes have proven to be a useful tool in understanding and unifying many constructions of crystals for finite-type Kac-Moody algebras. These polytopes arise naturally in many places, including the affine Grassmannian, pre-projective algebras, PBW bases, and KLR algebras. There has recently been progress in extending this theory to the affine Kac-Moody algebras. A definition of MV polytopes in symmetric affine cases has been proposed using pre-projective algebras. In the rank-2 affine cases, a combinatorial definition has also been proposed. Additionally, the theory of…
We construct convex PBW-type Lyndon bases for two-parameter quantum groups U_{r,s}({so}_{2n+1}) with detailed commutation relations. It turns out that under a certain condition, the restricted two-parameter quantum group u_{r,s}({so}_{2n+1}) (r, s are roots of unity) is of Drinfeld double. All of Hopf isomorphisms of u_{r,s}({so}_{2n+1}), as well as u_{r,s}({sl}_n) are determined. Finally, necessary and sufficient conditions for u_{r,s}({so}_{2n+1}) to be a ribbon Hopf algebra are singled out by describing the left and right integrals.
Let G be either of Mat(n), GL(n) or SL(n), let O_q(G) be the quantum function algebra - over Z[q,q^{-1}] - associated to G, and let O_e(G) be the specialisation of O_q(G) at a root of unity, of odd order l. Then O_e(G) is a module over the corresponding classical function algebra O(G) via the quantum Frobenius morphism, which embeds O(G) as a central subbialgebra of O_e(G). In this note we prove a PBW-like theorem for O_q(G) - more or less known in literature, but not in this form (to the best of the author's knowledge) - and we show that it yields explicit bases of O_e(G) over O(G) when G is …
An integral PBW-basis of type $A_1^{(1)}$ has been constructed by Zhang [Z] and Chen [C] using the Auslander-Reiten quiver of the Kronecker quiver. We associate a geometric order to elements in this basis following an idea of Lusztig [L1] in the case of finite type. This leads to an algebraic realization of a bar-invariant basis of $\uq2$. For any affine symmetric type, we obtain an integral PBW-basis of the generic composition algebra, by using an algebraic construction of the integral basis for a tube in [DDX], an embedding of the module category of the Kronecker quiver into the module categ…
Quotient Hopf algebras of the free bialgebra with PBW bases and GK-dimensions
本文研究由二维余代数生成的自由双代数T及其商，揭示其与非交换Faà di Bruno双代数的关系，并在正特征下构造具有有限GK维数的商Hopf代数链，分析其同调性质与余根滤过。
We introduce an odd double affine Hecke algebra (DaHa) generated by a classical Weyl group W and two skew-polynomial subalgebras of anticommuting generators. This algebra is shown to be Morita equivalent to another new DaHa which are generated by W and two polynomial-Clifford subalgebras. There is yet a third algebra containing a spin Weyl group algebra which is Morita (super)equivalent to the above two algebras. We establish the PBW properties and construct Verma-type representations via Dunkl operators for these algebras.
PBW deformations of a Fomin-Kirillov algebra and other examples
本文研究了与 Hopf 代数理论相关的分次代数的 PBW 形变，以 Fomin-Kirillov 代数 FK3 和 García Iglesias-Vay 的例子为对象，确定了形变的半单性条件，并构造了相应的 PBW 基和多项式恒等式。
We generalize Lusztig's geometric construction of the PBW bases of finite quantum groups of type $\mathsf{ADE}$ under the framework of [Varagnolo-Vasserot, J. reine angew. Math. 659 (2011)]. In particular, every PBW basis of such quantum groups is proven to yield a semi-orthogonal collection in the module category of the KLR-algebras. This enables us to prove Lusztig's conjecture on the positivity of the canonical (lower global) bases in terms of the (lower) PBW bases in the $\mathsf{ADE}$ case. In addition, we verify Kashiwara's problem on the finiteness of the global dimensions of the KLR-al…
Let ${\mathbf U}^-_q$ be the negative part of the quantum enveloping algebra associated to a simply laced Kac-Moody Lie algebra ${\mathfrak g}$, and $\underline{\mathbf U}^-_q$ the algebra corresponding to the fixed point subalgebra of ${\mathfrak g}$ obtained from a diagram automorphism $σ$ on ${\mathfrak g}$. Let ${\mathbf B}^σ$ be the set of $σ$-fixed elements in the canonical basis of ${\mathbf U}_q^-$, and $\underline{\mathbf B}$ the canonical basis of $\underline{\mathbf U}_q^-$. Lusztig proved that there exists a canonical bijection ${\mathbf B}^σ \simeq \underline{\mathbf B}$ based on …
PBW bases and centralisers for the $q$-Onsager algebra
该研究解决了q - 昂萨格代数PBW基与中心化子的五个猜想，通过证明特定根向量给出PBW基、建立多个PBW基及确定相关子代数中心化子，运用多种论证方法得出四个单族交替多项式子代数是极大交换的结论。
Let $G$ be a connected simply-connected simple complex algebraic group and $\mathfrak{g}$ the corresponding simple Lie algebra. In the first half of the present paper, we study the relation between the positive part $U_q(\mathfrak{n^+})$ of the quantized enveloping algebra $U_q(\mathfrak{g})$ and the specific irreducible representations of the quantized function algebra $\mathbb{Q}_q[G]$, taking into account the right $U_q(\mathfrak{g})$-algebra structure of $\mathbb{Q}_q[G]$. This work is motivated by Kuniba, Okado and Yamada's result together with Tanisaki and Saito's results. In the latter …
Combinatorial descriptions of the crystal structure on certain PBW bases
针对 PBW 基实现 B(∞) 晶体时算子算法复杂的问题，本文刻画了一类约化表达式，使晶体算子可由更简单的括号规则描述，并证明其在多数类型中存在。
For a finite-dimensional simple Lie algebra $\mathfrak{g}$, let $U^+_q(\mathfrak{g})$ be the positive part of the quantized universal enveloping algebra, and $A_q(\mathfrak{g})$ be the quantized algebra of functions. We show that the transition matrix of the PBW bases of $U^+_q(\mathfrak{g})$ coincides with the intertwiner between the irreducible $A_q(\mathfrak{g})$-modules labeled by two different reduced expressions of the longest element of the Weyl group of $\mathfrak{g}$. This generalizes the earlier result by Sergeev on $A_2$ related to the tetrahedron equation and endows a new represent…
In this paper, we study transition matrices of PBW bases of the nilpotent subalgebra of quantum superalgebras associated with all possible Dynkin diagrams of type A and B in the case of rank 2 and 3, and examine relationships with three-dimensional (3D) integrability. We obtain new solutions to the Zamolodchikov tetrahedron equation via type A and the 3D reflection equation via type B, where the latter equation was proposed by Isaev and Kulish as a 3D analog of the reflection equation of Cherednik. As a by-product of our approach, the Bazhanov-Sergeev solution to the Zamolodchikov tetrahedron …
The structure of iterated Hopf Ore extensions
本文研究特征零域上的迭代Hopf Ore扩张，证明其结构性质、PBW生成系继承定理，开发枚举其非平凡单侧余理想子代数的算法，并分类三维非余交换连通Hopf代数的非平凡右余理想子代数。
PBW bases of irreducible Ising modules
本文研究了伊辛模型中三种不可约模的PBW基，给出了显式单子基及与矩阵(8 3;3 2)相关的精炼特征函数公式。
Geometric realization of affine bases: the Kronecker quiver case
本文从几何角度研究Kronecker箭图量子包络代数负部分中PBW基与标准基之间的转移矩阵，通过旗层复形构造PBW基元素的几何实现，并证明转移系数由交上同调复形限制到小子层的局部系统重数决定。
We provide $\mathbb{N}$-filtrations on the negative part $U_q(\mathfrak{n}^-)$ of the quantum group associated to a finite-dimensional simple Lie algebra $\mathfrak{g}$, such that the associated graded algebra is a skew-polynomial algebra on $\mathfrak{n}^-$. The filtration is obtained by assigning degrees to Lusztig's quantum PBW root vectors. The possible degrees can be described as lattice points in certain polyhedral cones. In the classical limit, such a degree induces an $\mathbb{N}$-filtration on any finite dimensional simple $\mathfrak{g}$-module. We prove for type $\tt{A}_n$, $\tt{C}_n…
On SAGBI bases theory of skew Poincaré-Birkhoff-Witt extensions
本文首次探讨了斜 Poincaré-Birkhoff-Witt 扩展的 SAGBI 基理论，并研究了这些扩展子代数的 SAGBI 基的多项式复合问题。
We give a systematic description of many monomial bases for a given quantized enveloping algebra and of many integral monomial bases for the associated Lusztig $\mathbb Z[v,v^{-1}]$-form. The relations between monomial bases, PBW bases and canonical bases are also discussed.
Finite Gröbner-Shirshov bases for double extension regular algebras of type (14641)
本文用Matlab算法计算了(14641)型双重扩张正则代数的Gröbner-Shirshov基，并证明其具有PBW基。
The $q$-Onsager algebra $O_q$ is presented by two generators $W_0$, $W_1$ and two relations, called the $q$-Dolan/Grady relations. Recently Baseilhac and Koizumi introduced a current algebra $\mathcal A_q$ for $O_q$. Soon afterwards, Baseilhac and Shigechi gave a presentation of $\mathcal A_q$ by generators and relations. We show that these generators give a PBW basis for $\mathcal A_q$. Using this PBW basis, we show that the algebra $\mathcal A_q$ is isomorphic to $O_q \otimes \mathbb F \lbrack z_1, z_2, \ldots \rbrack$, where $\mathbb F$ is the ground field and $\lbrace z_n \rbrace_{n=1}^\in…
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