Generalizing the notion of Koszul algebra

We introduce a generalization of the notion of a Koszul algebra, which includes graded algebras with relations in different degrees, and we establish some of the basic properties of these algebras. This class is closed under twists, twisted tensor products, regular central extensions and Ore extensions. We explore the monomial algebras in this class and we include some well-known examples of algebras that fall into this class.     

Double ore extensions

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.     

Ore Extensions of Multiplier Hopf Algebras

The goal of this article is to generalize the theory of Hopf–Ore extensions on Hopf algebras to multiplier Hopf algebras. First the concept of a Hopf–Ore extension of a multiplier Hopf algebra is introduced. We give a necessary and sufficient condition for Ore extensions to become a multiplier Hopf algebra. Finally, *-structures are constructed on Hopf–Ore extensions, and certain isomorphisms between Hopf–Ore extensions are discussed.     

Embedding a quantum rank three quadric in a quantum P3

We continue the classification, begun in [11], [14] and [12], of quadratic Artin-Schelter regular algebras of global dimension 4 which map onto a twisted homogeneous coordinate ring of a quadric hypersurfcice in P³. In this paper, we consider those cases where the quadric has rank 3. We also give sufficient conditions for the point scheme of any quadratic regular algebra of global dimension 4 to be the graph of an automorphism.     

Hopf代数的二重Ore扩张

本文的目的 是定义Hopf二重Ore扩张,讨论这种扩张的基本性质并研究Hopf代数的分次与Hopf二重Ore扩张之间的关系.作者还研究了连通分次Hopf代数的结构及其Hopf二重Ore扩张的同调性质.     

张量代数上一种带辫子的Poisson结构

对应结合代数的R-冲积构造,考虑了相应半古典极限的Poisson结构构造.进而给出了张量代数上一种带辫子的Poisson结构,该结果推广了Poisson多项式环和双Poisson-Ore扩张.     

拟三角Hopf代数的Ore -扩张

该文主要考虑了拟三角Hopf代数的某种Ore -扩张问题. 对拟三角Hopf代数的Ore -扩张何时保持相同的拟三角结构给出了充分必要条件. 最后作为应用, 文章讨论了Sweedler Hopf代数和Lusztig小量子群的Ore -扩张结构.     

Quiver Algebras of Coverings and Twisted Tensor Products

Given a covering Γ of a quiver Δ, we show that the quiver algebra K[Γ] of Γ over a field K is a twisted tensor product of the quiver algebra of the fibre of the covering viewed as a trivial quiver and the quiver algebra K[Δ]. To make sense of this, we first extend the theory of twisted tensor products of algebras to include algebras without units.     

Classification of relation types of Ore extensions of dimension 5

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.     

L-R-Twisted Tensor Products of Nonlocal Vertex Algebras and Their Modules

In this article, we introduce and study a common generalization of the twisted tensor product construction of nonlocal vertex algebras and their modules. We investigate some properties of this new construction; for instance, we give the relations between L-R-twisted tensor product nonlocal vertex algebras and twisted tensor product vertex algebras. Furthermore, we find the conditions for constructing an iterated L-R-twisted tensor product nonlocal vertex algebra and its module.     

Enveloping algebras of double Poisson-Ore extensions

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.     

Twisted hopf comodule algebras

For k a commutative ring, H a k‐bialgebra and A a right H‐comodule k‐algebra, we define a new multiplication on the H‐comodule A to obtain a twisted algebra” A^T, T sumHom(H,End (A)). If ^T is convolution invertible, the categories of relative right Hopf modules over A and A^Tare isomorphic. Similarly a convolution invertible left twisting gives an isomorphism of the categories of relative left Hopf modules. We show that crossed products are invertible twistings of the tensor product, and obtain, as a corollary, a duality theorem for crossed products     

Bitwistor and Quasitriangular Structures of Bialgebras

In this article, we first introduce the notion of a bitwistor and discuss conditions under which such bitwistor forms a bialgebra as a generalization of the well-known Radford's biproduct. Then, in order to obtain new quasitriangular bialgebras, we consider a construction called twisted tensor biproduct, which is a special case of bitwistor bialgebra, and give a necessary and sufficient condition for such twisted tensor biproduct to admit quasitriangular structures.     

Module twistors for modules of nonlocal vertex algebras

We introduce the notion of module twistor for a module of a nonlocal vertex algebra. The aim of this paper is to use this concept to unify some deformed constructions of modules of nonlocal vertex algebras, such as twisted tensor products and iterated twisted tensor products of modules of nonlocal vertex algebras.     

关于二重Ore扩张的一点注释（英文）

The aim of this article is to summarize the relationship between double Ore extensions and iterated Ore extensions, and mainly describe the lifting of properties from an algebra A to a(right) double Ore extension B of A which can not be presented as iterated Ore extensions.     

Gelfand–Kirillov dimension of generalized Weyl algebras (In memory of Guenter Rudolf Krause (1941–2015))

Given a generalized Weyl algebra A of degree 1 with the base algebra D, we prove that the difference of the Gelfand–Kirillov dimension of A and that of D could be any positive integer or infinity. Under mild conditions, this difference is exactly 1. As applications, we calculate the Gelfand–Kirillov dimensions of various algebras of interest, including the (quantized) Weyl algebras, ambiskew polynomial rings, noetherian (generalized) down-up algebras, iterated Ore extensions, quantum Heisenberg algebras, universal enveloping algebras of Lie algebras, quantum GWAs, etc.     

Hochschild Homology of Twisted Tensor Products

We compute the Hochschild homology of some twisted tensor products of algebras, which are a natural generalization of the Ore extensions. We apply our result to the ring D _Q,P(X,/X) of differential operators of the multiparametric affine space, the ring of coordinates of the quantum symplectic 2v-dimensional space and the ring of coordinates of the quantum 2v-dimensional Euclidean space.     

Four dimensional regular algebras with point scheme,anonsingular quadric in P3

In [22], a class of four-dimensional, quadratic, Artin-Schelter regular algebras was introduced, whose point scheme is the graph of an automorphism of a nonsingular quadric in P³. These algebras are the first examples of quadratic Artin-Schelter regular algebras whose defining relations are not determined by the point scheme and, hence, not determined by the algebraic data obtained from the point modules. In this paper, we study these algebras via their line modules. In particular, the set of lines in P³ that correspond to left line modules is not the set of lines in P³ that correspond to right line modules. Our analysis focuses on a distinguished member R _λ of this class of algebras, where R _λ is a twist by a twisting system of the other algebras. We prove that R _λ is a finite module over its center and that its central Proj is a smooth quadric inP⁴.     

Poisson double extensions

A double Ore extension was introduced by Zhang and Zhang(2008) to study a class of ArtinShelter regular algebras. Here we give a definition of Poisson double extension which may be considered as an analogue of the 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.     

Isospectral deformation of the reduced quasi-classical self-dual Yang–Mills equation

We derive a new four-dimensional partial differential equation with the isospectral Lax representation by shrinking the symmetry algebra of the reduced quasi-classical self-dual Yang–Mills equation and applying the technique of twisted extensions to the obtained Lie algebra. Then we find a recursion operator for symmetries of the new equation and construct a Bäcklund transformation between this equation and the four-dimensional Martínez Alonso–Shabat equation. Finally, we construct extensions of the integrable hierarchies associated to the hyper-CR equation for Einstein–Weyl structures, the reduced quasi-classical self-dual Yang–Mills equation, the four-dimensional universal hierarchy equation, and the four-dimensional Martínez Alonso–Shabat equation.