On locally finite skew group algebras

3-dimensional algebras. Part 1. Skew-symmetric case

An algebra is called skew-symmetric if its multiplication operation is a skew-symmetric bilinear application. We determine all these algebras in dimension $3$ over a field of characteristic different from $2$. As an application, we determine the subvariety of $3$-dimensional Hom-Lie algebras. For this type of algebras, we study also the dimension $4$.

Classifying Quadratic Quantum P^2s by using Graded Skew Clifford Algebras

We prove that quadratic regular algebras of global dimension three on degree-one generators are related to graded skew Clifford algebras. In particular, we prove that almost all such algebras may be constructed as a twist of either a regular graded skew Clifford algebra or of an Ore extension of a regular graded skew Clifford algebra of global dimension two. In so doing, we classify all quadratic regular algebras of global dimension three that have point scheme either a nodal cubic curve or a cuspidal cubic curve in P^2.

On Hermitian and skew-Hermitian matrix algebras over octonions

We prove simplicity, and compute $δ$-derivations and symmetric associative forms of algebras in the title.

Skew derivations on down-up algebras

A class of skew derivations on complex Noetherian generalized down-up algebras $L=L(f,r,s,γ)$ is constructed.

Graded Skew Clifford Algebras that are Twists of Graded Clifford Algebras

We prove that if $A$ is a regular graded skew Clifford algebra and is a twist of a regular graded Clifford algebra $B$ by an automorphism, then the subalgebra of $A$ generated by a certain normalizing sequence of homogeneous degree-two elements is a twist of a polynomial ring by an automorphism, and is a skew polynomial ring. We also present an example that demonstrates that this can fail when $A$ is not a twist of $B$.

Looking for Groebner Basis Theory for (Almost) Skew 2-Nomial Algebras

In this paper, we introduce (almost) skew 2-nomial algebras and look for a one-sided or two-sided Gröbner basis theory for such algebras at a modest level. That is, we establish the existence of a skew multiplicative $K$-basis for every skew 2-nomial algebra, and we explore the existence of a (left, right, or two-sided) monomial ordering for an (almost) skew 2-nomial algebra. As distinct from commonly recognized algebras holding a Gröbner basis theory (such as algebras of the solvable type [K-RW] and some of their homomorphic images), a subclass of skew 2-nomial algebras that have a left Gröbn…

Restricted enveloping algebras whose skew and symmetric elements are Lie metabelian

Let $L$ be a restricted Lie algebra over a field of characteristic $p>2$ and denote by $u(L)$ its restricted enveloping algebra. We establish when the symmetric or skew elements of $u(L)$ under the principal involution are Lie metabelian.

Division algebras and MRD codes from skew polynomials

Let $D$ be a division algebra, finite-dimensional over its center, and $R=D[t;σ,δ]$ a skew polynomial ring. Using skew polynomials $f\in R$, we construct division algebras and a generalization of maximum rank distance codes consisting of matrices with entries in a noncommutative division algebra or field. These include a class of codes constructed by Sheekey (in particular, generalized Gabidulin codes), as well as Jha Johnson semifields.

Ext-algebras of graded skew extensions

In this paper, we study the Ext-algebras of graded skew extensions. For a connected graded algebra $A$ and a graded automorphism $σ$, we analyze the Yoneda product of the Ext-algebra of graded skew extension $A[z;σ]$, and prove this Ext-algebra is an $R$-smash product of the Ext-algebra of $A$ and the one of polynomial algebra $k[z]$ as an associative algebra.

Some homological properties of skew PBW extensions arising in non-commutative algebraic geometry

In this short paper we study for the skew PBW (Poincaré-Birkhoff-Witt) extensions some homological properties arising in non-commutative algebraic geometry, namely, Auslander-Gorenstein regularity, Cohen-Macaulayness and strongly noetherianity. Skew PBW extensions include a considerable number of non-commutative rings of polynomial type such that classical PBW extensions, quantum polynomial rings, multiplicative analogue of the Weyl algebra, some Sklyanin algebras, operator algebras, diffusion algebras, quadratic algebras in 3 variables, among many others. For some key examples we present the …

Simplicity of the Lie algebra of skew symmetric elements of a Leavitt path algebra

For a field $F$ of characteristic not 2 and a directed row-finite graph $Γ$ let $L(Γ)$ be the Leavitt path algebra with the standard involution $*.$ We study the Lie algebra $K=K(L(Γ),*)$ of $*-$skew-symmetric elements and find necessary and sufficient conditions for the Lie algebra $[K,K]$ to be simple.

Ore and Goldie theorems for skew PBW extensions

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…

Skew derivations on generalized Weyl algebras

A wide class of skew derivations on degree-one generalized Weyl algebras $R(a,φ)$ over a ring $R$ is constructed. All these derivations are twisted by a degree-counting extensions of automorphisms of $R$. It is determined which of the constructed derivations are $Q$-skew derivations. The compatibility of these skew derivations with the natural ${\mathbb Z}$-grading of $R(a,φ)$ is studied. Additional classes of skew derivations are constructed for generalized Weyl algebras given by an automorphism $φ$ of a finite order. Conditions that the central element $a$ that forms part of the structure of…

Universal property of skew $PBW$ extensions

In this paper we prove the universal property of skew $PBW$ extensions generalizing this way the well known universal property of skew polynomial rings. For this, we will show first a result about the existence of this class of non-commutative rings. Skew $PBW$ extensions include as particular examples 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. As a corollary we will give a new short proof of the Poincaré-Birkhoff-Witt theorem about the bas…

Koszulity for graded skew PBW extensions

Pre-Koszul and Koszul algebras were defined by Priddy. There exist some relations between these algebras and the skew PBW extensions defined. We have established conditions to guarantee that skew PBW extensions over fields it turns out homogeneous pre-Koszul or Koszul algebra. In this paper we complement these results defining graded skew PBW extensions and showing that if R is a finite presented Koszul K-algebra then every graded skew PBW extension of R is Koszul.

d-Hermite rings and skew PBW extensions

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 short paper we study the d-Hermite condition about stably free modules for skew PBW extensions. For this purpose, we estimate the stable rank of these non-commutative rings. In addition, and close related with these questions, …

Prime ideals of skew PBW extensions

In this paper we describe the prime ideals of some important classes of skew PBW extensions, using the classical technique of extending and contracting ideals. Skew PBW extensions include as particular examples 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.

Some homological properties of skew PBW extensions

We prove that if R is a left Noetherian and left regular ring then the same is true for any bijective skew PBW extension A of R. From this we get Serre's Theorem for such extensions. We show that skew PBW extensions and its localizations include a wide variety of rings and algebras of interest for modern mathematical physics such as PBW extensions, well known classes of Ore algebras, operator algebras, diffusion algebras, quantum algebras, quadratic algebras in 3-variables, skew quantum polynomials, among many others. We estimate the global, Krull and Goldie dimensions, and also Quillen's K-gr…

环的一般自由斜扩张的代数方面

Algebraic aspects of general free skew extensions of rings

本文研究环的自由斜扩张（即自由多元斜多项式环），给出其多种刻画，并分别给出其为整环、可嵌入级数环及素环的充分条件。

自由斜布尔代数

Free skew Boolean algebras

本文研究自由斜布尔代数的结构，给出有限生成时的乘积表示与基数公式，刻画原子与中心元素，并证明无限生成时中心平凡且所有自由代数有交集。

Cancellation and skew cancellation for Poisson algebras

We study the Zariski cancellation problem for Poisson algebras in three variables. In particular, we prove those with Poisson bracket either being quadratic or derived from a Lie algebra are cancellative. We also use various Poisson algebra invariants, including the Poisson Makar-Limanov invariant, the divisor Poisson subalgebra, and the Poisson stratiform length, to study the skew cancellation problem for Poisson algebras.

Linear commuting maps and skew-symmertric biderivations of the deformative Schrodinger-Virasoro Lie algebras

In this paper, we investigate the skew-symmertric biderivations of the deformative Schrodinger-Virasoro Lie algebras which contain the twisted and original deformative Schrodinger-Virasoro Lie algebras. As an application, we give the explicit form of each linear commuting map on the deformative Schrodinger-Virasoro Lie algebras. In particular, we obtain that there exist non-inner biderivations and non-standard linear commuting maps for the certain deformative Schrodinger-Virasoro Lie algebras.

On Rota-Baxter operators of non-zero weight induced by non skew-symmetric solutions of the classical Yang-baxter equation on simple anticommutative algebras

Let $L$ be a simple anti-commutative algebra. In this paper we prove that a non skew-symmetric solution of the classical Yang-Baxter equation on $L$ with $L$-invariant symmetric part induces on $L$ a Rota-Baxter operator of a non-zero weight.

Skew-symmetric identities of finitely generated alternative algebras

We prove that for every natural number n there exists a natural number N(n) such that every multilinear skew-symmetric polynomial on N(n) or more variables which vanishes in the free associative algebra vanishes as well in any n-generated alternative algebra over a field of characteristic 0. Before this was proved only for a series of such polynomials constructed by the author in [7].

Rota-Baxter operators and non-skew-symmetric solutions of the classical Yang-Baxter equation on quadratic Lie algebras

We study possible connections between Rota-Baxter operators of non-zero weight and non-skew-symmetric solutions of the classical Yang-Baxter equation on finite-dimensional quadratic Lie algebras. The particular attention is made to the case when for a solution $r$ the element $r+τ(r)$ is $L$-invariant.

Azumaya skew group algebras and an application to quantum Kleinian singularities

We provide easily-verified necessary and sufficient conditions for a skew group ring, or more generally, a crossed product ring, to be an Azumaya algebra. We use our results to show that (suitable localisations of) skew group rings associated to the quantum Kleinian singularities introduced by Chan, Kirkman, Walton, and Zhang in [CKWZ16a] are Azumaya, and use this to show that these algebras are maximal orders. We also give a new proof of Auslander's Theorem for quantum Kleinian singularities.

通过局部幂零导出物解决取消问题

Cancellation problem via locally nilpotent derivations

本文通过局部幂零导出物方法研究取消问题，探讨其在刚性环境中的有效性及非斜情况下的局限性。

Ring theoretic properties of partial skew groupoid rings with applications to Leavitt path algebras

Let $α=(A_g,α_g)_{g\in G}$ be a group-type partial action of a connected groupoid $G$ on a ring $A=\bigoplus_{z\in G_0}A_z$ and $B=A\star_αG$ the corresponding partial skew groupoid ring. In the first part of this paper we investigate the relation of several ring theoretic properties between $A$ and $B$. For the second part, using that every Leavitt path algebra is isomorphic to a partial skew groupoid ring obtained from a partial groupoid action $λ$, we characterize when $λ$ is group-type. In such a case, we obtain ring theoretic properties of Leavitt path algebras from the results on general…

Skew-Symmetric Elements of Rational Group Algebras

Let $RG$ be the group ring of a finite group $G$ over a commutative ring $R$ with $1$. An element $x$ in $RG$ is said to be skew-symmetric with respect to an involution $σ$ of $RG$ if $σ(x)=-x.$ A structure theorem for the skew-symmetric elements of $FG$ is given where $F$ is an algebraic extension of $\mathbb{Q}$ which generalizes some previously known results in this direction.

Skew group algebras, invariants and Weyl Algebras

The aim of this paper is two fold: First to study finite groups $G$ of automorphisms of the homogenized Weyl algebra $B_{n}$, the skew group algebra $B_{n}\ast G$, the ring of invariants $B_{n}^{G}$, and the relations of these algebras with the Weyl algebra $A_{n}$, with the skew group algebra $A_{n}\ast G$, and with the ring of invariants $A_{n}^{G}$. Of particular interest is the case $n=1$. In the on the other hand, we consider the invariant ring $\QTR{sl}{C}[X]^{G}$ of the polynomial ring $K[X]$ in $n$ generators, where $G$ is a finite subgroup of $Gl(n,\QTR{sl}{C}$) such that any element …

Enveloping Algebra and Skew Calabi-Yau algebras over Skew Poincaré-Birkhoff-Witt Extensions

In this paper we show that the tensor product of skew PBW extensions is a skew PBW ex- tension. We also characterize the enveloping algebra of a skew PBW extension. Finally, we establish sufficient conditions to guarantee the property of being skew Calabi-Yau algebra over skew PBW extensions.

Projective modules and Gröbner bases for skew PBW extensions

Many rings and algebras arising in quantum mechanics, algebraic analysis, and non-commutative algebraic geometry can be interpreted as skew PBW (Poincaré-Birkhoff-Witt) extensions. In the present paper we study two aspects of these non-commutative rings: its finitely generated projective modules from a matrix-constructive approach, and the construction of the Gröbner theory for its left ideals and modules. These two topics could be interesting in future eventual applications of skew $PBW$ extensions in functional linear systems and in non-commutative algebraic geometry.

斜自伴映射与二次李代数

Skew-adjoint maps and quadratic Lie algebras

本文利用双重扩张方法在任意域上引入广义振子代数，并基于斜自伴自同态的典范形式对二次幂零代数进行分类，从而重新得到 Medina 关于实振子代数的经典分类。

A refinement of Stone duality to skew Boolean algebras

We establish two duality theorems which refine the classical Stone duality between generalized Boolean algebras and locally compact Boolean spaces. In the first theorem we prove that the category of left-handed skew Boolean algebras whose morphisms are proper skew Boolean algebra homomorphisms is equivalent to the category of étale spaces over locally compact Boolean spaces whose morphisms are étale space cohomomorphisms over continuous proper maps. In the second theorem we prove that the category of left-handed skew Boolean $\cap$-algebras whose morphisms are proper skew Boolean $\cap$-algebr…

Structurable algebras of skew-dimension one and hermitian cubic norm structures

We study structurable algebras of skew-dimension one. We present two different equivalent constructions for such algebras: one in terms of non-linear isotopies of cubic norm structures, and one in terms of hermitian cubic norm structures. After this work was essentially finished, we became aware of the fact that both descriptions already occur in (somewhat hidden places in) the literature. Nevertheless, we prove some facts that had not been noticed before: (1) We show that every form of a matrix structurable algebra can be described by our constructions; (2) We give explicit formulas for the n…

Permutable subgroups in $\GL_n(D)$ and applications to locally finite group algebras

In this paper we study the existence of free non-abelian subgroups in non-central permutable subgroups of general skew linear groups and locally finite group algebras.

Values of Noncommutative Polynomials, Lie Skew-Ideals and the Tracial Nullstellensatz

A subspace of an algebra with involution is called a Lie skew-ideal if it is closed under Lie products with skew-symmetric elements. Lie skew-ideals are classified in central simple algebras with involution (there are eight of them for involutions of the first kind and four for involutions of the second kind) and this classification result is used to characterize noncommutative polynomials via their values in these algebras. As an application, we deduce that a polynomial is a sum of commutators and a polynomial identity of $d\times d$ matrices if and only if all of its values in the algebra of…

How a nonassociative algebra reflects the properties of a skew polynomial

Let $S$ be a unital associative ring and $S[t;σ,δ]$ be a skew polynomial ring, where $σ$ is an injective endomorphism of $S$ and $δ$ a left $σ$-derivation. For each $f\in S[t;σ,δ]$ of degree $m>1$ with a unit as leading coefficient, we construct a unital nonassociative algebra whose behaviour reflects the properties of $f$. The algebras obtained yield canonical examples of right division algebras when $f$ is irreducible. We investigate the structure of these algebras. The structure of their right nucleus depends on the choice of $f$. In the classical literature, this nucleus appears as the eig…

斜Laurent级数与一般循环卷积码

Skew Laurent Series and General Cyclic Convolutional Codes

本文研究了斜Laurent级数在构造一般循环卷积码中的作用，并探讨了如何处理斜导出带来的困难。

正特征多项式环上循环换位群作用的斜群代数的分级形变

Graded deformations of skew group algebras for cyclic transvection groups acting on polynomial rings in positive characteristic

本文在正特征且特征整除群阶的条件下，从多角度完整刻画了循环换位群作用斜群代数的Poincare-Birkhoff-Witt形变，给出了条件、构造方法、公式及组合分析。

Computing finite presentations of Tor and Ext over skew PBW extensions and some applications

In this paper we compute the $Tor$ and $Ext$ modules over skew $PBW$ extensions. If $A$ is a bijective skew $PBW$ extension of a ring $R$, we give presentations of $Tor_r^{A}(M,N)$, where $M$ is a finitely generated centralizing subbimodule of $A^m$, $m\geq 1$, and $N$ is a left $A$-submodule of $A^l$, $l\geq 1$. In the case of $Ext_A^{r}(M,N)$, $M$ is a left $A$-submodule of $A^m$ and $N$ is a finitely generated centralizing subbimodule of $A^l$. As application of these computations, we test stably-freeness, reflexiveness, and we will compute also the torsion, the dual and the grade of a give…

斜 Hecke 代数

Skew Hecke Algebras

本文定义并研究斜 Hecke 代数，证明其双陪集分解、与斜群代数及诱导表示自同态代数的关系，并验证其在多种代数运算下的相容性。

Point modules over regular graded skew Clifford algebras

Results of Vancliff, Van Rompay and Willaert in 1998 prove that point modules over a regular graded Clifford algebra (GCA) are determined by (commutative) quadrics of rank at most two that belong to the quadric system associated to the GCA. In 2010, Cassidy and Vancliff generalized the notion of a GCA to that of a graded skew Clifford algebra (GSCA). The results in this article show that prior results may be extended, with suitable modification, to GSCAs. In particular, using the notion of μ-rank introduced recently by the authors, the point modules over a regular GSCA are determined by (nonco…

关于q-斜阿米修尔定理

On a $q$-Skew Amitsur's Theorem

本文研究q-斜Ore扩张的Jacobson radical，证明其常数部分为 nil，部分回答Greenfeld等2019年问题，并在特征为零时运用Shin结果证明q-斜阿米修尔定理。

简单Leavitt路代数中斜对称元素的李代数

Lie Algebras of Skew-Symmetric Elements in Simple Leavitt path algebras

本文研究简单Leavitt路代数中斜对称元素构成的李代数，证明其不满足Herstein在1961年提出的问题中的性质，给出否定答案。

Leavitt path algebras as partial skew group rings

We realize Leavitt path algebras as partial skew group rings and give new proofs, based on partial skew group ring theory, of the Cuntz-Krieger uniqueness theorem and simplicity criteria for Leavitt path algebras.

分次斜Clifford代数的光滑几何

Smooth geometry of graded skew Clifford algebras

该论文聚焦分次斜Clifford代数领域，针对其微分光滑性展开研究，属于代数与微分几何交叉方向的理论探索。

斜Brauer图代数

Skew-Brauer graph algebras

本文提出斜Brauer图代数，推广Brauer图代数，并研究其表示类型与几何解释。

斜范畴代数、扭曲张量积代数与预层的诱导

Skew category algebras, twisted tensor product algebras and induction of precosheaves

本文证明斜范畴代数可嵌入扭曲张量积代数，并将Puig和Turull的概念从群代数推广到范畴代数，研究其在斜范畴代数下的性质。

Simplicity and chain conditions for ultragraph Leavitt path algebras via partial skew group ring theory

We realize Leavitt ultragraph path algebras as partial skew group rings. Using this realization we characterize artinian ultragraph path algebras and give simplicity criteria for these algebras.

Free group algebras in Malcev-Neumann skew fields of fractions

Let K be a skew field and (G,<) an ordered group. We show that the skew field generated by the group ring K[G] inside the Malcev-Neumann series ring K((G;<)) contains noncommutative free group algebras.

Monster型II的斜轴代数

Skew Axial Algebras of Monster Type II

该研究延续此前关于2生成Monster型本原轴代数的工作，对带有更大斜轴集的这类代数进行分类，推进了该类代数的完整分类进程。

On the representation dimension of skew group algebras, wreath products and blocks of Hecke algebras

We establish bounds for the representation dimension of skew group algebras and wreath products. Using this, we obtain bounds for the representation dimension of a block of a Hecke algebra of type A, in terms of the weight of the block. This includes certain blocks of group algebras of symmetric groups.

The interplay between Steinberg algebras and partial skew rings

We study the interplay between Steinberg algebras and partial skew rings: For a partial action of a group in a Hausdorff, locally compact, totally disconnected topological space, we realize the associated partial skew group ring as a Steinberg algebra (over the transformation groupoid attached to the partial action). We then apply this realization to characterize diagonal preserving isomorphisms of partial skew group rings, over commutative algebras, in terms of continuous orbit equivalence of the associated partial actions. Finally, we show that any Steinberg algebra, associated to a Hausdorf…