Gradings on simple algebras of finitary matrices
Yuri Bahturin, Mikhail Zaicev
本文讨论了特征为零的代数闭域上的无穷多个非零元素矩阵的结合代数上的有限阿贝尔群群的分次。
We describe gradings by finite abelian groups on the associative algebras of infinite matrices with finitely many nonzero entries, over an algebraically closed field of characteristic zero.
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Group gradings on finitary simple Lie algebras
我们分类，直到同构，由任意阿贝尔群上的线性变换（特殊的线性，正交和辛）的简单有限李代数无限维向量空间上的代数闭域的特征不同于2的所有分次。
Images of Multilinear Polynomials in the Algebra of Finitary Matrices Contain Trace Zero Matrices
本研究针对无限域上非零多重线性多项式在有限矩阵代数中的像问题，证明对任意正整数d存在s使其像含所有d阶迹零矩阵，进而得出其在全体有限矩阵代数中的像包含所有迹零有限矩阵的结论。
Group gradings on the Jordan algebra of upper triangular matrices
设G是任意群，K是特征不同于2的域。对K上n阶上三角形矩阵的Jordan代数上的G-分次进行了分类。事实证明，有，直到一个分次同构，两个家庭的分次：基本分次（类似于在结合的情况下），和所谓的镜像类型（MT）分次。此外，我们证明了这个代数上的G-分次是唯一确定的，直到一个分次同构，由分次恒等式，他们满足。
We investigate the group gradings on the algebra of upper triangular matrices over an arbitrary field, viewed as a Lie algebra. These results were obtained a few years early by the same authors. We provide streamlined proofs, and present a complete classification of isomorphism classes of the gradings. We also provide a classification of the practical isomorphism classes of the gradings, which is a better alternative way to consider these gradings up to being essentially the same object. Finally, we investigate in details the case where the characteristic of the base field is $2$, a topic that…
We classify up to isomorphism all gradings by an arbitrary group $G$ on the Lie algebras of zero-trace upper block-triangular matrices over an algebraically closed field of characteristic $0$. It turns out that the support of such a grading always generates an abelian subgroup of $G$. Assuming that $G$ is abelian, our technique also works to obtain the classification of $G$-gradings on the upper block-triangular matrices as an associative algebra, over any algebraically closed field. These gradings were originally described by A. Valenti and M. Zaicev in 2012 (assuming characteristic $0$ and $…
On graded identities of block-triangular matrices with the grading of Di Vincenzo-Vasilovsky
域$F$上的$n\times n$矩阵代数具有自然的$\mathbb{Z}_n$ -分次.它的分级身份已被描述的Vasilovsky谁延长了以前的工作Di Jienzo代数的$2\times 2$矩阵。本文研究了块三角形矩阵的分次恒等式，其分次由$M_n(F)$的分次所继承。我们证明了它的分次恒等式是从$M_n(F)$的分次恒等式和从它的度高达$2n-2$的单项式恒等式得到的.在块的大小$n-1$和1的情况下，我们给出了一个完整的描述，它的单项式的身份，并展示了一个最小的基础，其$T_{\mathbb{Z}_n}$ -理想。
It was proved by Valenti and Zaicev, in 2011, that, if $G$ is an abelian group and $K$ is an algebraically closed field of characteristic zero, then any $G$-grading on the algebra of upper block triangular matrices over $K$ is isomorphic to a tensor product $M_n(K)\otimes UT(n_1,n_2,\ldots,n_d)$, where $UT(n_1,n_2,\ldots,n_d)$ is endowed with an elementary grading and $M_n(K)$ is provided with a division grading. In this paper, we prove the validity of the same result for a non necessarily commutative group and over an adequate field (characteristic either zero or large enough), not necessaril…
Pauli gradings on Lie superalgebras and graded codimension growth
利用初等交换2-群对有限维$P(t)$型单李超代数进行分次.这种分次产生了Pauli矩阵，是$(2\times 2)$ -无迹矩阵的李代数上的$(\mathbb Z_2\times \mathbb Z_2)$ -分次的进一步推广.特别地，我们计算了对应于泡利分次的分次PI指数。
Let $F$ be a field of characteristic zero, $G$ be a group and $R$ be the algebra $M_n(F)$ with a $G$-grading. Bahturin and Drensky proved that if $R$ is an elementary and the neutral component is commutative then the graded identities of $R$ follow from three basic types of identities and monomial identities of length $\geq 2$ bounded by a function $f(n)$ of $n$. In this paper we prove the best upper bound is $f(n)=n$, more generally we prove that all the graded monomial identities of an elementary $G$-grading on $M_n(F)$ follow from those of degree at most $n$. We also study gradings which sa…
Let $G$ be an abelian group and $\mathbb{K}$ an algebraically closed field of characteristic zero. A. Valenti and M. Zaicev described the $G$-gradings on upper block-triangular matrix algebras provided that $G$ is finite. We prove that their result holds for any abelian group $G$: any grading is isomorphic to the tensor product $A\otimes B$ of an elementary grading $A$ on an upper block-triangular matrix algebra and a division grading $B$ on a matrix algebra. We then consider the question of whether graded identities $A\otimes B$, where $B$ is an algebra with a division grading, determine $A\o…
We determine the number of isomorphism classes of elementary gradings by a finite group on an algebra of upper block-triangular matrices. As a consequence we prove that, for a finite abelian group $G$, the sequence of the numbers $E(G,m)$ of isomorphism classes of elementary $G$-gradings on the algebra $M_{m}(\mathbb{F})$ of $m\times m$ matrices with entries in a field $\mathbb{F}$ characterizes $G$. A formula for the number of isomorphism classes of gradings by a finite abelian group on an algebra of upper block-triangular matrices over an algebraically closed field, with mild restrictions on…
Graded polynomial identities of the infinite-dimensional upper triangular matrices over an arbitrary field
本文计算了任意域上无限维上三角矩阵代数的分级多项式恒等式，证明了有限分级群下存在有限基，给出了满足非平凡恒等式的条件，并举例说明非同构分级可具有相同的恒等式集合。
A graph theoretic approach to graded identities for matrices
本文考虑复数上k × k矩阵代数M_k（C），通过正则表示将G嵌入对称群S_k中，并按通常的方法将S_k嵌入M_k（C）中，将M_k（C）看作与k阶群G的交叉积.这在M_k（C）上导出了一个自然的G-分次，我们称之为交叉积分次。这种分级是由任意k元组（g_1，g_2，.，g_k）的集合。本文研究了M_k（C）具有交叉积分次的分次多项式恒等式.对于自由分次代数中的每一个多线性单项式，我们关联一个有向标号图。这种方法使我们能够给出新的证明已知的结果Bahturin和Drensky的生成元的T-理想的身份和Amitsur-Levitsky定理。我们最重要的新结果是确定了M_k（C）的G-分次余维数的渐近公式。
On the asymptotic behaviour of the graded-star-codimension sequence of upper triangular matrices
本文研究了无限域上赋予群分级和齐次对合的上三角矩阵代数，计算了其分级星余维数序列的渐近行为，发现该渐近增长仅取决于矩阵代数的大小，与分级和对合无关。
On gradings of matrix algebras and descent theory
利用有限阿贝尔群对矩阵代数上的分次进行分类。如果所有的初等矩阵都是齐次的，则称一个分次是好的。对于循环群，代数闭域上的矩阵代数的所有分次都是好的。我们可以通过一个循环群来计算好的评分数。使用下降理论，我们分类非良好的评分后，成为良好的基础扩展的矩阵代数。
The Jordan algebra of the symmetric matrices of order two over a field $K$ has two natural gradings by $\mathbb{Z}_2$, the cyclic group of order 2. We describe the graded polynomial identities for these two gradings when the base field is infinite and of characteristic different from 2. We exhibit bases for these identities in each of the two cases. In one of the cases we perform a series of computations in order to reduce the problem to dealing with associators while in the other case one employs methods and results from Invariant theory. Moreover we extend the latter grading to a $\mathbb{Z}…
On homogeneous involutions on matrix algebras
本文研究了在特定分级下的矩阵代数中的同构反演分类及其在不同分级下的性质。
Graded polynomial identities for matrices with the transpose involution
设$G$是阶为$k$的群。我们考虑$k$的代数$M_k(\mathbb{C})$通过$k$矩阵在复数上，并认为它是一个交叉积关于$G$通过嵌入$G$在对称群$S_k$通过正则表示和嵌入$S_k$在$M_k(\mathbb{C})$在通常的方式。这在$M_k(\mathbb{C})$上产生自然的$G$分级，我们称之为交叉积分级。我们研究了具有这种交叉积分次和转置对合的$M_k(\mathbb{C})$的分次$*$ -恒等式。对于具有对合的自由分次代数中的每一个多线性单项式，我们都对应一个有向标号图。利用这些图，我们可以产生一组生成器的$(T,*)$理想的身份。这也导致了新的证明结果Kostant和Rowen的标准身份所满足的斜矩阵。最后我们确定了$M_k(\mathbb{C})$的$*$分次余维数的一个渐近公式.
On star-homogeneous-graded polynomial identities of upper triangular matrices over an arbitrary field
本文研究任意域上带精细群分次和齐次对合的上三角矩阵代数的分次多项式恒等式，给出基与渐近行为，并计算任意分次与对合下的指数和第二指数。
Our approach to structural matrix rings defines them over preordered directed graphs. A grading of a structural matrix ring is called a good grading if its standard unit matrices are homogeneous. For a group $G$, a $G$ -grading set is a set of arrows with the property that any assignment of these arrows to elements of $G$ uniquely determines an induced good grading. One of our main results is that a $G$-grading set exists for any transitive directed graph if $G$ is a group of prime order. This extends a result of Kelarev. However, an example of Molli Jones shows there are directed graphs which…
Graded identities with involution for the algebra of upper triangular matrices
本文研究上三角矩阵代数上带分次对合的群分次，证明其可归结为反射或辛对合，给出恒等式基与余维数增长，并确定指数，特别地证明 $UT_3(F)$ 的普通 $\ast$-指数为 $3$。
Graded Polynomial Identities for Matrices with the Transpose Involution over an Infinite Field
设$F$是一个无限域，$M_{n}(F)$是$F$上的$n\times n$矩阵代数.假设这个代数具有一个初等分次，其中性分量与主对角线重合。本文给出了$M_{n}(F)$的分次多项式恒等式在转置对合下的基.我们的结果推广到无限域的任意特征以前的结果在文献中得到的领域的复数和一类特殊的初等G-分次。
Graded polynomial identities for the Lie algebra of upper triangular matrices of order 3
本文计算了3阶上三角矩阵李代数在初等分次下的分次多项式恒等式及其分次余维数序列。
In this paper we study the images of multilinear graded polynomials on the graded algebra of upper triangular matrices UT_n. For positive integers q \leq n, we classify these images on UT_n endowed with a particular elementary Z_q-grading. As a consequence, we obtain the images of multilinear graded polynomials on UT_n with the natural Z_n-grading. We apply this classification in order to give a new condition for a multilinear polynomial in terms of graded identities so that to obtain the traceless matrices in its image on the full matrix algebra. We also describe the images of multilinear pol…
Gradings, graded identities, $*$-identities and graded $*$-identities of an algebra of upper triangular matrices
本文研究上三角矩阵代数子代数 $\mathcal{A}$ 的阿贝尔群分次，证明其为初等分次，并计算其 $\mathbb{Z}_2$-分次恒等式及带分次对合的分次恒等式的基，同时描述其余特征标。
Graded identities of block-triangular matrices
设$F$是无限域，$UT(d_1,\dots, d_n)$是$F$上块三角形矩阵代数.在本文中，我们描述了一个基础的$G$ -分次多项式的身份$UT(d_1,\dots, d_n)$，与一个基本的分次诱导的$n$ -元组的元素的一个组$G$，使中性分量对应于对角线的$UT(d_1,\dots,d_n)$。特别地，我们证明了这种代数的单项式恒等式是从直到$2n-1$的次数的单项式恒等式导出的。我们的结果推广到任意特征的无限域，以前的结果在文献中得到的领域的特征零和特殊的$G$ -分次。在特征为零的情况下，我们还推广了张量积分次代数$UT(d_1,\dots, d_n)\otimes C$的结果，其中$C$是生成所有色交换代数簇的色交换代数.
Values of multilinear graded $*$-polynomials on upper triangular matrices of small dimension
本文研究多重线性分次*-多项式在小维数上三角矩阵上的像，证明其在 UT_2 及非平凡分次 UT_3 上为齐次向量空间，并说明该结论最优，从而表明 L'vov-Kaplansky 猜想在带分次对合的代数中不成立。
A note on the image of graded multilinear polynomials on upper triangular matrices
研究上三角矩阵代数上分次变量多元多项式的像，证明在恒等分量交换时像总是子空间，并推广到逆极限和直极限。
Let $K$ be a field of characteristic different from 2 and let $G$ be a group. If the algebra $UT_n$ of $n\times n$ upper triangular matrices over $K$ is endowed with a $G$-grading $Γ: UT_n=\oplus_{g\in G}A_g$ we give necessary and sufficient conditions on $Γ$ that guarantees the existence of a homogeneous antiautomorphism on $A$, i.e., an antiautomorphism $φ$ satisfying $φ(A_g)=A_{θ(g)}$ for some permutation $θ$ of the support of the grading. It turns out that $UT_n$ admits a homogeneous antiautomorphism if and only if the reflection involution of $UT_n$ is homogeneous. Moreover, we prove that…
Graded polynomial identities and Specht property for the Lie algebra of upper triangular matrices of order 3
本文研究了任意域上三次上三角矩阵李代数在初等分次下的分次多项式恒等式，并探讨了相应簇的Specht性质。
Y-proper graded cocharacters of upper-triangular matrices od order m graded by the m-tuple ϕ=(0,0,1,...,m-2)
设F是特征为0的域。本文考虑由m元组phi=（0，0，1，.. m-2），然后计算了它的Y-proper分次余特征标序列，并给出了m= 2，3，4，5时重数的显式表达式.
A model for the relatively free graded algebra of block triangular matrices with entries from a graded algebra
设G是群，A是满足多项式恒等式的G-分次代数.建立了相对自由G-分次代数的一个模型，作为应用，得到了有限维Grassmann代数E中元素为有限维G-分次代数E的块三角形矩阵的T_G-理想的“因子分解”性质。
Graded polynomial identities and central polynomials of matrices over an infinite integral domain
设$K$是无限整环，$M_{n}(K)$是$K$上所有$n\times n$矩阵的代数.本文的主要目的是：当$M_{n}(K)$的中性分支与对角线重合时，给出$M_{n}(K)$中初等分次的分次恒等式的基;当$p$为素数时，给出$M_{p}(K)$的$\mathbb{Z}_{p}$ -分次中心多项式;给出$M_{n}(K)$的$\mathbb{Z}$ -分次中心多项式.
Matrices as graded BiHom-algebras and decompositions
本文通过引入连接概念构造典范分次理想，证明在Σ-乘性等条件下矩阵正则BiHom-结合代数可分解为分次单理想的直和，并推广至一般情形及经典分次。
On the graded automorphisms of upper triangular matrix algebras
我们计算了上三角形矩阵的分次自同构，将其看作结合代数、李代数和Jordan代数。我们还计算所谓的自我等价和Weyl和对角群的每一个评分。
Pseudo grading on cluster automorphism group with application to cluster algebras of rank $3$
本文提出簇自同构群上的伪 $\mathbb{N}$-分次，证明其由 $G_0\cup G_1$ 生成，并据此完全确定了秩 $3$ 且交换矩阵不可分解的簇代数的簇自同构群。
Primeness property for regular gradings
本文研究了正则分级的素性性质，证明了G-正则代数不满足素性，而Z_2-正则代数在普通意义上满足素性，且最小性对正则性无要求。
Let $ F $ be a finite field and consider $ UT_n $ the algebra of $ n\times n $ upper triangular matrices over $ F $. In [1], it was proved that every $ G $-grading is elementary. In [2], the authors classified all nonisomorphic elementary $ G $-gradings. They also described the set of all $G $-graded polynomial identities for $ UTn $ when $ F $ is an infinite field. In [3], was described the all $ G $-graded polynomial identities for $ UT_n $ when $ F $ is a finite field. In this work, we will discuss the case when $G = \mathbb{Z}_2$, $ n=2, 3 $ and $ F $ is a finite field.
Upper triangular matrices with superinvolution: identities and images of multilinear polynomials
本文完整刻画了UT_4(F)上由序列(0,1,0,1)诱导分次及超辛超级对合的恒等式，并证明UT_n(F)上多重线性多项式像为向量空间当且仅当n≤3，推进了L'vov-Kaplansky猜想的类比研究。
Let $K$ be a field (finite or infinite) of char$(K)\neq 2$ and let $UT_n=UT_n(K)$ be the $n\times n$ upper triangular matrix algebra over $K$. If $\cdot $ is the usual product on $UT_n$ then with the new product $a\circ b=(1/2)(a\cdot b +b\cdot a)$ we have that $UT_n$ is a Jordan algebra, denoted by $UJ_n=UJ_n(K)$. In this paper, we describe the set of all $\mathbb{Z}_2$-graded polynomial identities of $UJ_2$ with any nontrivial $\mathbb{Z}_2$-grading. Moreover, we describe a linear basis for the corresponding relatively free $\mathbb{Z}_2$-graded algebra.
The Schützenberger groups and maximal subgroups of tropical matrices
本研究分类了热带半环矩阵范畴与有限热带半环矩阵半群胚的Schützenberger群，完成了所有阶数热带矩阵幺半群极大子群的分类，推广并修正了已有结果，还证明存在可作为Schützenberger群但无法作为极大子群的群。
On the Codimension Sequence of G-Simple Algebras
80年代，Regev利用Formanek，Procesi和Razmyslov在不变量理论中的结果和Hilbert级数的结果，渐近地确定了特征为零的代数闭域上mXm矩阵的余维数序列.受Regev思想的启发，我们发现有限维G单代数A的G分次余维序列$c_{n}^{G}(A)$的渐近性等于$αn^{\frac{1-\dim(A_{e})}{2}}(\dim(A)^{n} $（这是E.Aljadeff，D.Haile和M. Natapov），其中α是尚未确定的数。此外，当A是具有任意初等G-分次的m × m矩阵的代数时，我们也计算了α。
Signed seeds and G-gradings on cluster algebras
本文引入簇代数上的G-分次种子概念，统一了符号种子、簇自同构群和分次簇代数，并应用于矩阵空间、positroid簇等例子，与热带突变建立联系。
Fine Gradings of Low-Rank Complex Lie Algebras and of Their Real Forms
在这篇评论文章中，我们处理的主题，精细分级的李代数。这个概念不仅对研究代数的结构性质很重要，而且在此之上，精细分次通常被用作研究代数的分次压缩或变形的起点。一个基本的问题解决的工作是术语之间的关系“分级”和“组分级”。虽然这些条款最初声称符合简单李代数，后来发现，证明这一主张是不正确的。因此，关于精细分级和MAD组之间一一对应的关键陈述必须进行修改，并重新制定精细组分级。然而，仍然有一个假设，即“分次”和“群分次”这两个术语在简单复李代数中是一致的。我们使用MAD-群作为寻找复李代数$A_3 \cong D_3$、$B_2 \cong C_2$和$D_2$的精细群分次的主要工具。此外，我们还开发了其他的方法来寻找细（组）分级。它们对于复代数的实形式特别有用，在这方面它们提供了比MAD-群更丰富的结果。系统地使用了三个李代数的$4\times 4$矩阵的忠实表示：$A_3 = sl(4,\mathbb C)$，$C_2 = sp(4,\mathbb C)$，$D_2 = o(4,\mathbb C)$。内含物$sl(4,\mathbb C)\supset sp(4,\mathbb C)$和$sl(4,\mathbb C) \supset o(4,\mathbb C)$在我们的介绍中是很重要的，因为它们允许采用的方法之一，大大简化了计算时，发现精细群分级的子代数$sp(4,\mathb…
Multiary gradings
本文提出多 ary 分级多项式代数理论，揭示了更高 arity 结构中的新现象，如更高幂分级和 arity 兼容性的约束。
Polynomial identities, central polynomials and cocharacters of $M_2(F)$ with $G$-graded involution
本文研究特征零域上矩阵代数 $M_2(F)$ 的 $G$-分次对合，确定其恒等式与中心多项式的生成元，并显式计算余特征与余维数。
Gelfand-Kirillov dimension and Jordan algebras
设A是被有限阿贝尔群G分次的结合代数，若分别用GKdim_k（A）和GKdim^G_k（A）表示其k元相对自由代数和相对自由G-分次代数的Gelfand-Kirillov维数，则GKdim_k（A）\leq GKdim^G_k（A）.我们显示了一个反例以前的结果为约旦代数（因此非结合）。特别地，存在$Z_2$ -分次在$UJ_n$上，$n\times n$上三角形矩阵的Jordan代数，n等于2或3，使得前面的不等式不成立.
Idempotent tropical matrices and finite metric spaces
在有限集上距离函数的三角不等式与热带半环上伴随矩阵的幂等性之间存在一个众所周知的对应关系。最近的研究揭示了新的光的结构（代数，组合和几何）的热带幂等元，在本文中，我们探讨了热带多面体的度量几何的后果。我们证明，例如，每一个n点度量空间是实现的希尔伯特射影度量的顶点上的纯n维热带多面体在热带n-空间。更一般地说，每个n点非对称距离函数都是由这样一个多面体的顶点上的剩余算子实现的。在对称情形下，我们证明了包含度量空间幂等元的热带矩阵的极大群是实数与度量空间等距群的直积;由此得出，有限群与实数的直积都是一个充分大的有限全热带矩阵半群的极大子群.在这个过程中，我们还证明了一些新的结果，热带幂等矩阵，并注意到一些半群理论的结果，可能是独立的利益。
Determinants over graded-commutative algebras, a categorical viewpoint
本文将线性超代数推广到由任意阿贝尔群和双特征标给出的高阶和交换因子。我们的中心工具是一个扩展，幺半范畴的模块，Nekludova-Scheunert忠实函子之间的范畴的分次交换和超交换代数。从而将分次交换代数上的分次矩阵的（超）迹、行列式和Berezian推广到分次矩阵上。例如，在齐次四元数矩阵上，我们得到了迪厄多内行列式到四元数斜域的提升。
Let $UT_n(F)$ be the algebra of the $n\times n$ upper triangular matrices and denote $UT_n(F)^{(-)}$ the Lie algebra on the vector space of $UT_n(F)$ with respect to the usual bracket (commutator), over an infinite field $F$. In this paper, we give a positive answer to the Specht property for the ideal of the $\mathbb{Z}_n$-graded identities of $UT_n(F)^{(-)}$ with the canonical grading when the characteristic $p$ of $F$ is 0 or is larger than $n-1$. Namely we prove that every ideal of graded identities in the free graded Lie algebra that contains the graded identities of $UT_n(F)^{(-)}$, is f…
Associative algebras and the representation theory of grading-restricted vertex algebras
本文通过构造与分级受限顶点代数关联的结合代数及其有限维子代数，建立了下界广义顶点代数模与相应分级结合代数模之间的等价对应，并将不可约性与完全可约性转化为结合代数模的相应性质。
Cyclotomic quiver Hecke algebras of type A
本章是基于我2013年4月在新加坡国立大学的一系列讲座。该说明调查的表示论的分圆Hecke代数的A型，重点是了解KLR分级和之间的连接“经典”ungraded表示论和迅速崛起的分级理论。他们是相当独立的，他们试图给一个悠闲地介绍这些代数，有许多例子和计算，没有出现在其他地方。我们广泛地利用了无等级和等级表示论之间的相互作用，并试图解释等级给我们带来了什么，我们以前没有。组合学和细胞代数技术被用于整个，与一些结果从几何和2-表示论被引用的文献。重点包括一个完整的描述的半单KLR代数的A型使用的KLR关系，广泛的讨论分次Specht模块，证明的Ariki-Brundan-Kleshchev分次分类定理的基础上的分次分支规则，细胞代数方法调整矩阵和（可能乐观）猜想的分次维度的简单模块。
Specht property for the graded identities of the pair $(M_2(D), sl_2(D))$
本文研究诺特无限整环上配对 $(M_2(D), sl_2(D))$ 的 $\mathbb{Z}_2$ 自然分次，给出其分次多项式恒等式的有限基，并证明其分次恒等式理想满足 Specht 性质。
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