The geometry of polynomial identities
Claudio Procesi
本文讨论了由Kemer理论导出的代数多项式恒等式的一些几何不变量，导出了余维数和余长度的一些定量信息
We discuss some geometric invariants of polynomial identities of algebras deduced from Kemer's theory and deduce some quantitative information on codimension and co--length
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Tensor polynomial identities generalize the concept of polynomial identities on $d \times d$ matrices to identities on tensor product spaces. Here we completely characterize a certain class of tensor polynomial identities in terms of their associated Young tableaux. Furthermore, we provide a method to evaluate arbitrary alternating tensor polynomials in $d^2$ variables.
Quite recently, Bremner et al. introduced a new approach to Rota's Classification Problem and classified some (new) operated polynomial identities. In this paper, we prove that all operated polynomial identities classified by Bremner et al. are Gröbner-Shirshov.
Polynomial identities of finite prime Universal algebras
本文证明了在相同环上满足相同多项式恒等式的有限素数Ω-代数必为同构。
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。
The identities and central polynomials of the finite dimensional unitary Grassmann algebras over a finite field
本文描述了有限域上酉有限维Grassmann代数的恒等式的T-理想和中心多项式的T-空间。
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。
Let $R$ be an associative algebra over a field $K$ generated by a vector subspace $V$. The polynomial $f(x_1,\ldots,x_n)$ of the free associative algebra $K\langle x_1,x_2,\ldots\rangle$ is a weak polynomial identity for the pair $(R,V)$ if it vanishes in $R$ when evaluated on $V$. We survey results on weak polynomial identities and on their applications to polynomial identities and central polynomials of associative and close to them nonassociative algebras and on the finite basis problem. We also present results on weak polynomial identities of degree three.
Let $A$ and $B$ be finite-dimensional simple algebras with arbitrary signature over an algebraically closed field. Suppose $A$ and $B$ are graded by a semigroup $S$ so that the graded identitical relations of $A$ are the same as those of $B$. Then $A$ is isomorphic to $B$ as an $S$-graded algebra.
Weak polynomial identities of small degree for the Weyl algebra
本文刻画了任意特征无限域上Weyl代数的弱多项式恒等式，完整描述了最小次数3、次数4和5以及两个变元情形的全部结果。
The identities and the central polynomials of the infinite dimensional unitary Grassmann algebra over a finite field
本文描述了有限域上无限维酉Grassmann代数的恒等式的T-理想和中心多项式的T-空间。
Homogeneous involution on graded division algebras and their polynomial identities
本文研究代数闭域上有限维分次可除代数的齐合对合，计算其带对合的分次多项式恒等式，指出齐合对合在分次多项式恒等式与相容对合研究中自然出现。
Let $A$ be a finite dimensional real algebra with a division grading by a finite abelian group $G$. In this paper we provide finite basis for the $T_G$-ideal of graded identities and for the $T_G$-space of graded central polynomials for $A$.
An algebra with identities $a(bc)=b(ac),$ $(ab)c=(ac)b$ is called bicommutative. We construct list of identities satisfied by commutator and anti-commutator products in a free bicommutative algebra. We give criterions for elements of a free bicommutative algebra to be Lie or Jordan.
We determine minimal Cayley--Hamilton and Capelli identities for matrices over a Grassmann algebra of finite rank. For minimal standard identities, we give lower and upper bounds on the degree. These results improve on upper bounds given by L.\ Márki, J.\ Meyer, J.\ Szigeti, and L.\ van Wyk in a recent paper.
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.
Simple G-graded algebras and their polynomial identities
设G是任意群，F是特征为零的代数闭域.证明了F上的任意G-分次有限维结合G-单代数代数都由其G-分次多项式恒等式确定为G-分次同构。Koshlukov和Zaicev在G是交换的情况下证明了这个结果。
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。
Codimensions of polynomial identities of representations of Lie algebras
考虑表示$ρ\colon L \to \mathfrak{gl}(V)$，其中$L$是李代数，而$V$是有限维向量空间。我们证明了类似的Amitsur猜想关于$ρ$的多项式恒等式的余维数的渐近行为。
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。
On the growth of graded polynomial identities of sl_n
设K是特征为0的域，L是G-分次李PI-代数，其中G是有限群.定义了L的分次Gelfand-Kirillov维数。然后我们度量了n × n无迹矩阵sl_n（K）的李代数的Z_n-分次多项式恒等式的增长性，给出了它的Z_n-分次Gelfand-Kirillov维数的精确值.
Let $\mathbb{F}$ be a field of characteristic $p$, and let $UT_n(\mathbb{F})$ be the algebra of $n \times n$ upper triangular matrices over $\mathbb{F}$ with an involution of the first kind. In this paper we describe: the set of all $*$-central polynomials for $UT_n(\mathbb{F})$ when $n\geq 3$ and $p\neq 2$ ; the set of all $*$-polynomial identities for $UT_3(\mathbb{F})$ when $\mathbb{F}$ is infinite and $p>2$.
Higher polynomial identities for mutations of associative algebras
该研究针对结合代数上的突变乘积，简化了4次多项式恒等式的生成条件，确定5次恒等式仅需新增一个即可，并发现6次情形下存在多个新的恒等式。
In the paper we provide some polynomial identities for finite-dimensional algebras. A list of well known single polynomial identities is exposed and the classification of all $2$-dimensional algebras with respect to these identities is given.
Graded polynomial identities and exponential growth
设A是特征为零的域上被有限阿贝尔群G分次的有限维代数.本文通过计算A的分次余维序列的指数增长率，研究了与A所满足的分次多项式恒等式有关的增长函数。我们证明了A的G-指数存在，并且是一个与A的适当半单子代数的维数显式相关的整数。
Let $F$ be a finite field of $char F > 3$ and $sl_{2}(F)$ be the Lie algebra of traceless $2\times 2$ matrices over $F$. In this paper, we find a basis for the $\mathbb{Z}_{2}$-graded identities of $sl_{2}(F)$.
Some new operated Lie polynomial identities and Gröbner-Shirshov bases
本文证明Bremner-Elgendy分类中新的带算子结合多项式恒等式所对应的李多项式恒等式均为Gröbner-Shirshov，肯定回答了Guo等人提出的问题，推进了李代数背景下Rota纲领的发展。
Multialternating graded polynomials and growth of polynomial identities
设G是有限群，A是特征为零的域上的有限维G-分次代数.当A是单G-分次代数时，利用Regev中心多项式构造了A上任意大次数非零的多重交错分次多项式.作为结果，我们计算了满足普通多项式恒等式的任意G-分次代数的分次余维序列的指数增长率。特别地，我们证明它是一个整数。在G是阿贝尔的情况下，结果是已知的。
Let $F$ be a finite field of $char F > 3$ and $sl_{2}(F)$ be the Lie algebra of traceless $2\times 2$ matrices over $F$. This paper aims for the following goals: Find a basis for the $\mathbb{Z}_{2}$-graded identities of $sl_{2}(F)$; Find a basis for the $\mathbb{Z}_{3}$-graded identities of $sl_{2}(F)$ when $F$ contains a primitive 3rd root of one; Find a basis for the $\mathbb{Z}_{2}\times \mathbb{Z}_{2}$-graded identities of $sl_{2}(F)$.
We survey a vast array of known results and techniques in the area of polynomial identities in pointed Hopf algebras. Some new results are proven in the setting of Hopf algebras that appeared in papers of D. Radford and N. Andruskiewitsch - H.-J. Schneider.
This is a short review of some recent results obtained by the author. These results are related the problem of obtaining polynomial identities (computational formulas) for some matrix functions by means of the known polarization theorem, including the case of noncommutative variables and of determinant of space matrices.
Let H be the Taft algebra over a finite commutative ring R. When N is a unit in R, we show that all H-cleft extensions over R are determined up to H-comodule algebra isomorphism by their polynomial H-identities.
When k is an algebraically closed field of characteristic 0 and H is a non-semisimple monomial Hopf algebra, we show that all Galois objects over H are determined up to H-comodule algebra isomorphism by their polynomial H-identities, extending a previous result by Kassel.
Hopf algebras and polynomial identities
这是与E. Aljadeff和发表在Adv.Math.218（2008），1453-1495中。阐述了如何建立霍普夫代数上余代数的多项式恒等式理论，并着重讨论了由给定余代数所满足的恒等式所构造的泛余代数.所有的概念都用各种例子来说明。
Examples of polynomial identities distinguishing the Galois objects over finite-dimensional Hopf algebras
定义了霍普夫代数H上余模代数的多项式H-恒等式，并建立了相应的T-理想的一般性质.在H是Taft代数或霍普夫代数E（n）的情况下，我们给出了一个有限集的多项式H-恒等式，它能区分H上的Galois对象直到同构.
Cocharacters of polynomial identities of upper triangular matrices
给出了特征为零的域上上三角形矩阵代数的多项式恒等式的T-理想的余特征标序列的母函数的一个简便算法。应用该算法，我们发现了显式形式的多重性在两种情况下：（i）的“最大”分区;（ii）的矩阵的小尺寸和所有分区。
We give an algorithm which calculates the generating function of the cocharacter sequence of the polynomial identities of the algebra of upper block triangular (p+2q) x (p+2q) matrices over a field of characteristic zero with diagonal consisting of p copies of 1 x 1 and q copies of 2 x 2 matrices. We have found the explicit form of the multiplicities and their asymptotic behaviour for small values of p and q.
Algebras, dialgebras, and polynomial identities
这是一个调查的结合和非结合双代数理论的一些最新发展，重点是多项式的身份和多线性运算。我们讨论结合，李，约旦，和交错代数，以及相应的双代数;的KP算法转换成身份的双代数的身份; BSO算法转换成操作代数中的操作双代数;李和约旦三元系，以及相应的双系统;和一个非交换版本的李三元系的基础上三线性操作abc-bca。本文最后提出了一个与KP算法和BSO算法相关的猜想，并对进一步的研究提出了一些建议。大部分的原始成果是与劳尔·费利佩、路易斯·A.佩雷西和胡安娜·桑切斯·奥尔特加
Derivations and identities for Fibonacci and Lucas polynomials
本文引入了多项式代数的Fibonacci和Lucas导子的概念，证明了导子核的任何元素都定义了Fibonacci和Lucas多项式的多项式恒等式。证明了Appel多项式的任何多项式恒等式都能得到Fibonacci多项式和Lucas多项式的多项式恒等式，并给出了相应的交织映射。
Polynomial identities for quivers via incidence algebras
研究有向图的路径代数与矩阵代数的多项式恒等式，证明其PI等价性。
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