H. Ahmed, U. Bekbaev, I. Rakhimov
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.
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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.
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.
Finitely generated algebras with involution and their identities
考虑零特征域上具有对合的结合代数。证明了在这种情况下，对任意一个带对合的有限维生成结合代数，存在一个带对合的有限维代数，它满足与带对合的恒等式完全相同的恒等式.
On identities in Hom-Malcev algebras
在Hom-Malcev代数中，找到了一个等价于Hom-Malcev恒等式的恒等式。
Let A and B be finite dimensional simple real algebras with division gradings by an abelian group G. In this paper we give necessary and sufficient conditions for the coincidence of the graded identities of A and B. We also prove that every finite dimensional simple real algebra with a G-grading satisfies the same graded identities as a matrix algebra over an algebra D with a division grading that is either a regular grading or a non-regular Pauli grading. Moreover we determine when the graded identities of two such algebras coincide. For graded simple algebras over an algebraically closed fie…
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。
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.
We consider associative algebras with involution graded by a finite abelian group G over a field of characteristic zero. Suppose that the involution is compatible with the grading. We represent conditions permitting PI-representability of such algebras. Particularly, it is proved that a finitely generated (Z/qZ)-graded associative PI-algebra with involution satisfies exactly the same graded identities with involution as some finite dimensional (Z/qZ)-graded algebra with involution for any prime q or q = 4. This is an analogue of the theorem of A.Kemer for ordinary identities, and an extension …
Group identities on the units of algebraic algebras with applications to restricted enveloping algebras
一个代数称为GI-代数，如果它的单位群满足一个群恒等式。我们对以下两个未决问题提供积极支持。1.每个代数GI-代数都满足多项式恒等式吗？2.每个代数生成的GI-代数都是局部有限的吗？
Graded Identities of the Virasoro Algebra
本研究完成了对无限域上Virasoro代数分次多项式恒等式的分析，相关成果补充了Fidelis等人的前期研究。
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。
We study polynomial identities of algebras with adjoined external unit. For a wide class of algebras we prove that adjoining external unit element leads to increasing of PI-exponent precisely to 1. We also show that any real number from the interval [2,3] can be realized as PI-exponent of some unital algebra.
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 $A$.
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。
In 1970 Vaughan-Lee established that over an infinite field of characteristic two the ideal T[gl(2)] of all polynomial identities for the Lie algebra gl(2) is not finitely generated as a T-ideal. But a generating set for this ideal of polynomial identities was not found. We establish some generating set for the T-ideal T[gl(2)].
Some identities in algebras obtained by the Cayley-Dickson process
本文证明了Hall恒等式在所有由Cayley-Dickson过程得到的代数中成立，并且在一定条件下，Hall恒等式的逆也成立。
Some Graded Identities of The Cayley-Dickson Algebra
我们的工作是为八元数代数找到一个分次恒等式的基。我们这样做的$\mathbb{Z}_2^2$和$\mathbb{Z}_2^3$分级，他们都来自凯莱-迪克森过程，后分级是可能的，只有当标量的特征是不是两个。
Gelfand--Dorfman bialgebras (GD-algebras) are nonassociative systems with two bilinear operations satisfying a series of identities that express Hamiltonian property of an operator in the formal calculus of variations. The paper is devoted to the study of GD-algebras related with differential Poisson algebras. As a byproduct, we obtain a general description of identities that hold for operations $a\succ b = d(a)b$ and $a\prec b = ad(b)$ on a (non-associative) differential algebra with a derivation~$d$.
On the Graded Identities for Elementary Gradings in Matrix Algebras over Infinite Fields
我们找到了特征$p>0$的无限域$K$上的$n\times n$矩阵代数$M_n(K)$的$G$ -分次恒等式的一个基，使得中性分量对应于$M_n(K)$的对角线.
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。
We search an identity basis for the adjoint Lie algebra of the algebra $M_{1,1}(K)$ over a field, where $K$ is either the infinite generated Grassmann algebra $E$ or $E^1$, the variant of the algebra with $1$. In particular, we prove that in the case of an infinite base field of characteristic different from two the identities of $M_{1,1}(E^1)$ are exactly all the consequences of the identity $[x,y,[z,t],v]=0$. We also find an identity basis of $M_{1,1}(E)$ consisting of three identities.
Let $\mathbb F$ be an algebraically closed field, $G$ be an abelian group, and let $A$ and $B$ be arbitrary finite-dimensional $G$-graded simple algebras over $\mathbb F$. We prove that $A$ and $B$ are isomorphic if, and only if, they satisfy the same graded polynomial identities.
We consider associative algebras with involution over a field of characteristic zero. We proved that any algebra with involution satisfies the same identities with involution as the Grassmann envelope of some finite dimensional $Z_4$-graded algebra with graded involution. As a consequence we obtain the positive solution of the Specht problem for identities with involution: any associative algebra with involution over a field of characteristic zero has a finite basis of identities with involution. These results are analogs of theorems of A.R.Kemer for ordinary identities.
Complete solutions of functional identities $\sum_{k\in K}F_k(\bar{x}_m^k)x_k = \sum_{l\in L}x_lG_l(\bar{x}_m^l)$ on the matrix algebra $M_n(\mathbb{F})$ are given. The nonstandard parts of these solutions turn out to follow from the Cayley-Hamilton identity.
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。
Let $K$ be an infinite field of characteristic different from two and let $U_1$ be the Lie algebra of the derivations of the algebra of Laurent polynomials $K[t,t^{-1}]$. The algebra $U_1$ admits a natural $\mathbb{Z}$-grading. We provide a basis for the graded identities of $U_1$ and prove that they do not admit any finite basis. Moreover, we provide a basis for the identities of certain graded Lie algebras with a grading such that every homogeneous component has dimension $\leq 1$, if a basis of the multilinear graded identities is known. As a consequence of this latter result we are able to…
Let $P_n=k[x_1,x_2,\ldots,x_n]$ be the polynomial algebra over a field $k$ of characteristic zero in the variables $x_1,x_2,\ldots,x_n$ and $\mathscr{L}_n$ be the left-symmetric Witt algebra of all derivations of $P_n$. We describe all right operator identities of $\mathscr{L}_n$ and prove that the set of all algebras $\mathscr{L}_n$, where $n\geq 1$, generates the variety of all left-symmetric algebras. We also describe a class of general (not only right operator) identities for $\mathscr{L}_n$.
Novikov algebras in low dimension: identities, images and codimensions
本文研究复数域上二维Novikov代数的多项式恒等式，确定其T-理想极小生成集与相对自由代数基，证明多项式恒等式可区分非结合二维Novikov代数，得出余维数序列至多线性增长并描述多重线性像。
The asymptotic behaviour is studied of exponentially bounded sequences of codimensions of identities of algebras with unity. A series of algebras is constructed for which the base of the exponential increases by exactly one when an outer unity is adjoined to the original algebra. We show that the PI-exponents of unital algebras can take any value greater than two, and the exponents of finite-dimensional unital algebras form a dense subset of the domain [2,$\infty$).
On the trace identities of 2-dimensional tr-prime algebras over a finite field
本文研究了带迹映射的结合代数，在任意域上分类了二维迹素代数，并在有限域情形下为每个代数给出了迹多项式恒等式 T-理想的有限基。
Consider a representation $ρ\colon L \to \mathfrak{gl}(V)$ where $L$ is a Lie algebra and $V$ is a finite dimensional vector space. We prove the analog of Amitsur's conjecture on asymptotic behavior for codimensions of polynomial identities of $ρ$.
We study identities of finite dimensional algebras over a field of characteristic zero, graded by an arbitrary groupoid $Γ$. First we prove that its graded colength has a polynomially bounded growth. For any graded simple algebra $A$ we prove the existence of the graded PI-exponent, provided that $Γ$ is a commutative semigroup. If $A$ is simple in a non-graded sense the existence of the graded PI-exponent is proved without any restrictions on $Γ$.
Let L be a restricted Lie algebra over a field of characteristic p > 2 and denote by u(L) its restricted enveloping algebra. We determine the conditions under which the set of symmetric elements of u(L) with respect to the principal involution is Lie solvable, Lie nilpotent, or bounded Lie Engel.
Let $F$ be a finite field with characteristic $p > 2$ and let $G$ be the unitary Grassmann algebra generated by an infinite dimensional vector space $V$ over $F$. In this paper, we determine a basis of the $\mathbb{Z}_{2}$-graded polynomial identities for any non-trivial $\mathbb{Z}_{2}$-grading such that a basis of $V$ is homogeneous in this grading.
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。
Polynomial identities for tangent algebras of monoassociative loops
本文通过引入n次Sabinin代数并利用交换子、结合子与四元子的多项式恒等式刻画4次情形，构造了满足单结合性a^2 a = a a^2的解析环切空间上的自然代数结构，并给出一个不能由低次恒等式推出的5次恒等式。
$2$-Graded Identities for the Tensor Square of the Grassmann Algebra
考虑无限域上的代数$E\otimes E$，它具有$\mathbb{Z}_2$ -分次，其中正则基是齐次的，并证明了在各种情况下分次恒等式都是普通恒等式.如果分级是一个非典型的分级作为商分级的自然$\mathbb{Z}_2\times\mathbb{Z}_2$ -分级，我们展示了一个基础的分级身份。
Let $K$ be any field of characteristic two and let $U_1$ and $W_1$ be the Lie algebras of the derivations of the algebra of Laurent polynomials $K[t,t^{-1}]$ and of the polynomial ring $K[t]$, respectively. The algebras $U_1$ and $W_1$ are equipped with natural $\mathbb{Z}$-gradings. In this paper, we provide bases for the graded identities of $U_1$ and $W_1$, and we prove that they do not admit any finite basis.
Let $F$ be a finite field with the characteristic $p > 2$ and let $G$ be the unitary Grassmann algebra generated by an infinite dimensional vector space $V$ over $F$. In this paper, we determine a basis for $\mathbb{Z}_{2}$-graded polynomial identities for any non-trivial $\mathbb{Z}_{2}$-grading such that its underlying vector space is homogeneous.
Let P a locally finite partially ordered set, F a field, G a group, and I(P,F) the incidence algebra of P over F. We describe all the inequivalent elementary G-gradings on this algebra. If P is bounded, F is a infinite field of characteristic zero, and A, B are both elementary G-graded incidence algebras satisfying the same G-graded polynomial identities, and the automorphisms group of P acts transitively on the maximal chains of P , we show that A and B are graded isomorphic.
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].
Identities for subspaces of the Weyl algebra
研究任意特征无限域上 Weyl 代数 A_1 的特定子空间，给出其四次多项式恒等式的刻画。
Graded polynomial identities, group actions, and exponential growth of Lie algebras
考虑特征为0的域上的有限维李代数L具有有限群G的作用。我们证明了类似于Amitsur猜想的关于L的多项式G-恒等式的余维数的渐近性质。作为结果，我们证明了类似Amitsur猜想的分次余维数的任何有限维李代数分次由有限阿贝尔群。
Exploring Functional Identities: From Division Rings to Matrix Algebras
本文通过严格分析与定理证明，解决了Ferreira等人提出的未决问题，深入探究了除环上矩阵代数中加性函数的复杂行为，并揭示了其对代数结构的影响。
Superpolynomial identities of finite-dimensional simple algebras
研究有限维Z2-分次单代数的Grassmann包络，刻画其超多项式恒等式，并分类具有相同超多项式恒等式的代数条件，同时推广至Z2-代数并描述其分次单代数。
In 2013 Benkart, Lopes and Ondrus introduced and studied in a series of papers the infinite-dimensional unital associative algebra $\A_h$ generated by elements $x,y,$ which satisfy the relation $yx-xy=h$ for some $0\neq h\in \FF[x]$. We generalize this construction to $\A_h(\B)$ by working over the fixed $\FF$-algebra $\B$ instead of $\FF$. We describe the polynomial identities for $\A_h(\B)$ over the infinite field $\FF$ in case $h\in\B[x]$ satisfies certain restrictions.
We describe the weak polynomial identities of the Jordan algebra of symmetric $2\times 2$ matrices over a field of characteristic zero. The corresponding weak verbal ideal is generated by the standard identity of degree four and the metabelian identity.
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 $p>2$.
Multihomogeneous Normed Algebras and Polynomial Identities
本文考虑$\R$或$\C$上的PI-代数$A$.众所周知，一般来说，这样的代数不是赋范代数。事实上，存在一个幂零的交换代数，它不是一个赋范代数，见[1]。在这里，我们解决的问题是，是否有可能找到一个赋范PI-代数$B$具有相同的多项式恒等式作为$A$，而且，是否有一些Banach PI-代数具有这一性质。我们的主要定理为这个问题提供了一个肯定的答案，此外，我们还证明了存在一个Banach代数具有相同的多项式身份的$A$。作为一个副产品，我们证明了如果$A$是一个赋范PI-代数，它的完成是零，那么$A$是幂零的。通过引入多重齐次范数的概念，作为我们主要结果的一个应用，我们得到：如果$\FX$是多重齐次赋范代数，$A$是PI-代数，使得商空间$\FX/Id(A)$的完备化为零，则$A$是幂零的.这两个应用程序都是[4]中启动的研究的扩展。
Multipliers, $W$-algebras and the growth of generalized polynomial identities
研究W-代数的广义恒等式理论，证明k×k矩阵代数生成的广义变种具有近多项式增长，并反驳了零特征下的Specht性质。
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。
On the special identities of Gelfand--Dorfman algebras
本文证明了特殊 Gelfand--Dorfman 代数在同态下封闭并构成簇，且所有 2 维 GD-algebra 均特殊，同时提出寻找特殊恒等式的方法并计算了次数为 6 的 Gröbner 基分量。
Graded identities for matrix algebras of order two over a finite field
本文确定了有限域上二阶矩阵代数 $M_2(\mathbb{F})$ 在所有可能群分次下的分次多项式恒等式的 $T_G$-理想基，基由有限个非平凡分次恒等式组成，且当群有限时基有限。
In this paper we study matrix algebras with a degenerate trace in the framework of the theory of polynomial identities. The first part is devoted to the study of the algebra $D_n$ of $n \times n$ diagonal matrices. We prove that, in case of a degenerate trace, all its trace identities follow by the commutativity law and by pure trace identities. Moreover we relate the trace identities of $D_{n+1}$ endowed with a degenerate trace, to those of $D_n$ with the corresponding trace. This allows us to determine the generators of the trace T-ideal of $D_3$. In the second part we study commutative suba…
Graded identities of the first Weyl algebra and its generalizations
研究无限域上第一Weyl代数W1的分次多项式恒等式，构造其Z-分次恒等式的基（仅含一个恒等式），并推广到多种代数，如分次Galois环、量子Weyl代数、量子平面及sl_2的泛包络代数。
Let $K$ be a field of characteristic zero and let $\mathfrak{M}_5$ be the variety of associative algebras over $K$, defined by the identity $[x_1,x_2][x_3,x_4,x_5]$. It is well-known that such variety is a minimal variety and that is generated by the algebra $$A=\begin{pmatrix} E_0 & E\\ 0 & E\\ \end{pmatrix},$$ where $E=E_0\oplus E_1$ is the Grassmann algebra. In this paper, for any positive integer $k$, we describe the polynomial identities of the relatively free algebras of rank $k$ of $\mathfrak{M}_5$, \[F_k(\mathfrak{M}_5)=\dfrac{K\langle x_1,\dots, x_k \rangle}{K\langle x_1,\dots, x_k \r…
Numerical characteristics of polynomial identities of left nilpotent algebras are examined. Previously, we came up with a construction which, given an infinite binary word, allowed us to build a two-step left nilpotent algebra with specified properties of the codimension sequence. However, the class of the infinite words used was confined to periodic words and Sturm words. Here the previously proposed approach is generalized to a considerably more general case. It is proved that for any algebra constructed given a binary word with subexponential function of combinatorial complexity, there exis…
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