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.
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.
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
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.
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.
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.
Let $K$ be an infinite integral domain and $M_{n}(K)$ be the algebra of all $n\times n$ matrices over $K$. This paper aims for the following goals: Find a basis for the graded identities for elementary grading in $M_{n}(K)$ when the neutral component and diagonal coincide; Describe the $\mathbb{Z}_{p}$-graded central polynomials of $M_{p}(K)$ when $p$ is a prime number; Describe the $\mathbb{Z}$-graded central polynomials of $M_{n}(K)$.
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 $ρ$.
Let K be a field of characteristic 0 and L be a G-graded Lie PI-algebra, where G is a finite group. We define the graded Gelfand-Kirillov dimension of L. Then we measure the growth of the Z_n-graded polynomial identities of the Lie algebra of n x n traceless matrices sl_n(K) giving an exact value of its Z_n-graded Gelfand-Kirillov dimension.
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 $p>2$.
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.
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 $sl_{2}(F)$.
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.
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.
