Related papers: Graded identities of block-triangular matrices

We give a closed formula for the graded decomposition numbers of the blob algebra over a field of characteristic zero at a root of unity.

Let $\mathbb{F}$ be a field. Denote by $t_n(\mathbb{F})$ the greatest possible dimension for a vector space of $n$-by-$n$ matrices over $\mathbb{F}$ in which every element is triangularizable over $\mathbb{F}$. It was recently proved that…

We study the structure of a $3-$Leibniz algebra $T$ graded by an arbitrary abelian group $G,$ which is considered of arbitrary dimension and over an arbitrary base field $\bbbf.$ We show that $T$ is of the form $T=\uu\oplus\sum_jI_j,$ with…

We study algebras k[x_1,...,x_n]/I which admit a grading by a subsemigroup of N^d such that every graded component is a one-dimensional k-vector space. V.I.~Arnold and coworkers proved that for d = 1 and n <= 3 there are only finitely many…

The goal of the paper is to give a complete description of the images of noncommutative polynomials with zero constant term on upper triangular matrix algebras over an algebraically closed field. This is a variation of the old and famous…

For each commutative, graded algebra with finite dimension in each degree, we construct a graded cohomology theory for graphs whose graded Euler characteristic is the chromatic polynomial of the graph. This extends our previous work which…

In [2], an exhaustive construction is achieved for the class of all 4-dimensional unital division algebras over finite fields of odd order, whose left nucleus is not minimal and whose automorphism group contains Klein's four-group. We…

For $0<\alpha\le 1$, we say that a sequence $(X_k)_{k>0}$ of $d$-regular graphs has property $D_\alpha$ if there exists a constant $C>0$ such that $\mathrm{diam}(X_k)\ge C\cdot|X_k|^\alpha$. We investigate property $D_\alpha$ for arithmetic…

We consider certain quotient algebras of tensor algebras of bimodules $M$ over a finite-dimensional algebra $R$, and we investigate Frobenius type properties of such algebras. Our main interest is in the case where $M=R^*$, the linear dual…

Graded-division algebras are building blocks in the theory of finite-dimensional associative algebras graded by a group G. If G is abelian, they can be described, using a loop construction, in terms of central simple graded-division…

We define a new grading, that we call the "level grading", on the algebra of polynomials generated by the derivatives $u_{k+i}=\partial^{k+i}u/\partial x^{k+i}$ over the ring $K^{(k)}$ of $C^{\infty}$ functions of $u,u_1,...,u_k$. This…

The "2-variable general-$\lambda$-matrix polynomials (2VG$\lambda$MP)" is a new family of matrix polynomials, introduced and studied in this article. These matrix polynomials are constructed using umbral and symbolic methods. We delve into…

We consider quotients of the unit cube semigroup algebra by particular $\mathbb{Z}_r\wr S_n$-invariant ideals. Using Gr\"obner basis methods, we show that the resulting graded quotient algebra has a basis where each element is indexed by…

We describe a connection between the subjects of cluster algebras, polynomial identity algebras and discriminants. For this, we define the notion of root of unity quantum cluster algebras and prove that they are polynomial identity…

Let $n>1$ and let $R$ be a commutative ring with identity $1\ne 0$ and $R[x_1,\ldots,x_n]^n$ the set of all $n$-tuples of polynomials of the form $(f_1,\ldots,f_n),$ where $f_1,\ldots,f_n\in R[x_1,\ldots,x_n]$. We call these $n$-tuples…

Denote by C_{n,d} the nilpotency degree of a relatively free algebra generated by d elements and satisfying the identity x^n=0. Under assumption that the characteristic p of the base field is greater than n/2, it is shown that…

In this short note, we describe the so-called homogeneous involution on finite-dimensional graded-division algebra over an algebraically closed field. We also compute their graded polynomial identities with involution. As pointed out by L.…

For an infinite family of monogenic trinomials $P(X) = X^3\pm 3rbX-b$ in $\mathbb{Z}\lbrack X\rbrack$, arithmetical invariants of the cubic number field $L = \mathbb{Q}(\theta)$, generated by a zero $\theta$ of $P(X)$, and of its Galois…

Classifying Frobenius algebras is a key question that has been addressed in various contexts. The structure of finite-dimensional Frobenius algebras depends on the base field and the dimension of the algebra, leading to different…

Let A be a finitely generated connected graded k-algebra defined by a finite number of monomial relations. Then there is a finite directed graph, Q, the Ufnarovskii graph of A, for which the categories QGr(A) and QGr(kQ) are equivalent:…