Related papers: Bivariate polynomial mappings associated with simp…

We study central extensions E of elementary abelian 2-groups by elementary abelian 2-groups. Associated to such an extension is a quadratic map which determines the extension uniquely. The components of the map determine a quadratic ideal…

Two proper polynomial maps $f_1, f_2 \colon \mathbb{C}^2 \longrightarrow \mathbb{C}^2$ are said to be \emph{equivalent} if there exist $\Phi_1, \Phi_2 \in \textrm{Aut}(\mathbb{C}^2)$ such that $f_2=\Phi_2 \circ f_1 \circ \Phi_1$. We…

A subspace of an algebra with involution is called a Lie skew-ideal if it is closed under Lie products with skew-symmetric elements. Lie skew-ideals are classified in central simple algebras with involution (there are eight of them for…

This article is the second in the series and is devoted to the type G_2. The work consists of two parts. In the first part we calculate the structure constants of the complex simple Lie algebra of type G_2. All structure constants are…

Bicommutative algebras are nonassociative algebras satisfying the polynomial identities of right- and left-commutativity (xy)z=(xz)y and x(yz)=y(xz). We study subvarieties of the variety of all bicommutative algebras over a field of…

The variety of bicommutative algebras is the class of all nonassociative algebras satisfying the polynomial identities $(x_1x_2)x_3=(x_1x_3)x_2$ and $x_1(x_2x_3)=x_2(x_1x_3)$. In this paper we provide a complete description of varieties of…

In this article, the matrix elements of a representation of the 5-dimensional Lie algebra K5 are obtained for the first time. The bivariate degenerate Hermite polynomials Hm(z1, z2|{\tau} ) are considered within the context of this…

Fomin and Zelevinsky show that a certain two-parameter family of rational recurrence relations, here called the (b,c) family, possesses the Laurentness property: for all b,c, each term of the (b,c) sequence can be expressed as a Laurent…

We study a family of bivariate orthogonal polynomials associated to the deltoid curve. These polynomials arise when classifying bivariate diffusion operators that have discrete spectral decomposition given by orthogonal polynomials with…

Some polynomials $P$ with rational coefficients give rise to well defined maps between cyclic groups, $\Z_q\longrightarrow\Z_r$, $x+q\Z\longmapsto P(x)+r\Z$. More generally, there are polynomials in several variables with tuples of rational…

Polynomial relations for generators of $su(2)$ Lie algebra in arbitrary representations are found. They generalize usual relation for Pauli operators in spin 1/2 case and permit to construct modified Holstein-Primakoff transformations in…

Let $\mathbf{O}(\mathbb{F})$ be the split octonion algebra over an algebraically closed field $\mathbb{F}$. For positive integers $k_1, k_2\geq 2$, we study surjectivity of the map $A_1(x^{k_1}) + A_2(y^{k_2}) \in…

A non-zero constant Jacobian polynomial map $F=(P,Q):\mathbb{C}^2 \longrightarrow \mathbb{C}^2$ has a polynomial inverse if the component $P$ is a simple polynomial, i.e. if, when $P$ extended to a morphism $p:X\longrightarrow \mathbb{P}^1$…

Motivated by many recent constructions of permutation polynomials over $\mathbb{F}_{q^2}$, we study permutation polynomials over $\mathbb{F}_{q^3}$ in terms of their coefficients. Based on the multivariate method and resultant elimination,…

We reexamine different examples of reduction chains $\mathfrak{g} \supset \mathfrak{g}'$ of Lie algebras in order to show how the polynomials determining the commutant with respect to the subalgebra $\mathfrak{g}'$ leads to polynomial…

Let $f,g\in\mathbb{C}[z]\setminus\mathbb{C}$ and $c\in\mathbb{C}[z]$. Suppose that $\mathrm{deg}(c)=1$ if $\mathrm{deg}(f)=\mathrm{deg}(g)=1$. Using the theory of Presburger arithmetic, we prove that the rank-two recurrence set…

Bivariate P-polynomial association scheme of type $(\alpha,\beta)$ are defined as a generalization of the P-polynomial association schemes. This generalization is shown to be equivalent to a set of conditions on the intersection parameters.…

We construct certain integral structures for the cores of reduced tame extended affine Lie algebras of rank at least 2. One of the main tools to achieve this is a generalization of Chevalley automorphisms in the context of extended affine…

In this article, the 2-iterated q-Appell family is introduced. Certain 2-iterated q-Appell and mixed type q-special polynomials are considered as members of this family. The numbers related to these polynomials are obtained. The determinant…

For each 3-dimensional non-Lie Leibniz algebra over the complex numbers, we describe the algebra of polynomial invariants and determine its group of automorphisms. As a consequence, we establish that any two non-nilpotent 3-dimensional…