Related papers: Lattices and Parameter Reduction in Division Algeb…

Let $A$ be a finite-dimensional division algebra containing a base field $k$ in its center $F$. We say that $A$ is defined over a subfield $F_0$ of $F$ if $A = A_0\otimes_{F_0} F$ for some $F_0$-subalgebra $A_0$ of $A$. We show that: (1) In…

A $k$-regular graph is called a divisible design graph (DDG for short) if its vertex set can be partitioned into $m$ classes of size $n$, such that two distinct vertices from the same class have exactly $\lambda_1$ common neighbors, and two…

We present algorithms for parametrizing by radicals an irreducible curve, not necessarily plane, when the genus is less o equal to 4 and they are defined over an algebraically closed field of characteristic zero. In addition, we also…

Let $k$ be a field containing an algebraically closed field of characteristic zero. If $G$ is a finite group and $D$ is a division algebra over $k$, finite dimensional over its center, we can associate to a faithful $G$-grading on $D$ a…

Lie group theory states that knowledge of a $m$-parameters solvable group of symmetries of a system of ordinary differential equations allows to reduce by $m$ the number of equations. We apply this principle by finding some \emph{affine…

Let $R$ be a finite-dimensional algebra over an algebraically closed field $F$ graded by an arbitrary group $G$. We prove that $R$ is a graded division algebra if and only if it is isomorphic to a twisted group algebra of some finite…

We introduce the notion of radical parametrization of a surface, and we provide algorithms to compute such type of parametrizations for families of surfaces, like: Fermat surfaces, surfaces with a high multiplicity (at least the degree…

Let $\K$ be an algebraic number field of degree $d$ and discriminant $\Delta$ over $\Q$. Let $\A$ be an associative algebra over $\K$ given by structure constants such that $\A\cong M_n(\K)$ holds for some positive integer $n$. Suppose that…

A graded-division algebra is an algebra graded by a group such that all nonzero homogeneous elements are invertible. This includes division algebras equipped with an arbitrary group grading (including the trivial grading). We show that a…

Consider a one-parameter family of algebraic varieties degenerating to a reducible one. Our main result is a formula for the fundamental cycle of the limit subscheme of any family of effective Cartier divisors. The formula expresses this…

We completely classify Cartan subalgebras of dimension drop algebras with coprime parameters. More generally, we classify Cartan subalgebras of arbitrary stabilised dimension drop algebras that are non-degenerate in the sense that the…

Let K be an algebraic number field of degree d and discriminant D over Q. Let A be an associative algebra over K given by structure constants such that A is isomorphic to the algebra M_n(K) of n by n matrices over K for some positive…

This paper studies a problem of Erd\"{o}s concerning lattice cubes. Given an $N \times N \times N$ lattice cube, we want to find the maximum number of vertices one can select so that no eight corners of a rectangular box are chosen…

We consider Artinian level algebras arising from the whiskering of a graph. Employing a result by Dao-Nair we show that multiplication by a general linear form has maximal rank in degrees 1 and $n-1$ when the characteristic is not two,…

Let $k$ be a field of characteristic not two or three. We classify up to isomorphism all finite-dimensional Lie superalgebras $\mathfrak{g}=\mathfrak{g}_0\oplus \mathfrak{g}_1$ over $k$, where $\mathfrak{g}_0$ is a three-dimensional simple…

Let $p$ be a prime integer, $1\leq s\leq r$ integers, $F$ a field of characteristic $p$. Let $\cat{Dec}_{p^r}$ denote the class of the tensor product of $r$ $p$-symbols and $\cat{Alg}_{p^r,p^s}$ denote the class of central simple algebras…

In this paper we study the lattice of restricted subalgebras of a restricted Lie algebra. In particular, we consider those algebras in which this lattice is dually atomistic, lower or upper semimodular, or in which every restricted…

Let $D$ be a division ring and $K$ a subfield of $D$ which is not necessarily contained in the center $F$ of $D$. In this paper, we study the structure of $D$ under the condition of left algebraicity of certain subsets of $D$ over $K$.…

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…

We generalize Amitsur's construction of central simple algebras over a field $F$ which are split by field extensions possessing a derivation with field of constants $F$ to nonassociative algebras: for every central division algebra $D$ over…