Related papers: Arithmetic of algebraic groups
Braid groups are an important and flexible tool used in several areas of science, such as Knot Theory (Alexander's theorem), Mathematical Physics (Yang-Baxter's equation) and Algebraic Geometry (monodromy invariants). In this note we will…
This paper classifies the derivations of group algebras in terms of the generators and defining relations of the group. If $RG$ is a group ring, where $R$ is commutative and $S$ is a set of generators of $G$ then necessary and sufficient…
Intended for mathematical physicists interested in applications of the division algebras to physics, this article highlights some of their more elegant properties with connections to the theories of Galois fields and quadratic residues.
Replacing finite groups by linear algebraic groups, we study an algebraic-geometric counterpart of the theory of free profinite groups. In particular, we introduce free proalgebraic groups and characterize them in terms of embedding…
The aim of the paper is to discuss the relations between the three kinds of objects named in the title. In a sense, this is a survey of such relations; however, some new directions are also considered. This relates, especially, to sections…
Finite fields form an important chapter in abstract algebra, and mathematics in general. We aim to provide a geometric and intuitive model for finite fields, involving algebraic numbers, in order to make them accessible and interesting to a…
An algebraic technique adapted to the problems of the fundamental theoretical physics is presented. The exposition is an elaboration and an extension of the methods proposed in previous works by the aut
The survey contains a brief description of the ideas, constructions, results, and prospects of the theory of hypergroups and generalized translation operators. Representations of hypergroups are considered, being treated as continuous…
The classification of gradings by abelian groups on finite direct sums of simple finite-dimensional nonassociative algebras over an algebraically closed field is reduced, by means of the use of loop algebras, to the corresponding problem…
We establish several finiteness properties of groups defined by algebraic difference equations. One of our main results is that a subgroup of the general linear group defined by possibly infinitely many algebraic difference equations in the…
We show that all spin groups of non-definite, quinary quadratic forms over a field with characteristic 0 can be represented as 2 by 2 matrices with entries in an associated quaternion algebra. Over local and global fields, we further study…
First, we study the linear equations in general. Second, we focus our attention in periodic sequences over finite fields and de Bruijn directed graph.
We introduce the notion of ends for algebras. The definition is analogous to the one in geometric group theory. We establish some relations to growth conditions and cyclic cohomology.
This article is a short introduction to the theory of the groups of points of elliptic curves over finite fields. It is concerned with the elementary theory and practice of elliptic curves cryptography, the new generation of public key…
In this paper we study Cartan subalgebras in general and special linear algebras over a field of positive characteristic. We determined the conjugacy classes of Cartan subalgebras under the general linear groups, and count the explicit…
Algebraic statistics uses tools from algebra (especially from multilinear algebra, commutative algebra and computational algebra), geometry and combinatorics to provide insight into knotty problems in mathematical statistics. In this survey…
Tropical algebraic geometry is an active new field of mathematics that establishes and studies some very general principles to translate algebro-geometric problems into purely combinatorial ones. This expository paper gives an introduction…
A complete classifications, up to isomorphism, of two-dimensional associative and diassociative algebras over any basic field are given.
Passing from arithmetic schemes to algebraic schemes, in a similar manner we will have the computation of the \'etale fundamental group of an algebraic scheme and then will define and discuss the qc fundamental group of an algebraic scheme…
The work is devoted to the variety of $2$-dimensional algebras over an algebraically closed field. Firstly, we classify such algebras modulo isomorphism. Then we describe the degenerations and the closures of principal algebra series in the…