Related papers: Geometry of word equations in simple algebraic gro…

We present a survey of results on word equations in simple groups, as well as their analogues and generalizations, which were obtained over the past decade using various methods, group-theoretic and coming from algebraic and arithmetic…

We give a brief survey of recent results on word maps on simple groups and polynomial maps on simple associative and Lie algebras. Our focus is on parallelism between these theories, allowing one to state many new open problems and giving…

The paper is a short survey of recent developments in the area of word maps evaluated on groups and algebras. It is aimed to pose questions relevant to Kac--Moody theory.

We study dynamical systems arising from word maps on simple groups. We develop a geometric method based on the classical trace map for investigating periodic points of such systems. These results lead to a new approach to the search of…

We extend Borel's theorem on the dominance of word maps from semisimple algebraic groups to some perfect groups. In another direction, we generalize Borel's theorem to some words with constants. We also consider the surjectivity problem for…

Algebraic geometry for groups and Lie algebraic has been recently defined and studied by many authors on the purpose to study set defined by algebraic equations on abstract groups and Lie algebras. The purpose of this paper is to present a…

We develop a new tool, namely polynomial and linear algebraic methods, for studying systems of word equations. We illustrate its usefulness by giving essentially simpler proofs of several hard problems. At the same time we prove extensions…

This paper focuses on the derivations and automorphism groups of certain finite-dimensional associative algebras over the field of complex numbers. Using classification results for algebras of dimensions two, three, and four, along with…

This thesis studies arithmetic of linear algebraic groups. It involves studying the properties of linear algebraic groups defined over global fields, local fields and finite fields, or more generally the study of the linear algebraic groups…

In this short survey we describe recent advances on word equations with non-rational constraints in groups and monoids, highlighting the important role that formal languages play in this area.

This is a survey article on some recent developments in the arithmetic theory of linear algebraic groups over higher-dimensional fields, written for the Notices of the AMS.

We study topological group theoretic properties of algebraic groups over local fields. In particular, we find conditions under which such groups have closed images under arbitrary continuous homomorphisms into arbitrary topological groups.

We discuss some recent results by a number of authors regarding word maps on algebraic groups and finite simple groups, their mixing properties and the geometry of their fibers, emphasizing the role played by equidistribution results in…

We consider word maps and word maps with constants on a simple algebraic group. We present results on the images of such maps, in particular, we prove a theorem on the dominance of general word maps with constants, which can be viewed as an…

We determine the structure of the semisimple group algebra of certain groups over the rationals and over those finite fields where the Wedderburn decompositions have the least number of simple components. We apply our work to obtain similar…

On the one hand the algebras of linear operators here act on finite-dimensional vector spaces, and on the other hand the point of view is generally an analysts'. Also, one might think of algebras as being used to add more data to basic…

This paper presents an alternative approach to simplify the proofs of some important results related to polynomial mappings in Computational Algebraic Geometry such as Polynomial Implicitization, Image Closure and some properties of the…

In this paper, we study group equations with occurrences of automorphisms. We describe equational domains in this class of equations. Moreover, we solve a number of open problem posed in universal algebraic geometry.

The group gradings on the symmetric composition algebras over arbitrary fields are classified. Applications of this result to gradings on the exceptional simple Lie algebras are considered too.

We study groups, exponential groups and ordered groups equipped with valuations. We investigate algebraic and topological features of such valued structures, and apply our findings in order to solve regular equations over groups using…