Matrix problems and noexpand representations of algebras

Abstract

This paper is devoted to the theory of matrix problems, a new branch of modern algebra created and developed to a large extent by the Kyiv algebraic school. It originated from the questions of the theory of representations, but now has proved its efficiency in many areas, such as algebraic geometry, algebraic topology, linear algebra, theory of groups etc.

Certainly, I could not embraced all achievements or even all directions of investigation, so their choice in the paper is rather subjective. Moreover, I only consider the <<classical results>>, not involving the new investigations and applications to algebraic geometry and algebraic topology (see surveys~\cite{cm,top}).