Kruglyak S. Coxeter functors for Representations of Quivers, Algebras and Categories in Hilbert spaces

The object of study are representations of associative algebras and quivers in the category of finite-dimensional linear and Hilbert spaces. The goal of study are criterions of finiteness and wildness of type of the object under research and algorithms of construction of its indecomposable representations. During this study methods of linear algebra and, in particular, of the theory of matrix problems, methods of the theory of representations of algebras and quivers are employed. The criterion of the finiteness of a type of a finite-dimensional algebra, which squared radical equals zero, is obtained. The relation of majorization on categories on the complexity of the category of representations is introduced. An algorithm of differentiation of quivers and Coxeter functors are constructed. The theorem, analogous to Gabriel's theorem, is proved.