Coxeter Functors and Gabriel's Theorem

It has recently become clear that a whole range of problems of linear algebra can be formulated in a uniform way, and in this common formu-lation there arise general effective methods of investigating such problems. It is interesting that these methods turn out to be connected with such ideas as the Coxeter—Weyl group and the Dynkin diagrams. We explain these connections by means of a very simple problem. We assume no preliminary knowledge. We do not touch on the connections between these questions and the theory of group representations or the theory of infinite—dimensional Lie algebras. For this see 3—5. Let Γ be a finite connected graph; we denote the set of its vertices by Γ ο and the set of its edges by ΓΊ (we do not exclude the cases where two vertices are joined by several edges or there are loops joining a vertex to itself). We fix a certain orientation Λ of the graph Γ; this means that for each edge / e Γι we distinguish a starting-point a(/) e Γ ο and an end-point With each vertex a G Γ ο we associate a finite-dimensional linear space