Coxeter elements and periodic Auslander-Reiten quiver

In this paper we show that for a simply-laced root system a choice of a Coxeter element C gives rise to a natural construction of the Dynkin diagram, in which vertices of the diagram correspond to C-orbits in R; moreover, it gives an identification of R with a certain subset over(I, ̂) of I × Z_{2 h}, where h is the Coxeter number. The set over(I, ̂) has a natural quiver structure; we call it the periodic Auslander-Reiten quiver. This gives a combinatorial construction of the root system associated with the Dynkin diagram I: roots are vertices of over(I, ̂), and the root lattice and the inner product admit an explicit description in terms of over(I, ̂). Finally, we relate this construction to the theory of quiver representations.