In Chapter VIII of Volume 1, we have started to study the Auslander– Reiten quiver Γ(mod A) of any hereditary K-algebra A of Euclidean type,

that is, the path algebra A = KQ of an acyclic quiver Q whose underlying graph Q is one of the Euclidean diagrams Ãm, with m ≥ 1, Dm,, with m ≥ 4,E6,, E7,7, and E 8,8. We recall that any such an algebra A is representationinfinite.

We have shown in (VIII.2.3) that the quiver Γ(modA) contains a unique postprojective component ρ(A) containing all the indecomposable projective A-modules, a unique preinjective component Q(A) containing all the indecomposable injective A-modules, and the family R(A) of the remaining components being called regular (see (VIII.2.12)). This means that Γ(mod A) has the disjoint union form

Γ(modA) = P(A)?R(A)⊂ Q(A).

The indecomposable modules in R(A) are called regular. We have shown in (VIII.4.5) that there is a similar structure of Γ(modB), for any concealed algebra B of Euclidean type, that is, the endomorphism algebra

B = End TA

of a postprojective tilting module TA over a hereditary algebra A = KQ of Euclidean type. The algebra B is representation-infinite.

The objective of Chapters XI-XIII is to describe the structure of regular components of the Auslander-Reiten quiver γ(modB) of any concealed algebra B of Euclidean type.

We introduce in this chapter a special type of a translation quiver, which we call a stable tube. The main aim of Section 1 is to describe special properties of irreducible morphisms between indecomposable modules in stable tubes of the Auslander Reiten quiver Γ(mod B) of an algebra B and their compositions with arbitrary homomorphisms in the module category modB.