Distinguished Bases of Exceptional Modules

Abstract

An indecomposable representation M of a quiver Q = (Q 0 , Q 1 ) is said to be exceptional provided Ext 1 (M, M ) = 0. And it is called a tree module provided one can choose a set B of bases of the vector spaces M x (x ∈ Q 0 ) such that the coefficient quiver Γ(M, B) is a tree quiver; we call B a tree basis of M . It is known that exceptional modules are tree modules. A tree module usually has many tree bases and the corresponding coefficient quivers may look quite differently. The aim of this note is to introduce a class of indecomposable modules which have a distinguished tree basis, the "radiation modules" (generalizing an inductive construction considered already by Kinser). For a Dynkin quiver, nearly all indecomposable representations turn out to be radiation modules, the only exception is the maximal indecomposable module in case E 8 . Also, the exceptional representations of the generalized Kronecker quivers are given (via the universal cover) by radiation modules. Consequently, with the help of Schofield induction one can display all the exceptional modules of an arbitrary quiver in a nice way.Let Q = (Q 0 , Q 1 ) be a locally finite quiver. We will consider finite-dimensional representations of Q (thus kQ-modules, where kQ is the path algebra of Q). An indecomposable representation M of Q is said to be exceptional provided Ext 1 (M, M ) = 0. And it is called a tree module provided one can choose a set B of bases B x of the vector spaces M x (x ∈ Q 0 ) such that the coefficient quiver Γ(M, B) is a tree quiver; in this case, we call B a tree basis of M . Let me recall the definition of the coefficient quiver Γ(M, B) as introduced in [R3], it is a quiver whose vertices and arrows are labeled by elements of Q 0 and Q 1 , respectively. Its vertex set is the disjoint union of the sets B x , the elements of B x being labeled by x. The arrows of Γ(M, B) are obtained as follows: For an arrow α :It is known [R3] that exceptional modules are tree modules. But even if we know that a module M is a tree module, it often seems to be difficult to find directly a tree basis. The aim of this note is to provide for the exceptional modules an algorithm for obtaining a tree basis. It turns out that we should start by looking at indecomposable representations M with a thin vertex (a vertex x is said to be thin for M provided the vector space M x is one-dimensional). The first modules which we will consider are what we call the radiation modules, see sections 2 and 3. They are inductively defined, in any step one constructs an 1 indecomposable module with a thin vertex. This generalizes a construction introduced by Kinser [K] for rooted tree quivers (using the name "reduced representations").As we will see in section 4, nearly all indecomposable representations of a Dynkin quiver are radiation modules, the only exception is the maximal indecomposable module in case E 8 . The radiation modules which are our main concern are the preprojective and preinjective representations of a tree without leaves with bipartite orientation (...