Indecomposable Representations of Finite-Dimensional Algebras

Let 1c "be a field and A a finite-dimensional 7c-algebra (associative, with 1). AVe consider representations of A as rings of endomorpliisms of finitedimensional 7c-spaces, and thus JL-modules, and we ask for a classification of such representations. More generally, we may consider the following problem: given an abelian category # and simple ( = irreducible) objects F(l),..., B(n) in #, what are the objects in # of finite length with all composition factors of the form jßf(l),..., 22/(w). Problems of this kind arise naturally in many branches of mathematics, in particular, classification problems for linear representations of other algebraic structures (groups, Lie algebras, etc.) may be reinterpreted in this way. We will always assume that we know the simple -A-modules B(i), 1 < i < w, and also their first extension groups Ext(5/(i), F(j)), and thus the modules of length 2, and our aim is to study the indecomposable modules of greater length. Note that any (finite-dimensional) A-module can be written as a direct sum of indecomposable modules, and the Krull-Schmidt theorem asserts that these indecomposable direct summands, as well as their multiplicities, are uniquely determined. Besides the semisimple algebras (with all indecomposable modules being simple), there are other algebras with only finitely many (isomorphism classes of) indecomposable modules (they are said to be representation finite). However, there will usually be large families of (pairwise non-isomorphic) indecomposable modules, indexed over suitable algebraic varieties. As Drozd [16] has shown, for representation infinite algebras, there is a strict distinction between the tame and the wild representation type, the tame algebras being characterized by the property that there are at most one-parameter families of indecomposable modules. Por a wild algebra, it seems difficult to obtain a complete classification of all indecomposable modules, since it would involve the (unsolved) problem of classifying pairs of square matrices