Nazarova, Lioudmila A.; Roiter, Andrei V.; Gabriel, Pierre Représentations indécomposables: Un algorithme. (Indecomposable representations: An algorithm). (French) Zbl 0662.16019 C. R. Acad. Sci., Paris, Sér. I 307, No. 13, 701-706 (1988). Let \(k\) be an algebraically closed field. An additive category \({\mathcal A}\) together with a homomorphism of \(k\) into the center of \({\mathcal A}\) is considered. For an \({\mathcal A}\)-module M, that is an additive functor form \({\mathcal A}\) into the category of abelian groups, an M-subspace is a couple (W,X) formed by an object \(X\in {\mathcal A}\) and a \(k\)-subspace W of M(X). One says that for a finite set \({\mathcal C}\) of submodules of M, an M- subspace avoids \({\mathcal C}\), if \(W\cap L(X)=\{0\}\) for all \(L\in {\mathcal C}\). For given \({\mathcal A}\), M and \({\mathcal C}\), the aim of the paper is to study the indecomposable objects of the category of M-subspaces avoiding \({\mathcal C}\). This eventually provides an algorithm yielding all indecomposable representations of representation-finite algebras and posets. It naturally leads to the reduction process for posets as introduced by L. A. Nazarova and A. V. Rojter [in Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova 28, 5-31 (1972; Zbl 0336.16031)] and also gives new evidence for the first Brauer-Thrall conjecture, first proved by A. V. Rojter. Reviewer: A.Wiedemann Cited in 3 Documents MSC: 16Gxx Representation theory of associative rings and algebras 16B50 Category-theoretic methods and results in associative algebras (except as in 16D90) 18E05 Preadditive, additive categories Keywords:indecomposable objects; category of M-subspaces; algorithm; indecomposable representations; representation-finite algebras and posets; first Brauer-Thrall conjecture Citations:Zbl 0336.16031 PDFBibTeX XMLCite \textit{L. A. Nazarova} et al., C. R. Acad. Sci., Paris, Sér. I 307, No. 13, 701--706 (1988; Zbl 0662.16019)