# UNZERLEGBARE DARSTELLUNGEN I # Peter Gabriel # Herrn Professor E. Witt zum 60. Geburtstag Let K be the structure obtained by forgetting the composition law of morphisms in a given category. A linear representation of K is given by a map V associating with any morphism φ: a → e of K a linear vector space map V(φ): V(a) → V(e). We classify those K having only finitely many isomorphism classes of indecomposable linear representations. This classification is related to an old paper by Yoshii [3]. ### 1. Introduction 1.1. In this work, we consider 4-tuples K = (F(K), P(K), n_K, s_K) consisting of a set of points P(K), a set of arrows F(K), and two maps n_K, s_K: F(K) → P(K), assigning to each arrow φ ∈ F(K) its tail n_K(φ) ∈ P(K) and its head s_K(φ) ∈ P(K). For such a 4-tuple, we propose the term "quiver" instead of "graph," as the latter term is already associated with many related concepts. The set of all arrows with tail a and head e is denoted by K[a, e] or [a, e], and we write φ: a → e instead of φ ∈ [a, e], as is customary in categories. However, no composition is given for the arrows of a quiver. Quivers can be schematically illustrated as in the following example. In our classification theorems, the following classes of quivers play an important role: $$ \underline{A}_n\qquad1\quad\text{—}\quad2\quad\text{—}\quad3\quad\text{—}\quad\cdots\quad\text{—}\quad n-1\quad\text{—}\quad n,\quad n\quad\gg1 $$ $$ \underbrace{D}_{n}-\underbrace{\frac{1}{I}}_{0}-\underbrace{1}_{1}-\cdots-\underbrace{n-4}_{n-3},\;n\geq4 $$ $$ \underset{\mathbb{E}_{6}}{2^{\prime}}\longrightarrow1^{\prime}\longrightarrow\underset{0}{\overset{1}{}}\longrightarrow1\longrightarrow2 $$ $$ \begin{array}{r l r}{\mathbb{E}_{7}}&{{}}&{2^{\prime}\mathrm{~-~}1^{\prime}\mathrm{~-~}\stackrel{1}{0}\mathrm{~-~}1\mathrm{~-~}2\mathrm{~-~}3}\end# UNZERLEGBARE DARSTELLUNGEN I # Peter Gabriel # Herrn Professor E. Witt zum 60. Geburtstag Let K be the structure obtained by forgetting the composition law of morphisms in a given category. A linear representation of K is given by a map V associating with any morphism φ: a → e of K a linear vector space map V(φ): V(a) → V(e). We classify those K having only finitely many isomorphism classes of indecomposable linear representations. This classification is related to an old paper by Yoshii [3]. ### 1. Introduction 1.1. In this work, we consider 4-tuples K = (F(K), P(K), n_K, s_K) consisting of a set of points P(K), a set of arrows F(K), and two maps n_K, s_K: F(K) → P(K), assigning to each arrow φ ∈ F(K) its tail n_K(φ) ∈ P(K) and its head s_K(φ) ∈ P(K). For such a 4-tuple, we propose the term "quiver" instead of "graph," as the latter term is already associated with many related concepts. The set of all arrows with tail a and head e is denoted by K[a, e] or [a, e], and we write φ: a → e instead of φ ∈ [a, e], as is customary in categories. However, no composition is given for the arrows of a quiver. Quivers can be schematically illustrated as in the following example. In our classification theorems, the following classes of quivers play an important role: $$ \underline{A}_n\qquad1\quad\text{—}\quad2\quad\text{—}\quad3\quad\text{—}\quad\cdots\quad\text{—}\quad n-1\quad\text{—}\quad n,\quad n\quad\gg1 $$ $$ \underbrace{D}_{n}-\underbrace{\frac{1}{I}}_{0}-\underbrace{1}_{1}-\cdots-\underbrace{n-4}_{n-3},\;n\geq4 $$ $$ \underset{\mathbb{E}_{6}}{2^{\prime}}\longrightarrow1^{\prime}\longrightarrow\underset{0}{\overset{1}{}}\longrightarrow1\longrightarrow2 $$ $$ \begin{array}{r l r}{\mathbb{E}_{7}}&{{}}&{2^{\prime}\mathrm{~-~}1^{\prime}\mathrm{~-~}\stackrel{1}{0}\mathrm{~-~}1\mathrm{~-~}2\mathrm{~-~}3}\end