Path Algebras and Indecomposables

Path algebras provide the algebraic framework for quiver representations, while indecomposable representations form the building blocks.

DefinitionPath Algebra

The path algebra $\mathbb{F}Q$ of a quiver $Q$ over a field $\mathbb{F}$ is the associative algebra with:

Basis: all paths in $Q$ (including trivial paths $e_i$ at each vertex $i$ )
Multiplication: composition of paths (or zero if paths don't compose)
Unit: $\sum_{i \in Q_0} e_i$

Formally, $\mathbb{F}Q$ is the free algebra generated by arrows, modulo the relations from path composition.

TheoremRepresentations as Modules

There is an equivalence of categories: $\text{Rep}(Q) \cong \text{Mod}(\mathbb{F}Q)$

A quiver representation $V$ corresponds to a left $\mathbb{F}Q$ -module via: $M_V = \bigoplus_{i \in Q_0} V_i$ where paths act by composition of the corresponding linear maps.

ExamplePath Algebra of $A_2$

For $Q = (1 \xrightarrow{\alpha} 2)$ , the path algebra $\mathbb{F}Q$ has basis $\{e_1, e_2, \alpha\}$ with multiplication:

$e_1 \alpha = \alpha$ , $\alpha e_2 = \alpha$
$e_1 e_1 = e_1$ , $e_2 e_2 = e_2$
$e_1 e_2 = e_2 e_1 = 0$ , $\alpha \alpha = 0$ , $e_2 \alpha = \alpha e_1 = 0$

This is isomorphic to the algebra of upper triangular $2 \times 2$ matrices with zeros on the diagonal.

DefinitionIndecomposable Representation

A representation $V$ is indecomposable if it cannot be written as a direct sum $V = V_1 \oplus V_2$ with both $V_1, V_2$ non-zero.

A representation is simple (irreducible) if it has no proper non-zero subrepresentations.

For quivers: simple $\Rightarrow$ indecomposable, but the converse may fail (indecomposables need not be simple).

ExampleIndecomposables for $A_2$

For $1 \to 2$ , the indecomposable representations are:

$S_1$ : $\mathbb{F} \xrightarrow{0} 0$ (simple at vertex 1)
$S_2$ : $0 \xrightarrow{0} \mathbb{F}$ (simple at vertex 2)
$I$ : $\mathbb{F} \xrightarrow{1} \mathbb{F}$ (indecomposable but not simple, has $S_2$ as subrepresentation)

These are all indecomposables. Any representation decomposes as direct sums of these.

Remark

Path algebras are hereditary algebras: their global dimension is at most 1. This gives:

Krull-Schmidt: Every representation decomposes uniquely into indecomposables
Homological algebra: Clean Ext and Tor computations
Gabriel's theorem: Finite representation type quivers are Dynkin
Auslander-Reiten theory: Combinatorial methods via AR quivers

The path algebra perspective connects quiver representations to module theory, homological algebra, and derived categories, providing powerful computational tools.