Quiver Representations — Mathematics Knowledge System

Overview

A quiver is a directed graph, and a representation of a quiver Q assigns a vector space to each vertex and a linear map to each arrow. Quiver representations provide a unified framework for studying linear algebra problems (Jordan normal form, flags, etc.) and encode module categories over path algebras. Gabriel's theorem classifies quivers of finite representation type: they are exactly those whose underlying undirected graph is a Dynkin diagram of type A, D, or E.

Intuition

A quiver representation is like a system of vector spaces connected by linear maps — think of it as a 'wiring diagram' for linear algebra. The path algebra kQ encodes all composition of arrows, and a quiver representation is exactly a module over kQ. Gabriel's theorem is a beautiful surprise: the indecomposable representations of a Dynkin quiver are in bijection with positive roots of the root system.

Formal Definition

Definition

A quiver Q = (Q₀, Q₁, s, t) consists of a set Q₀ of vertices, a set Q₁ of arrows, and source/target maps s, t: Q₁ → Q₀. A representation M of Q over a field k assigns:

M: Q_0 \to \{\text{k-vector spaces}\}, \quad M: Q_1 \to \{\text{k-linear maps}\}, \quad \text{i.e.} \; M_i \in \mathbf{Vect}_k, \; M_\alpha: M_{s(\alpha)} \to M_{t(\alpha)}

Quiver representation: vector spaces and linear maps

\mathrm{Rep}(Q) \cong kQ\text{-mod} \quad (\text{category of representations} \cong \text{modules over the path algebra})

Equivalence with path algebra modules

\mathbf{dim}\, M = (\dim_k M_i)_{i \in Q_0} \in \mathbb{Z}_{\geq 0}^{Q_0} \quad (\text{dimension vector})

Dimension vector

\chi_Q(\mathbf{d}, \mathbf{e}) = \sum_{i \in Q_0} d_i e_i - \sum_{\alpha \in Q_1} d_{s(\alpha)} e_{t(\alpha)} \quad (\text{Euler form})

Tits/Euler form of the quiver

Notation

Notation	Meaning
$Q$	Quiver (directed graph)
$kQ$	Path algebra of Q over k
$\mathbf{dim}\, M$	Dimension vector of representation M
$\mathrm{Rep}(Q)$	Category of k-representations of Q

Properties

Simple representations

For each vertex i ∈ Q₀, there is a simple representation Sᵢ with (Sᵢ)ᵢ = k, (Sᵢ)ⱼ = 0 for j ≠ i, and all maps zero.

Projective indecomposables

The indecomposable projective kQ-module Pᵢ corresponding to vertex i has dimension vector given by the number of paths from i to each vertex.

Theorems

Theorem 1: Gabriel's Theorem

A connected quiver Q (with no oriented cycles) is of finite representation type (finitely many indecomposable representations up to isomorphism) if and only if the underlying undirected graph is a Dynkin diagram of type Aₙ, Dₙ (n≥4), E₆, E₇, or E₈. In this case, the indecomposable representations are in bijection with positive roots of the corresponding root system.

Theorem 2: Krull–Schmidt Theorem for Quivers

Every finite-dimensional representation of a finite quiver decomposes uniquely (up to isomorphism and permutation of summands) as a direct sum of indecomposable representations.

Worked Examples

1
A representation assigns vector spaces V₁, V₂ and a linear map f: V₁ → V₂. By the classification of linear maps (rank-nullity), indecomposables arise from f being 0, injective, or the identity on 1-dim spaces.
$\text{Rep}(A_2): \quad V_1 \xrightarrow{f} V_2$
2
Indecomposables: S₁ = (k → 0) with dim vector (1,0), S₂ = (0 → k) with dim vector (0,1), P₁ = (k →^{id} k) with dim vector (1,1).
$\text{Indecomposables:} \quad S_1: k \to 0, \quad S_2: 0 \to k, \quad P_1: k \xrightarrow{\mathrm{id}} k$
3
There are 3 indecomposables, matching the 3 positive roots of A₂: α₁, α₂, α₁+α₂ (Gabriel's theorem).
$\text{Positive roots of } A_2: \alpha_1,\; \alpha_2,\; \alpha_1+\alpha_2 \quad (3 \text{ roots} = 3 \text{ indecomposables})$

✓ Answer

A₂ has 3 indecomposable representations, corresponding to the 3 positive roots of A₂.

Practice Problems

Mediumfree response

For the quiver D₄ (one central vertex with 3 outgoing arrows to leaves), how many positive roots does the D₄ root system have, and therefore how many indecomposable representations does the D₄ quiver have?

Hardfree response

Describe the path algebra kQ for the quiver Q: 1 → 2 → 3 (type A₃). List a basis and explain the multiplication.

Quiz

Summary

A quiver representation assigns vector spaces to vertices and linear maps to arrows.
Rep(Q) ≅ kQ-mod: representations are modules over the path algebra.
Gabriel's theorem: finite representation type iff underlying graph is a Dynkin diagram (A, D, E).
For Dynkin quivers, indecomposables are in bijection with positive roots.
The Jordan quiver (one vertex, one loop) has infinitely many indecomposables (Jordan blocks).

References

BookAssem, I., Simson, D. & Skowroński, A. — Elements of the Representation Theory of Associative Algebras (2006)
BookKirillov, A. Jr. — Quiver Representations and Quiver Varieties (2016)