# nLab quiver representation

Contents

### Context

#### Representation theory

representation theory

geometric representation theory

# Contents

## Idea

Given a quiver $Q$, a linear representation of $Q$ (over some ground field $\mathbb{K}$) is:

1. a $\mathbb{K}$-vector space $V_v$ for each vertex $v \in Q_0$,

2. a linear map $V_v \xrightarrow{\rho(e)} V_{v'}$ for each edge $e \in Q_1$.

Notice that there is no further compatibility condition, in particular if there are edges forming a triangle, then the associated linear maps are not required to be related under composition.

A homomorphism between two quiver representations is a linear map $V_v \xrightarrow{\phi_v} V'_v$ for each vertex, such that for each edge $v \xrightarrow{e} v'$ the following diagram commutes in Vect${}_{\mathbb{K}}$:

This makes a category of quiver representations of $Q$ over $\mathbb{K}$, typically denoted $Rep_{\mathbb{K}}(Q)$, or similar.

In other words, if one regards $Q$ as the directed graph that it is, and considers its free category $FrCat(Q)$, then

1. a quiver representation is a functor

$FrCat(Q) \xrightarrow{\rho} Vect_{\mathbb{K}} \,,$
2. a morphism of quiver representations is a natural transformation

$\phi \;\colon\; \rho \Rightarrow \rho' \,,$
3. and the category of quiver representations is equivalently the functor category from the free category of the quiver to the category of vector spaces:

$Rep_{\mathbb{K}}(Q) \;\simeq\; Func \big( FrCat(Q) ,\, Vect_{\mathbb{K}} \big) \,.$

## References

Review:

• Harm Derksen, Jerzy Weyman, Quiver Representations, Notices of the AMS 52 2 (2005) $[$pdf$]$

Textbook accounts:

Discussion for A-type quivers in the context of (zigzag) persistent homology and topological data analysis:

Last revised on May 20, 2022 at 04:14:29. See the history of this page for a list of all contributions to it.