nLab quiver representation

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:

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

Related concepts

• Gabriel's theorem

References

Review:

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

Textbook accounts:

• Ralf Schiffler, Quiver Representations, CMS Books in Mathematics, Springer (2014) $[$doi:10.1007/978-3-319-09204-1$]$

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

• Gunnar Carlsson, Vin de Silva, Zigzag Persistence, Found Comput Math 10 (2010) 367–405 $[$arXiv:0812.0197, doi:10.1007/s10208-010-9066-0$]$

• Steve Y. Oudot, Persistence Theory: From Quiver Representations to Data Analysis, Mathematical Surveys and Monographs 209 AMS (2015) $[$ISBN:978-1-4704-3443-4, pdf$]$