nLab quiver representation




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

  1. a 𝕂\mathbb{K}-vector space V vV_v for each vertex vQ 0v \in Q_0,

  2. a linear map V vρ(e)V vV_v \xrightarrow{\rho(e)} V_{v'} for each edge eQ 1e \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 vϕ vV vV_v \xrightarrow{\phi_v} V'_v for each vertex, such that for each edge vevv \xrightarrow{e} v' the following diagram commutes in Vect 𝕂{}_{\mathbb{K}}:

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

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

  1. a quiver representation is a functor

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

    ϕ:ρρ, \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 𝕂(Q)Func(FrCat(Q),Vect 𝕂). Rep_{\mathbb{K}}(Q) \;\simeq\; Func \big( FrCat(Q) ,\, Vect_{\mathbb{K}} \big) \,.



  • 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.