Quiver representation

A $𝕂$ -representation of a quiver $Γ$ may be characterized in several different ways: #m/def/quiv

A quiver homomorphism from $Γ$ onto a $𝕂$ -linear quiver;
A functor from the free category $Γ ――$ to $𝖵 𝖾 𝖼 𝗍 𝕂$ ;
A module over the Path algebra $𝕂 Γ ――$ .

where the equivalence of ^R2 and ^R3 follows from Module over a category ring. Generally, it is useful to think of a quiver representation $𝑉$ as a $𝕂 Γ ――$ -representations which is also a $Γ 0$ -graded vector space.

Often we are only interested in finite-dimensional representations, i.e. those of the form $Γ ―― \to 𝖥 𝗂 𝗇 𝖵 𝖾 𝖼 𝗍 𝕂$ . We might also consider a Matrix quiver representation $Γ ―― \to S k (𝖵 𝖾 𝖼 𝗍 𝕂)$ .

#state/tidy | #lang/en | #SemBr