Quiver representation theory MOC

Quiver representation

A 𝕂-representation of a quiver Γ may be characterized in several different ways: quiv

1. A quiver homomorphism from Γ onto a 𝕂-linear quiver;
2. A functor from the free category Γ―― to Category of vector spaces;
3. 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 Γ𝐸-graded vector space.

Often we are only interested in finite-dimensional representations, i.e. those of the form Γ―― →𝖥𝗂𝗇𝖵𝖾𝖼𝗍𝕂. We might also consider a Matrix quiver representation Γ―― →Sk⁡(𝖵𝖾𝖼𝗍𝕂).

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