Quivers: A year of linear algebra by drawing arrows

4 days ago

Representations map objects to matrices or permutations, with linear representations being particularly common and well-understood.
Quivers are directed graphs allowing multiple edges and loops, studied through the lens of representation theory.
A quiver representation assigns a vector space to each vertex and a linear map to each edge.
Indecomposable representations cannot be decomposed into a direct sum of smaller representations.
Classifying representations involves finding indecomposable ones, akin to simplifying matrices via basis changes.
Simple quivers like • and •→• correspond to basic linear algebra concepts: vector spaces with bases and matrix decompositions via Gauss elimination.
The quiver •⟲ (one vertex with a loop) relates to Jordan normal forms, showing infinite indecomposable representations.
More complex quivers, like •→• with two parallel arrows, lead to Kronecker canonical forms used in control theory.
The quiver •⟲⟲ (two loops) represents a wild classification problem, considered intractable and linking to all finite-dimensional algebras.

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