Quiver Representation

A quiver turns a directed graph into algebra. Freely generate its paths and a category appears, then represent that category in vector spaces.

A quiver is a directed graph, given by a set of vertices , a set of arrows , and source/target maps , with loops and parallel arrows allowed. Its paths are finite composable strings of arrows together with one empty path at each vertex, and they compose by concatenation whenever endpoints match. This is the free category on the graph, the path category , with as objects and paths as morphisms. Linearising, formal -combinations of paths under concatenation form the path algebra , whose identity is .

A representation of is a functor , that is, a vector space at each vertex and a linear map along each arrow, with functoriality automatic because the paths are freely generated. Equivalently a representation is a left -module and a morphism of representations is a natural transformation, so quiver representation theory is the module theory of . Dimension vectors, indecomposables, and Auslander–Reiten theory all live here.

Gabriel's Theorem

A connected quiver has finite representation type, meaning finitely many indecomposable representations up to isomorphism, iff its underlying undirected graph is a disjoint union of simply-laced ADE Dynkin diagrams . Moreover the indecomposables then correspond bijectively to the positive roots of the associated root system, independent of how the arrows are oriented.

Remark on the free-generation ladder

Freely generating structure from a graph climbs a ladder of one-object-to-many analogues: a single binary generator gives a magma, imposing associativity a monoid, allowing several objects the free category (the paths of a quiver), adding directed 2-cells a 2-category, and collapsing objects into a tensor a strict monoidal category. A quiver representation then sits one rung further out, a functor out of the free category into .

References

• Gabriel, “Unzerlegbare Darstellungen I,” Manuscripta Mathematica (1972)
• Assem, Simson & Skowronski, Elements of the Representation Theory of Associative Algebras, Vol. 1 (Cambridge, 2006)