First-year graduateAlgebraPlanned

Quivers and path algebras

Quivers and their path algebras: paths as a basis, multiplication by concatenation, representations of quivers as modules over the path algebra, and simple, projective and indecomposable representations.

math.algebra.path-algebras

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Builds on

Modules

Required prerequisites, all the way down: 29

You will learn to

Compute the path algebra kQ of a small quiver and show that it is finite-dimensional exactly when Q is finite with no oriented cycles.
Prove that representations of Q correspond to kQ-modules, giving an equivalence of categories.
Classify the indecomposable representations of a linearly oriented quiver of type Aₙ.

Definition

A quiver Q is a directed graph, possibly with loops and multiple arrows.
The path algebra kQ has a basis consisting of all paths in Q, including a trivial path at each vertex, with product given by concatenation when the paths compose and zero otherwise.
A representation of Q assigns a vector space to each vertex and a linear map to each arrow.

Theorem

For a finite quiver, the category of representations of Q is equivalent to the category of kQ-modules.
For a finite quiver Q, kQ is finite-dimensional if and only if Q has no oriented cycles.
The path algebra of a finite acyclic quiver is hereditary: submodules of projective modules are projective.

Common misconceptions

Mixing conventions: texts compose paths left-to-right or right-to-left, which interchanges left and right modules.
Treating a quiver as an undirected graph; orientation and multiple arrows matter.

Leads to

Quiver representations

Optional discoveries

Categories The paths in a quiver form the free category generated by that quiver, and the path algebra is its linearisation.
← Graphs

Learning pathways

Research preparation