An associative algebra whose basis consists of all directed paths in a quiver (directed graph), with multiplication given by concatenation of paths when the target of the first equals the source of the second and defined to be zero otherwise, and with vertex idempotents corresponding to trivial paths.

Principle

Encode the combinatorial data of a directed graph as an algebra by taking paths as basis elements and making composition of compatible paths the multiplication, thereby turning graph-theoretic concatenation into associative algebra structure.

Demonstration

For a quiver with vertices v1 → v2 → v3 and arrows a:v1→v2, b:v2→v3, the path algebra basis includes idempotents e_{v1}, e_{v2}, e_{v3}, the arrows a, b and the length-two path ab; multiplication satisfies a·b = ab, b·a = 0, and e_{v1}·a = a = a·e_{v2}.

Misapplication

Treating the path algebra as commutative, concatenating non-composable paths and ignoring the zero product in those cases, or identifying it with the free algebra on the arrow set without accounting for vertex idempotents and source/target constraints.

Consequence

Modules over the path algebra correspond to representations of the quiver; finite-dimensional quotients by relations present a wide class of finite-dimensional algebras, making path algebras a central tool for encoding and studying representation-theoretic information.

Reversal

Reversing every arrow of the quiver produces the opposite algebra; dually, considering formal linear combinations of paths with deconcatenation coalgebra structure yields the path coalgebra, which inverts the direction of algebraic composition.

Boundary

Requires a directed graph (quiver) as input; infinite quivers or infinite path lengths raise convergence or cardinality issues and often demand additional finiteness hypotheses; the construction differs from a free associative algebra because vertex idempotents and composability constraints impose extra structure; imposing relations yields quotients that may lose the naive path basis.

Semantic Tension

The term 'quiver algebra' is often used interchangeably, but one must distinguish path algebras from other algebras associated to a quiver (e.g., bound quiver algebras with relations, path coalgebras) and from free algebras generated by arrows without vertex idempotents.

Synthesis

A path algebra is the associative algebra built from a directed graph by taking directed paths as a basis and defining multiplication by concatenation when composable, thereby converting combinatorial path composition into algebraic structure that encodes representations of the quiver.