QuiverIsAcyclic

Whether the quiver has no directed cycle — equivalently, whether its path algebra is finite-dimensional at all.

WikipediaDirected acyclic graph↗MathWorldAcyclicDigraph↗WikidataQ1195339↗

QuiverIsAcyclic(quiver)true when has a finite basis

Domain: Path algebras

Details

• is finite-dimensional exactly when is acyclic; one loop gives the paths and the Jordan quiver's path algebra is
• For a cyclic quiver, `Basis` and `AlgebraDimension` have no answer and leave the call standing rather than enumerating forever
• is the incidence algebra of a chain: its paths are the pairs , so both have dimension

Examples

See also: QuiverPath, QuiverCompose, MoebiusFunction