QuiverIsAcyclic

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

QuiverIsAcyclic(quiver)true when

kQ

has a finite basis

Typed signatures

Type	Package	Applies
`(expression<LinearQuiver> \| expression<Quiver> \| symbol) -> boolean`	`quiver`	always

Domain: Path algebras

Details

$kQ$ is finite-dimensional exactly when $Q$ is acyclic; one loop gives the paths $e, a, a^2, \dots$ and the Jordan quiver's path algebra is $k[x]$
For a cyclic quiver, Basis and AlgebraDimension have no answer and leave the call standing rather than enumerating forever
$kA_n$ is the incidence algebra of a chain: its paths are the pairs $i \le j$ , so both have dimension $\binom{n+1}{2}$

Examples

Basic #

\mathrm{QuiverIsAcyclic}(\mathrm{LinearQuiver}(4))

Out

\mathrm{True}

Properties #

a loop is a cycle

\mathrm{QuiverIsAcyclic}(\mathrm{JordanQuiver})

Out

\mathrm{False}

$\binom{7}{2}$ — the same as IncidenceAlgebra(Chain(6))

\mathrm{AlgebraDimension}(\mathrm{PathAlgebra}(\mathrm{LinearQuiver}(6)))

Out

21

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