QuiverPath

A directed path in a quiver: where it starts, and the arrow indices it follows. The basis of the path algebra $kQ$, with one trivial path per vertex.

WikipediaQuiver (mathematics)↗WikidataQ493980↗Encyclopedia of MathematicsQuiver↗nLabquiver↗

QuiverPath(v, [])the trivial path at vertex $v$ — a local identity

QuiverPath(v, [i, j, …])the path from $v$ along arrows $i$, $j$, …

Typed signatures

Type	Package	Applies
(integer, list<integer>) -> number	quiver	always

Domain: Path algebras

Details

• Arrows are indexed by position in the quiver's arrow list, so PARALLEL arrows are distinct basis elements
• Composition is QuiverCompose, which needs the quiver: a path alone does not know which quiver it belongs to
• The trivial paths are local identities: $e_{\text{start}}p = p = pe_{\text{end}}$
• Quivers: LinearQuiver(n) ($A_n$), JordanQuiver (one loop), KroneckerQuiver, or Quiver(n, [[from,to],…])

Examples

Basic #

from vertex 1 along two arrows

live ↗

In$\mathrm{QuiverPathEnd}(\mathrm{LinearQuiver}(4), \mathrm{QuiverPath}(1, \bigl\lbrack0, 1\bigr\rbrack))$Out$3$

Scope #

the Kronecker quiver: two trivial paths and two distinct arrows

live ↗

In$\mathrm{AlgebraDimension}(\mathrm{PathAlgebra}(\mathrm{Quiver}(2, \bigl\lbrack\bigl\lbrack1, 2\bigr\rbrack, \bigl\lbrack1, 2\bigr\rbrack\bigr\rbrack)))$Out$4$

Possible issues #

arrow 1 starts at vertex 2, so this is not a path

live ↗

In$\mathrm{QuiverPath}(1, \bigl\lbrack1\bigr\rbrack)∈\mathrm{PathAlgebra}(\mathrm{LinearQuiver}(4))$Out$\mathrm{False}$

About

Examples	3
Crosswalk	WikipediaWikidataEncyclopedia of MathematicsnLab

See also: QuiverCompose, QuiverIsAcyclic, Basis