@enumeratio/quiver

Path-algebra extensions for compute-engine: quivers, their paths, and the path algebra $kQ$ — finite-dimensional exactly when the quiver is acyclic. $kQ$'s basis is all directed paths (one trivial path per vertex included), and like the incidence algebra most products are zero: p · q is pq only when q starts where p ends. Unlike every other family in this section, $kQ$ need not be finite-dimensional. See the guide.

Declaring

import { ComputeEngine } from "@cortex-js/compute-engine";
import { declareQuiver } from "@enumeratio/quiver";

const ce = new ComputeEngine();
declareQuiver(ce);

Heads

• Quivers — Quiver, LinearQuiver, QuiverIsAcyclic
• Paths — QuiverPath, QuiverPathEnd, QuiverCompose
• The algebra — PathAlgebra

src/quiver.ts also exports jordanQuiver and kroneckerQuiver (the single-loop and two-parallel-arrow quivers used to illustrate finite vs. infinite dimension), plus adjacency, allPaths and hasCycle for computing on quivers directly.

Usage

QuiverIsAcyclic(LinearQuiver(4))              // True
QuiverIsAcyclic(JordanQuiver)                 // False
AlgebraDimension(PathAlgebra(LinearQuiver(4))) // has a value
QuiverPathEnd(LinearQuiver(4), QuiverPath(1, [0, 1]))   // 3

Next

A single loop gives the Jordan quiver, whose path algebra is $k[x]$ — no finite basis, so Basis and AlgebraDimension leave the call standing rather than enumerating forever; the guide walks through why.

Pages

• Path Algebras

---

• Source: packages/symbols/algebras/quiver
• Depends on: boxes, engine, structures
• In development, also: config, manifest