---
tag: 9NZ9FY1S7C
kind: concept
name: Leavitt path algebra
msc: [16S88, 16S10]
specializes: [3CTW1P4HBP]
sources:
  - url: https://en.wikipedia.org/wiki/Leavitt_path_algebra
same_as:
  wikidata: Q104849200
  wikipedia: Leavitt path algebra
---

## Summary

Let $E$ be a [[directed graph|0KFYGSHHF5]] and $K$ a field.
The Leavitt path algebra $L_K(E)$ is generated by the vertices and edges of $E$ and by a "ghost" edge $e^*$ for each edge $e$.
The generators satisfy the Cuntz–Krieger relations.
It contains the [[path algebra|2PYANBGXFD]] of $E$ and adds partial inverses of the edges.
These algebras include the matrix rings $M_n(K)$, the Laurent polynomial ring $K[x, x^{-1}]$ and Leavitt's algebras $L(1, n)$ without invariant basis number.
They are the algebraic counterparts of graph C*-algebras, and many ring-theoretic properties of $L_K(E)$ can be read off from the graph.

## Explanations

### L1 human

Start with a directed graph and walk along its edges.
For each edge $e$ add a ghost edge $e^*$ that walks back the opposite way.
Walking back along $e^*$ and then forward along $e$ returns you to where you started, and at each vertex the ways out add up to standing still.
For one vertex with one loop, the loop becomes an invertible variable $x$ with $x^* = x^{-1}$, and the algebra is $K[x, x^{-1}]$.