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  "body": "\n## Summary\n\nLet $E$ be a [[directed graph|0KFYGSHHF5]] and $K$ a field.\nThe Leavitt path algebra $L_K(E)$ is generated by the vertices and edges of $E$ and by a \"ghost\" edge $e^*$ for each edge $e$.\nThe generators satisfy the Cuntz–Krieger relations.\nIt contains the [[path algebra|2PYANBGXFD]] of $E$ and adds partial inverses of the edges.\nThese 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.\nThey are the algebraic counterparts of graph C*-algebras, and many ring-theoretic properties of $L_K(E)$ can be read off from the graph.\n\n## Explanations\n\n### L1 human\n\nStart with a directed graph and walk along its edges.\nFor each edge $e$ add a ghost edge $e^*$ that walks back the opposite way.\nWalking 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.\nFor one vertex with one loop, the loop becomes an invertible variable $x$ with $x^* = x^{-1}$, and the algebra is $K[x, x^{-1}]$.\n",
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