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  "body": "\n## Definition\n\nLet $K$ be a [[field|04C5ZBG1WT]] and let $E = (E^0, E^1, s, r)$ be a [[directed graph|0KFYGSHHF5]], with vertex set $E^0$, edge set $E^1$, and source and range maps $s, r \\colon E^1 \\to E^0$.\nFor each edge $e$ take a new symbol $e^*$, called a ghost edge.\nThe *Leavitt path algebra* $L_K(E)$ is the quotient of the [[free algebra|2P3Y2FRCSZ]] over $K$ on the set $E^0 \\cup E^1 \\cup \\set{e^* : e \\in E^1}$ by the [[two-sided ideal|0CD87SFM2N]] of these relations:\n\n1. $v w = \\delta_{v, w} v$ for all $v, w \\in E^0$.\n2. $s(e) e = e = e \\, r(e)$ for all $e \\in E^1$.\n3. $r(e) e^* = e^* = e^* s(e)$ for all $e \\in E^1$.\n4. $e^* f = \\delta_{e, f} \\, r(e)$ for all $e, f \\in E^1$.\n5. $v = \\sum_{e \\in E^1, \\, s(e) = v} e e^*$ for every vertex $v$ that is the source of at least one and at most finitely many edges.\n\nHere $\\delta_{a, b}$ is $1$ if $a = b$ and $0$ otherwise.\n\n## Remarks\n\nRelations 4 and 5 are the Cuntz–Krieger relations.\nThe algebra has an identity, $\\sum_{v \\in E^0} v$, exactly when $E^0$ is finite.\nRelations 1 and 2 alone define the [[path algebra|2PYANBGXFD]] of $E$, which maps into $L_K(E)$.\nExample: let $E$ have one vertex $v$ and one loop $e$.\nRelation 4 gives $e^* e = v$ and relation 5 gives $e e^* = v$, with $v$ the identity, so $L_K(E)$ is the [[Laurent polynomial ring|QMY5JTZZB8]] $K[x, x^{-1}]$ with $x = e$ and $x^{-1} = e^*$.\nExample: if $E$ is $u \\to v$ with one edge $e$, then $L_K(E)$ is isomorphic to the [[matrix ring|0FHVPRAE60]] $M_2(K)$, via $u \\mapsto E_{11}$, $v \\mapsto E_{22}$, $e \\mapsto E_{12}$ and $e^* \\mapsto E_{21}$.\nExample: one vertex with $n \\ge 2$ loops gives the Leavitt algebra $L(1, n)$, in which the free module of rank $1$ is isomorphic to the free module of rank $n$.\nNon-example: in the path algebra of the one-loop graph, the element $e$ has no inverse, since every product of $e$ with a combination of paths has no constant term.\nRelation 4 needs the ghost edge $e^*$.\n",
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