Leavitt path algebra

Remarks

Relations 4 and 5 are the Cuntz–Krieger relations. The algebra has an identity, $\sum_{v \in E^0} v$, exactly when $E^0$ is finite. Relations 1 and 2 alone define the path algebra of $E$, which maps into $L_K(E)$. Example: let $E$ have one vertex $v$ and one loop $e$. Relation 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 $K[x, x^{-1}]$ with $x = e$ and $x^{-1} = e^*$. Example: if $E$ is $u \to v$ with one edge $e$, then $L_K(E)$ is isomorphic to the matrix ring $M_2(K)$, via $u \mapsto E_{11}$, $v \mapsto E_{22}$, $e \mapsto E_{12}$ and $e^* \mapsto E_{21}$. Example: 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 $n$. Non-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. Relation 4 needs the ghost edge $e^*$.