Leavitt path algebra

Summary

Let $E$ be a directed graph 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 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.