The Leavitt path algebra of a graph

TL;DR

This paper constructs Leavitt path algebras from graphs over any field, explores their properties, and characterizes when these algebras are simple, linking graph structure to algebraic simplicity.

Contribution

It introduces a general construction of Leavitt path algebras for row-finite graphs over any field and characterizes their simplicity based on graph conditions.

Findings

Leavitt path algebras generalize matrix rings and Leavitt algebras.

Necessary and sufficient conditions for simplicity of L(E) are provided.

L(E) is an algebraic analog of Cuntz-Krieger algebras for complex fields.

Abstract

For any row-finite graph $E$ and any field $K$ we construct the {\its Leavitt path algebra} $L(E)$ having coefficients in $K$. When $K$ is the field of complex numbers, then $L(E)$ is the algebraic analog of the Cuntz Krieger algebra $C^*(E)$ described in [8]. The matrix rings $M_n(K)$ and the Leavitt algebras L(1,n) appear as algebras of the form $L(E)$ for various graphs $E$. In our main result, we give necessary and sufficient conditions on $E$ which imply that $L(E)$ is simple.