Prime spectrum and primitive Leavitt path algebras

TL;DR

This paper establishes a correspondence between prime ideals of Leavitt path algebras and certain graph and algebraic structures, and characterizes when these algebras are primitive.

Contribution

It introduces a bijection linking prime ideals of Leavitt path algebras to graph maximal tails and the prime spectrum of Laurent polynomial rings, and provides conditions for primitivity.

Findings

Bijection between prime ideals and graph/algebraic structures

Necessary and sufficient conditions for primitivity of Leavitt path algebras

Characterization of prime spectrum in terms of graph properties

Abstract

In this paper a bijection between the set of prime ideals of a Leavitt path algebra $L_K(E)$ and a certain set which involves maximal tails in $E$ and the prime spectrum of $K[x,x^{-1}]$ is established. Necessary and sufficient conditions on the graph $E$ so that the Leavitt path algebra $L_K(E)$ is primitive are also found.