When twist of a Leavitt path algebra is again a Leavitt path algebra

Abstract

In this paper, we study Zhang twist of Leavitt path algebra $L_K(E)$ using a graded automorphism $σ$ induced by a graph automorphism such that the twisted algebra is a Leavitt path algebra over another graph $E_σ$ which is a twisted graph obtained from the original graph $E$. We establish combinatorial connection between the graphs $E$ and $E_σ$ and as a consequence, we show that many ring-theoretic properties are invariant for Leavitt path algebras under the twist by $σ$. We also define the notion of noncommutative projective scheme for $\mathbb Z$-graded algebras that coincides with the notion of noncommutative projective scheme for connected $\mathbb N$-graded algebras defined by Artin and Zhang and study it in the context of twists of Leavitt path algebras.