Graded isomorphisms of Leavitt path algebras and Leavitt inverse semigroups

TL;DR

This paper establishes a combinatorial criterion for classifying Leavitt path algebras and Leavitt inverse semigroups of certain finite graphs up to graded isomorphism, linking graph structure to algebraic isomorphisms.

Contribution

It provides a necessary and sufficient combinatorial condition for graded isomorphisms of Leavitt path algebras and inverse semigroups of specific finite graphs.

Findings

The combinatorial condition characterizes graded isomorphisms.

Graph isomorphism coincides with algebraic isomorphism under the condition.

Applicable to connected finite graphs with vertices of out-degree at most one.

Abstract

Leavitt inverse semigroups of directed finite graphs are related to Leavitt graph algebras of (directed) graphs. Leavitt path algebras of graphs have the natural $\mathbb Z$-grading via the length of paths in graphs. We consider the $\mathbb Z$-grading on Leavitt inverse semigroups. For connected finite graphs having vertices out-degree at most $1$, we give a combinatorial sufficient and necessary condition on graphs to classify the corresponding Leavitt path algebras and Leavitt inverse semigroups up to graded isomorphisms. More precisely, the combinatorial condition on two graphs coincides if and only if the Leavitt path algebras of the two graphs are $\mathbb Z$-graded isomorphic if and only if the Leavitt inverse semigroups of the two graphs are $\mathbb Z$-graded isomorphic.