Nonexistence of graded unital homomorphisms between Leavitt algebras and their Cuntz splices

TL;DR

This paper proves that, under mild conditions, there are no graded unital homomorphisms between Leavitt algebras of a graph with one vertex and loops and their Cuntz splices, highlighting a fundamental nonexistence result.

Contribution

It establishes the nonexistence of graded unital homomorphisms between Leavitt algebras of certain graphs and their Cuntz splices, extending understanding of their algebraic structure.

Findings

No graded unital homomorphisms between L_n and L_{n^-} under mild conditions.

Results apply when the ring is a field or PID.

Highlights fundamental structural differences between these Leavitt algebras.

Abstract

Let $n\ge 2$, let $\mathcal{R}_n$ be the graph consisting of one vertex and $n$ loops and let $\mathcal{R}_{n^-}$ be its Cuntz splice. Let $L_n=L(\mathcal{R}_n)$ and $L_{n^-}=L(\mathcal{R}_{n^-})$ be the Leavitt path algebras over a unital ring $\ell$. Let $C_m$ be the cyclic group on $2\le m\le \infty$ elements. Equip $L_n$ and $L_{n^-}$ with their natural $C_m$-gradings. We show that under mild conditions on $\ell$, which are satisfied for example when $\ell$ is a field or a PID, there are no unital $C_m$-graded ring homomorphisms $L_n\to L_{n^-}$ nor in the opposite direction.