The graded Grothendieck group ${K}_0^{\mathrm{gr}}$ is full for weighted Leavitt path algebras

Abstract

The Graded Classification Conjecture asserts that the graded Grothendieck group $K_0^{\mathrm{gr}}$ is a complete invariant for the classes of Leavitt path algebras and graph $C^*$-algebras. The conjecture remains open, as neither a proof nor a counterexample is currently known. In this article, we extend the study of this invariant to the class of vertex-weighted Leavitt path algebras. We show that $K_0^{\mathrm{gr}}$ distinguishes weighted Leavitt path algebras from ordinary (unweighted) Leavitt path algebras. We further prove that an isomorphism between the graded Grothendieck groups of weighted Leavitt path algebras induces an isomorphism between the corresponding semilattices of vertex-generated ideals. In addition, we show that $K_0^{\mathrm{gr}}$ classifies the classical Leavitt algebras $L_K(n,n+k)$. Next, we prove that $K_0^{\mathrm{gr}}$ is a full functor on the category of all weighted Leavitt path algebras. Consequently, in the special case where all weights are equal to $1$, we recover the lifting theorem established independently by Arnone and Vas for Leavitt path algebras. This confirms one direction of the Graded Classification Conjecture for Leavitt path algebras.