The graded classification conjectures hold for various finite representations of Leavitt path algebras

Abstract

The Graded Classification Conjecture states that for finite directed graphs E and F, the associated Leavitt path algebras $L_k\left(E\right)$ and $L_k\left(F\right)$ are graded Morita equivalent, i.e., $\mathrm{Gr-}{L}_k\left(E\right){\approx }_{\mathrm{gr}}\mathrm{Gr-}{L}_k\left(F\right)$, if and only if, their graded Grothendieck groups are isomorphic $K_0^{\mathrm{gr}}\left(L_k\right(E\left)\right)\cong K_0^{\mathrm{gr}}\left(L_k\right(F\left)\right)$ as order-preserving $Z[x,x^{-1}]$-modules. Furthermore, if under this isomorphism, the class $\left[L_k\right(E\left)\right]$ is sent to $\left[L_k\right(F\left)\right]$ then the algebras are graded isomorphic, i.e., $L_k\left(E\right){\cong }_{\mathrm{gr}}{L}_k\left(F\right)$.

In this note we show that, for finite graphs E and F with no sinks and sources, an order-preserving $Z[x,x^{-1}]$-module isomorphism $K_0^{\mathrm{gr}}\left(L_k\right(E\left)\right)\cong K_0^{\mathrm{gr}}\left(L_k\right(F\left)\right)$ gives that the categories of locally finite dimensional graded modules of $L_k\left(E\right)$ and $L_k\left(F\right)$ are equivalent, i.e., $g{r}^Z-L_k\left(E\right){\approx }_{\mathrm{gr}}g{r}^Z-L_k\left(F\right)$. We further obtain that the category of finite dimensional (graded) modules is equivalent, i.e., $\mathrm{mod-}{L}_k\left(E\right)\approx \mathrm{mod-}{L}_k\left(F\right)$ and $\mathrm{gr-}{L}_k\left(E\right){\approx }_{\mathrm{gr}}\mathrm{gr-}{L}_k\left(F\right)$.