Morita theory of finite representations of Leavitt path algebras

TL;DR

This paper explores the graded Morita equivalence of Leavitt path algebras associated with finite graphs, establishing connections between their graded Grothendieck groups and categories of graded modules, thus advancing classification theory.

Contribution

It demonstrates that isomorphisms of graded Grothendieck groups imply equivalences of module categories for finite graphs without sinks or sources, extending classification results.

Findings

Categories of graded modules are equivalent under certain Grothendieck group isomorphisms.

Finite dimensional graded modules categories are equivalent for such algebras.

Results hold for graphs with no sinks or sources, linking algebraic invariants to module categories.

Abstract

The Graded Classification Conjecture states that for finite directed graphs $E$ and $F$, the associated Leavitt path algebras $L_\K(E)$ and $L_\K(F)$ are graded Morita equivalent, i.e., $\Gr L_\K(E) \approx_{\gr} \Gr L_\K(F)$, if and only if, their graded Grothendieck groups are isomorphic $K_0^{\gr}(L_\K(E)) \cong K_0^{\gr}(L_\K(F))$ as order-preserving $\mathbb Z[x,x^{-1}]$-modules. Furthermore, if under this isomorphism, the class $[L_\K(E)]$ is sent to $[L_\K(F)]$ then the algebras are graded isomorphic, i.e., $L_\K(E) \cong _{\gr} L_\K(F)$. In this note we show that, for finite graphs $E$ and $F$ with so sinks and sources, an order-preserving $\mathbb Z[x,x^{-1}]$-module isomorphism $K_0^{\gr}(L_\K(E)) \cong K_0^{\gr}(L_\K(F))$ gives that the categories of locally finite dimensional graded modules of $L_\K(E)$ and $L_\K(F)$ are equivalent, i.e., $\fGr[\mathbb{Z}] L_\K(E)\approx_{\gr} \fGr[\mathbb{Z}]L_\K(F).$ We further obtain that the category of finite dimensional (graded) modules are equivalent, i.e., $\fModd L_\K(E) \approx \fModd L_\K(F)$ and $\fGr L_\K(E) \approx_{\gr} \fGr L_\K(F)$.