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

The Graded Classification Conjecture states that for finite directed graphs E and F, the associated Leavitt path algebras Lk(E) and Lk(F) are graded Morita equivalent, i.e., Gr-Lk(E) approximate to gr Gr-Lk(F), if and only if, their graded Grothendieck groups are isomorphic K0gr(Lk(E)) congruent to K0gr(Lk(F)) as order-preserving Z[x,x-1]-modules. Furthermore, if under this isomorphism, the class [Lk(E)] is sent to [Lk(F)] then the algebras are graded isomorphic, i.e., Lk(E) congruent to gr Lk(F). 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 K0gr(Lk(E)) congruent to K0gr(Lk(F)) gives that the categories of locally finite dimensional graded modules of Lk(E) and Lk(F) are equivalent, i.e., grZ-Lk(E) approximate to grgrZ-Lk(F). We further obtain that the category of finite dimensional (graded) modules is equivalent, i.e., mod- Lk(E) approximate to mod Lk(F) and gr-Lk(E) approximate to gr gr-Lk(F).