Locally finite triangulated categories

A k-linear triangulated category A is called locally finite provided ∑X∈indAdimkHomA(X,Y)<∞ for any indecomposable object Y in A. It has Auslander-Reiten triangles. In this paper, we show that if a (connected) triangulated category has Auslander-Reiten triangles and contains loops, then its Auslander-Reiten quiver is of the form Lˆn: By using this, we prove that the Auslander-Reiten quiver of any locally finite triangulated category A is of the form ZΔ→/G, where Δ is a Dynkin diagram and G is an automorphism group of ZΔ→. For most automorphism groups G, the triangulated categories with ZΔ→/G as their Auslander-Reiten quivers are constructed. In particular, a triangulated category with Lˆn as its Auslander-Reiten quiver is constructed.