Non‐Strictly Wild Algebras

Finite‐dimensional algebras over an algebraically closed field are divided into two disjoint classes, called tame and wild respectively, by Drozd's tame and wild dichotomy (see [ 5 ] and [ 2 ]). A tame algebra, roughly speaking, has its n ‐dimensional indecomposable modules parametrized by finitely many one‐parameter families, for all natural numbers n , but a wild algebra has more indecomposable modules and it is considered hopeless to classify them. In [ 2 ], Crawley‐Boevey showed that all but finitely many n ‐dimensional indecomposable modules over a tame algebra are τ‐invariant, for all natural numbers n , and conjectured that the converse would be true, where τ := DTr is the Auslander–Reiten translation (see [ 1 ]) and we call an indecomposable module X τ‐ invariant if X ≅ τ X .