Wild canonical algebras

Abstract

Canonical algebras ￿=￿(p; ￿); depending on a weight sequence p=(p1; : : : ; pt) of positive integers, and a parameter sequence ￿ = (￿3; : : : ; ￿t) of pairwise distinct non-zero elements from the base ￿eld k; were introduced and studied in [23]. The ￿nite dimensional representation theory mod(￿) is completely controlled by the category coh(X) of coherent sheaves on a non-singular weighted projective line X = X(p; ￿); since the derived categories D(mod(￿)) and D(coh(X)) are equivalent as triangulated categories [9]. The curve X attached to ￿ has (virtual) genus gX = 1 + 2((t − 2)p − ￿ t i=1 p=pi); where p=l:c:m:(p1; : : : ; pt): The complexity of the classi￿cation problem for coh(X); and hence for mod(￿); is essentially determined by gX: For gXi1; the algebra ￿ is concealed of extended Dynkin type; accordingly the classi￿cation problem for coh(X) and mod(￿) is equivalent to the classi￿cation of indecomposable modules over a tame hereditary algebra (cf. [18], [9]) or, according to [10], to the classi￿cation problem for Cohen–Macaulay modules over a simple surface singularity. For gX = 1; the algebra ￿ is of tubular type, its representation theory is known from [23], while the classi￿cation problem for coh(X) [9, 20] relates to Atiyah’s classi￿cation [1] of vector bundles over an elliptic curve. Very little was known on mod(￿) or coh(X) if gX?1. In this case, ￿= ￿0[R] is the one-point extension of a wild hereditary algebra ￿0; accordingly the category mod(￿0) embeds into coh(X) as a full, exact and extension-closed subcategory, implying that the classi￿cation problems for coh(X) and mod(￿) are wild. From now on we assume gX ? 1: Combining the sheaf theory [9] with the spectral investigation of wild hereditary algebras [6], we obtain a fairly complete picture on the structure of Auslander–Reiten components for mod(￿); the asymptotic behaviour of the Auslander–Reiten translations ￿￿ and