Exceptional modules over wild canonical algebras

Abstract

We show that in a certain sense "almost all" exceptional modules over wild canonical algebra Λ(p, λ) can be described by matrices with entries of the form λi − λj, where λi, λj are elements from the parameter sequence λ.The proof is based on Schofield induction for sheaves in the associated categories of weighted projective lines (Kędzierski and Meltzer 2013) and an extended version of Ringel's proof of the 0, 1 matrix property of exceptional representations for finite acyclic quivers.2020 Mathematics Subject Classification: 16G20, 14F05, 16G60. the sequence of parameters λ = (λ 2 , . . . , λ t ). We can assume that λ 2 = 0 and λ 3 = 1. We write Λ = Λ(p, λ). Concerning the complexity of the module category over Λ there are three types of canonical algebras: domestic, tubular and wild (see [ASS06] for definitions). It was shown in [Rin76] that the canonical algebra Λ is of wild type if and only if the Euler characteristicbe a canonical algebra and let Q = (Q 0 , Q 1 ) be its quiver, where Q 0 is the set of vertices and Q 1 the set of arrows. Denote by mod(Λ) the category of finitely generated right Λ-modules. Then a right module M in mod(Λ) can be viewed as a k-is a k-linear map for any arrow β : j → i, such that the canonical relations are satisfied. We will usually identify linear maps with matrices.We recall that a right module M over an algebra A is defined to be exceptional if its endomorphism algebra End A (M ) is a skew field and M does not have self-extensions, i.e. Ext i A (M, M ) = 0 for i > 0. For a canonical algebra Λ the second condition can be reduced to Ext 1 Λ (M, M ) = 0. Moreover, since k is algebraically closed, the first condition implies that End Λ (M ) = k.Our aim is to study the possible entries of the matrices of the k-linear maps M β : M i → M j of an exceptional module M over a wild canonical algebra Λ.The issue of possible coefficients that appear in the matrices describing exceptional modules over various algebras, or more generally exceptional objects in some category, has been considered many times before. In 1998 C. M. Ringel [Rin88] proved that each exceptional representation of a finite acyclic quiver can be realized by 0, 1 matrices. The same result was shown by P. Dräxler [Drä01] for indecomposable modules over representation-finite algebras. In the case of the path algebra of a Dynkin quiver, P. Gabriel [Gab71] computed 0, 1 matrices for all indecomposable representations. A similar result was shown by M. Kleiner [Kle72] for indecomposable representations of some representation-finite posets, and a complete list is presented by D. Simson [Sim91] and by Arnold-Richman [AHR92] (see also a result of K. J. Bäckström [Bäc72] for orders over lattices). For the path algebra of an extended Dynkin quiver indecomposable representations allow 0, ±1 matrices (see [KuM07b], [KęM11]). Among new results we mention a paper of M. Grzecza, S. Kasjan and A. Mróz [GKM12].The problem of possible entries in matrices of exceptional representations was well researched in the case of canonical algebras ...