Moduli Spaces for Representations of Concealed-Canonical Algebras

Abstract

Continuing M. Domokos and H. Lenzing [ J. Algebra 228 (2000), 738–762], we investigate the relative invariants and moduli spaces introduced in A. D. King [ Quart. J. Math. Oxford Ser. (2) 45 (1994), 515–530] which are related to a separating family of stable tubes in the module category of a concealed-canonical algebra. The class of concealed-canonical algebras consists of representation-infinite algebras of tame or wild representation type and contains in particular the representation theory of extended Dynkin quivers. We present a uniform characteristic-free approach avoiding case-by-case analysis. We introduce the notion of admissible weight for such algebras; these are the weights (i.e., elements of the dual of the Grothendieck group of the module category) attached to the infinite moduli spaces for families of modules from the separating subcategory. By using an explicit description of the semigroup of admissible weights the corresponding notion of semistability is analyzed and is related to the formation of perpendicular categories. We construct relative invariants belonging to admissible weights and prove that the constructed system is complete in the case of canonical algebras. It turns out that for any concealed-canonical algebra and for any admissible weight all the corresponding moduli spaces are isomorphic to some projective space. Moreover, any such moduli space can be naturally identified with the moduli space belonging to a special weight, called the rank. As a consequence we show that in the case of a tame concealed algebra, any infinite moduli space for families of modules is a projective space, and all fields of rational invariants on irreducible components of representation spaces are purely transcendental.