Rigid modules and Schur roots

Christof Geiss (1) , Bernard Leclerc (2) , Jan Schröer (3)

1 Instituto de Matematicas [México]
2 LMNO - Laboratoire de Mathématiques Nicolas Oresme
3 Mathematisches Institut der Universität Bonn

Let C be a symmetrizable generalized Cartan matrix with symmetrizer D and orientation Ω. In Geiß et al. (Invent Math 209(1):61–158, 2017) we constructed for any field F an F-algebra H:=HF(C,D,Ω), defined in terms of a quiver with relations, such that the locally free H-modules behave in many aspects like representations of a hereditary algebra H˜ of the corresponding type. We define a Noetherian algebra Hˆ over a power series ring, which provides a direct link between the representation theory of H and of H˜. We define and study a reduction and a localization functor relating the module categories of Hˆ, H˜ and H. These are used to show that there are natural bijections between the sets of isoclasses of tilting modules over the three algebras Hˆ, H˜ and H. We show that the indecomposable rigid locally free H-modules are parametrized, via their rank vectors, by the real Schur roots associated to (C,Ω). Moreover, the left finite bricks of H, in the sense of Asai, are parametrized, via their dimension vectors, by the real Schur roots associated to (CT,Ω).

Domains: Mathematics [math]; Representation Theory [math.RT]

Christof Geiss, Bernard Leclerc, Jan Schröer. Rigid modules and Schur roots. Mathematische Zeitschrift, 2019, ⟨10.1007/s00209-019-02396-5⟩. ⟨hal-02290592⟩