Séance Séminaire

Séminaire Algèbre Géométrie Algébrique Topologie Algébrique

Maximal orthogonal modules for selfinjective algebras and generalized cluster complexes

Maximal m-orthogonal modules for finite-dimensional algebras have recently been introduced by O. Iyama, in the context of a “higher Auslander-Reiten theory”. The existence of such modules has far reaching consequences, on the other hand there are only few algebras known for which maximal orthogonal modules exist.
After discussing the basic definitions and some examples, we shall first consider selfinjective algebras of finite representation type. Our aim here is to present a complete classification (for Dynkin types A, B/C and D) of all maximal m-orthogonal modules, for all m>0. It turns out that the maximal m-orthogonal modules for these algebras are in bijection with the maximal simplices of the m-cluster complexes (of the corresponding types A, B/C and D), as defined recently by S. Fomin and N. Reading. For m=1 these simplicial complexes are (dual to) the exchange graphs of cluster algebras of finite type. Our above classification is work in progress with my student G. Murphy, and generalizes results by O. Iyama for m=1.
Secondly, we will show that for arbitrary selfinjective algebras, maximal orthogonal modules rarely exist. More precisely, if a selfinjective algebra admits a maximal m-orthogonal module (for some m>0), then all A-modules have complexity at most one. (This is joint work with K. Erdmann; arXiv:math.RT/0603672.) This somehow explains why the only known examples so far occurred for algebras of finite representation type and for preprojective algebras of Dynkin type.