Séminaire Algèbre Géométrie Algébrique Topologie Algébrique :
Le 14 septembre 2006 à 11:15 - salle 431
Présentée par Holm Thorsten - University of Leeds
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.