11 research outputs found
Tilting mutation of weakly symmetric algebras and stable equivalence
We consider tilting mutations of a weakly symmetric algebra at a subset of
simple modules, as recently introduced by T. Aihara. These mutations are
defined as the endomorphism rings of certain tilting complexes of length 1.
Starting from a weakly symmetric algebra A, presented by a quiver with
relations, we give a detailed description of the quiver and relations of the
algebra obtained by mutating at a single loopless vertex of the quiver of A. In
this form the mutation procedure appears similar to, although significantly
more complicated than, the mutation procedure of Derksen, Weyman and Zelevinsky
for quivers with potentials. By definition, weakly symmetric algebras connected
by a sequence of tilting mutations are derived equivalent, and hence stably
equivalent. The second aim of this article is to study these stable
equivalences via a result of Okuyama describing the images of the simple
modules. As an application we answer a question of Asashiba on the derived
Picard groups of a class of self-injective algebras of finite representation
type. We conclude by introducing a mutation procedure for maximal systems of
orthogonal bricks in a triangulated category, which is motivated by the effect
that a tilting mutation has on the set of simple modules in the stable
category.Comment: Description and proof of mutated algebra made more rigorous (Prop.
3.1 and 4.2). Okuyama's Lemma incorporated: Theorem 4.1 is now Corollary 5.1,
and proof is omitted. To appear in Algebras and Representation Theor
Torsion pairs and simple-minded systems in triangulated categories
Let T be a Hom-finite triangulated Krull-Schmidt category over a field k.
Inspired by a definition of Koenig and Liu, we say that a family S of pairwise
orthogonal objects in T with trivial endomorphism rings is a simple-minded
system if its closure under extensions is all of T. We construct torsion pairs
in T associated to any subset X of a simple-minded system S, and use these to
define left and right mutations of S relative to X. When T has a Serre functor
\nu, and S and X are invariant under \nu[1], we show that these mutations are
again simple-minded systems. We are particularly interested in the case where T
is the stable module category of a self-injective algebra \Lambda. In this
case, our mutation procedure parallels that introduced by Koenig and Yang for
simple-minded collections in the derived category of \Lambda. It follows that
the mutation of the set of simple \Lambda-modules relative to X yields the
images of the simple \Gamma-modules under a stable equivalence between \Gamma\
and \Lambda, where \Gamma\ is the tilting mutation of \Lambda\ relative to X.Comment: Minor corrections. To appear in Applied Categorical Structures. The
final publication is available at springerlink.com:
http://link.springer.com/article/10.1007%2Fs10485-014-9365-
Representation-finite special algebras
The article contains no abstrac