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 (Q. J. Math 63(3), 653-674, 2012), we say that a family oe'(R) aS dagger oe'- of pairwise orthogonal bricks is a simple-minded system if its closure under extensions is all of oe'-. We construct torsion pairs in oe'- associated to any subset oe'(3) of a simple-minded system oe'(R), and use these to define left and right mutations of oe'(R) relative to oe'(3). When oe'- has a Serre functor nu and oe'(R) and oe'(3) are invariant under nu a similar to [1], we show that these mutations are again simple-minded systems. We are particularly interested in the case where oe'- = mod-I > for a self-injective algebra I >. In this case, our mutation procedure parallels that introduced by Koenig and Yang for simple-minded collections in D (b) (mod-I >) (Koenig and Yang, 2013). It follows that the mutation of the set of simple I >-modules relative to oe'(3) yields the images of the simple I"-modules under a stable equivalence mod-I" -> mod-I >, where I" is the tilting mutation of I > relative to chi.