David Pauksztello (Lancaster), Simple-minded systems and positive noncrossing partitions.

Abstract:

Module categories have two important types of generators: projective modules and simple modules. Morita theory describes equivalences of module categories in terms of images of projective modules. Tilting theory is the generalisation of Morita theory to derived categories describing equivalences of derived categories in terms of tilting objects. Tilting, silting and cluster-tilting objects, can be thought of as `projective-minded objects'.

`Simple-minded objects' are generalisations of simple modules. They satisfy Schur's lemma and a version of the Jordan-Holder theorem, depending on context, giving rise to `simple-minded collections' and `simple-minded systems'. Although the theory of simple-minded objects shows many parallels with that of projective-minded objects, it remains relatively undeveloped and is technically more challenging. In this talk I will explain the connection between simple-minded systems in negative Calabi-Yau orbit categories of hereditary algebras, simple-minded collections in their bounded derived category, and positive noncrossing partitions, generalising results of Buan-Reiten-Thomas, Coelho Simoes, and Iyama-Jin. This is a report on joint work with Raquel Coelho Simoes and David Ploog.