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Periodicity for finite-dimensional selfinjective algebras

Presented by: 
Karin Erdmann University of Oxford
Date: 
Tuesday 28th March 2017 -
11:30 to 12:30
Venue: 
INI Seminar Room 1
Abstract: 
 We give a survey on finite-dimensional selfinjective algebras which are periodic as bimodules, with respect to syzygies, and hence are stably Calabi-Yau. These include preprojective algebras of Dynkin types ADE and deformations, as well a class of algebras which we call mesh algebras of generalized Dynkin type. There is also a classification of the selfinjective algebras of polynomial growth which are periodic. Furthermore, we introduce weighted surface algebras, associated to triangulations of compact surfaces, they are tame and symmetric, and have period 4 (they are 3-Calabi-Yau). They generalize Jacobian algebras, and also blocks of finite groups with quaternion defect groups.
 
In general, for such an algebra, all one-sided simple modules are periodic. One would like to know whether the converse holds: Given a finite-dimensional selfinjective algebra A for which all one-sided simple modules are periodic. It is known that then some syzygy of A is isomorphic as a bimodule to some twist of A by an automorphism. It is open whether then A must be periodic.
University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons