Agbebra Seminar: J-equivalence for algebras (or: Splitting the regular bimodule)

Abstract:  In the study of the representation theory of (finitary Krull-Schmidt) bicategories as initiated by Mazorchuk and Miemietz, a key is the combinatorial structure of the 1-morphisms in terms of left-, right-, and two-sided cells. A natural class of bicategories to study are the bicategories of A-A-bimodules for fixed algebras A. Such bicategory has one object, morphism category A-mod-A, and composition given by tensor product over A. The crux is that, with a few exceptions, bimodule categories are of wild type, making them hard to study explicitly. A relevant question is therefore: can we compare the cell structures of the bimodule categories for different algebras?  To this end, we begin by an attempt to compare the regular bimodules (i.e. identity 1-morphisms). The regular A-A-bimodule canonically belong to the minimal two-sided cell in the bicategory of A-A-bimodules, so this gives some useful information. Explicitly, we ask, given algebras A and B, are there A-B- and B-A-bimodules whose tensor product over B contains the regular A-A-bimodule as a direct summand? If the answer is yes, and the same holds for the regular B-B-bimodule over A, then we call A and B J-equivalent.  In this talk we shall discuss different aspects of this scenario: requirements on the bimodules inducing J-equivalence, constructions resulting in J-equivalent algebras, and properties which are invariant under J-equivalence. Furthermore, we will see a number of examples of J-equivalent, and inequivalent but J-related, algebras.  

Welcome to join on site or via Zoom link (Meeting ID: 645 5572 6999, please contact the organizer for passcode)

This is a seminar in our algebra seminar series.