Kaveh Mousavand, Queen's University at Kingston
From bricks over minimal representation-infinite algebras to minimal brick-infinite algebras

Abstract: Motivated by the celebrated Brauer-Thrall conjectures (now theorems), the family of minimal representation-infinite algebras were extensively studied between 1960-90's. Over such algebras, one is concerned with the behavior of all indecomposable modules and their distribution with respect to their length. Inspired by the modern subject of $\tau$-tilting theory, and more specifically to initiate a systematic treatment of $\tau$-tilting finiteness of algebras, in my earlier work I focused on the behavior of a special subfamily of indecomposable modules, so-called bricks (i.e, those modules whose endomorphism algebra is a skew field). In particular, as the first family of interest, I studied the behavior of bricks over minimal representation-infinite algebras. That is, for every such algebra A, one first needs to decide whether A admits an infinite family of isoclasses of bricks, and if so, how bricks of A are distributed with respect to their length. In the first part of this talk, I address these questions for a large family of minimal representation-infinite algebras. Then, inspired by an open conjecture on the behavior of bricks, I introduce the modern family of minimal brick-infinite algebras, and view them as a conceptual counterpart of minimal representation-infinite algebras. In our more recent work with Charles Paquette, we showed that many important problems about the behavior of bricks (in particular our open conjecture) can be reduced to a special subfamily of minimal brick-infinite algebras. Moreover, we will see that these modern algebras manifest several important properties that can be used in other areas of research, including the classical tilting theory.