Algebra Seminar: Classifying 2-representation infinite algebras of type Ã

Abstract:  n-representation infinite algebras play an important role in higher AR-theory. They relate to (n+1)-Calabi-Yau algebras via the higher preprojective algebra, and higher type à was defined by Herschend-Iyama-Oppermann in terms of the higher preprojective algebra. We therefore need to study the preprojective algebras when trying to classify n-representation infinite algebras. In this talk, I will begin by giving a brief overview of n-representation infinite algebras, their preprojective algebras, and how they fit into higher AR-theory. Next, we introduce some skew-group algebras R*G of polynomial rings R = C[x_0, ... , x_n] by finite subgroups G < SL_{n+1}(C). Every R*G is (n+1)-Calabi-Yau, but not every R*G is higher preprojective, and we will discuss the (in general open) problem of deciding which ones are. Setting n=2 and G abelian, I will then present a classification of the 3-preprojective algebras of type Ã, and explain how this leads to a classification of all 2-representation infinite algebras of type à up to a certain kind of mutation. Time permitting I will point towards possible generalisations for n>2 and discuss the main obstruction. 

This is based on current joint work with Oleksandra Gasanova  

This is a seminar in our algebra seminar series.