Algebra seminar
Álvaro Sánchez Campillo (Univ. Murcia): Abstract representation theory via coherent Auslander-Reiten diagrams
Abstract: In this talk we explain a method to study representations of quivers over arbitrary stable \infty-categories in terms of Auslander-Reiten diagrams. Our techniques allow us to internally visualize (a significant piece of) the Auslander-Reiten quiver of the derived category of a hereditary finite-dimensional algebra inside much more general (\infty-)categories of representations, such as representations over arbitrary rings, schemes, dg algebras, or ring spectra. This is provided by an equivalence with a certain mesh subcategory of representations of the repetitive quiver, which we build inductively using abstract reflection functors. As an application we obtain that the automorphism group of the non-regular components of the Auslander-Reiten quiver acts on the \infty-category of representations over any stable \infty-category. When specialized to representations of trees over a field, this action gives an isomorphism with the derived Picard group as shown by Miyachi and Yekutieli.
The Algebra Seminar was founded by Vladimir Korinek in the early 1950's and continued by Karel Drbohlav until 1981. The seminar resumed its activities in 1990 under the guidance of Jaroslav Jezek and Tomas Kepka. Since 1994, the seminar is headed by Jan Trlifaj.
Presently, the seminar is supported by GACR. It serves primarily as a platform for presentation of recent research of the visitors to the Department of Algebra as well as members of the Department and their students.
