Rothschild Seminar Series

Mathematics

Audience: Researchers in the topic
Seminar series time: Wednesday 14:00-15:00 in your time zone, UTC
Organizers: Isaac Newton Institute Cambridge, Sibylle Schroll*
*contact for this listing

Rothschild Seminar Series: Cluster algebras with coefficients and relative cluster categories - 3 talks by Bernhard Keller

Organized by the Isaac Newton Institute Cambridge

6th, 13th and 20th October 2021 at 2pm London time

Zoom Link for 6th October:

newton-ac-uk.zoom.us/j/85293526487?pwd=RDRTKzROcmxwR1ZMWWk4UHQ2U0RKZz09 Meeting ID: 852 9352 6487 Passcode: +pGQ9rWi Zoom Link for 13th October:

newton-ac-uk.zoom.us/j/86242077224?pwd=L0JHYlZMVTVsUGdRUm9sZm91RDBvdz09 Meeting ID: 862 4207 7224 Passcode: gS&H^BP3 Zoom Link for 20th October:

newton-ac-uk.zoom.us/j/81357723426?pwd=STlxM0hKMmtjNE84dHRiKzBMYlVFdz09 Meeting ID: 813 5772 3426 Passcode: A*7a0$8Y

====================== Titel: Cluster algebras with coefficients and relative cluster categories ======================

Abstract: Coefficients are crucial when dealing with cluster algebras arising in geometry and Lie theory like the Grassmannians, double Bruhat cells, the base affine space, ... . They also arise naturally for cluster algebras associated with bordered marked surfaces. In additive categorification, there are two main methods for incorporating coefficients:

1) one "freezes" some indecomposable summands of a cluster-tilting object in a 2-Calabi-Yau triangulated category or

2) one uses a Frobenius category instead of a triangulated category.

The second method is used notably in the work of Geiss-Leclerc-Schroeer and, more recently, of Jensen-King-Su. Its drawback is that most ice quivers (=quivers some of whose vertices are frozen) do not allow such a Frobenius categorification. In this series of lectures, our aim is to show how this problem can be solved using relative cluster categories. Such a category can be associated with any ice quiver and, for the ice quivers arising in Geiss-Leclerc-Schroeer's setup, is equivalent to the derived category of Frobenius category they use. The Frobenius category itself is generalized by the Higgs category.

In the first lecture, we will review categorification based on Frobenius categories including cluster characters and the link to generalized cluster categories.

The second lecture will be devoted to the construction of relative cluster categories, Higgs categories and their comparison with Frobenius categories. This will be based on Yilin Wu's recent Ph. D. thesis.

In the third lecture, we will introduce the corresponding cluster characters and present an application to the categorification of Fraser's quasi cluster isomorphisms. In particular, using results of Yilin Wu, we will see how quasi cluster isomorphisms arise from mutations at frozen vertices.

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