Cluster Colloquium

Next talk: Thursday 09.Oct.2025, 17:00 Central European Time

Quantum cluster structures on simple algebraic groups and the Berenstein–-Zelevinsky conjecture

Fan Qin (Beijing Normal University).

Abstract. We present a general method to construct (partially compactified) quantum cluster structures on quantized coordinate rings from those defined on localizations, in a setting where the standard codimension-two arguments used in the classical case are no longer available.

As an application, we obtain natural quantum cluster structures on the quantized coordinate rings of all simple algebraic groups.  Along the way, we settle in full the Berenstein–Zelevinsky conjecture: every quantum double Bruhat cell has a cluster structure indexed by arbitrary signed words, and these seeds are mutation-connected.

This is a joint work with Milen Yakimov.