Fundamental relations in quantum cluster algebras

Speaker：Xueqing Chen (University of Wisconsin-Whitewater)

Time：2026-7-22 10:00

Location：Conference Room S105 at Experiment Building at Haiyun Campus

Abstract:

Ringel discovered a remarkable way to construct the positive part of quantum group using the Hall algebra. Shortly thereafter, Lusztig observed that the Hall algebra can be understood in terms of functions on certain representation spaces. Via the sheaf-function correspondence, Lusztig obtained the canonical basis of the positive part of quantum group.

Cluster algebra was invented by Fomin and Zelevinsky in their purpose to study the total positivity and Lusztig’s dual canonical basis in coordinate ring and its q-deformation. Ringel proved that the Hall algebra satisfied so-called fundamental relations, which are similar to the quantum Serre relations. In this talk we will address the fundamental relations in an arbitrary quantum cluster algebra with principal coefficients, immediately and directly, we obtain an algebra homomorphism from the corresponding (untwisted) quantum group to this quantum cluster algebra.

This talk is based on the joint work with J. Huang, M. Ding and F. Xu.