5 papers
Based cluster algebras of infinite rank
Fan Qin
We extend based cluster algebras from the finite rank case to the infinite rank case. By extending (quantum) cluster algebras whose initial seeds are associated with signed words (…
Integral cluster structures on quantized coordinate rings
Hironori Oya, Fan Qin, Milen Yakimov
We develop (quantum) cluster algebra structures over arbitrary commutative unital rings and prove that the (quantized) coordinate rings of connected simply-connected comple…
An introduction to representation-theoretic canonical bases of cluster algebras
Fan Qin
This is an introduction to cluster algebras and their common triangular bases. These bases are Kazhdan-Lusztig-type and serve as the canonical bases of cluster algebras from the re…
Partially compactified quantum cluster structures on simple algebraic groups and the full Berenstein--Zelevinsky conjecture
Fan Qin, Milen Yakimov
The construction of partially compactified cluster algebras on coordinate rings is handled by using codimension 2 arguments on cluster covers. An analog of this in the quantum situ…
Analogs of the dual canonical bases for cluster algebras from Lie theory
Fan Qin
We construct common triangular bases for almost all the known (quantum) cluster algebras from Lie theory. These bases provide analogs of the dual canonical bases, long anticipated…