Analogs of the dual canonical bases for cluster algebras from Lie theory

Abstract

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 in cluster theory. In cases where the generalized Cartan matrices are symmetric, we show that these cluster algebras and their bases are quasi-categorified. We base our approach on the combinatorial similarities among cluster algebras from Lie theory. For this purpose, we introduce new cluster operations to propagate structures across different cases, which allow us to extend established results on quantum unipotent subgroups to other such algebras. We also obtain fruitful byproducts. First, we prove A=U for these quantum cluster algebras. Additionally, we discover rich structures of the locally compactified quantum cluster algebras arising from double Bott-Samelson cells, including T-systems, standard bases, and Kazhdan-Lusztig type algorithms. Notably, in type ADE, we obtain their monoidal categorifications via monoidal categories associated with positive braids. As a special case, these categories provide monoidal categorifications of the quantum function algebras in type ADE.