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

For almost all the known (quantum) cluster algebras from Lie theory, we construct the common triangular bases. These bases provide analogs of the dual canonical bases which have long been anticipated in cluster theory. Moreover, when the generalized Cartan matrices are symmetric, these cluster algebras and bases are quasi-categorified. We employ a unified approach based on the flexibility of cluster algebras, introducing and investigating various cluster operations for this purpose. We obtain fruitful byproducts. First, we prove A=U for these quantum cluster algebras. Furthermore, we discover rich structures of the locally compactified quantum cluster algebras arising from double Bott-Samelson cells: T-systems, standard bases, Kazhdan-Lusztig type algorithms, and, in type ADE, monoidal categorifications. The monoidal categories appearing are associated with positive braids and consist of modules of quantum affine algebras.