Based cluster algebras of infinite ranks

We extend based cluster algebras to infinite ranks. By extending (quantum) cluster algebras associated with double Bott-Samelson cells, we recover infinite rank cluster algebras arising from representations of (shifted) quantum affine algebras. As the main application, we show that the fundamental variables of the cluster algebras associated with double Bott-Samelson cells could be computed via a braid group action when the Cartan matrix is of finite type. We also obtain the result A=U for the associated infinite rank (quantum) cluster algebras. Additionally, several conjectures regarding quantum virtual Grothendieck rings by Jang-Lee-Oh and Oh-Park follow as consequences. Finally, we quantize cluster algebras arising from representations of shifted quantum affine algebras.