Monoidal Categorification and Quantization of Braid Varieties

Let $G$ be a simple and simply connected algebraic group of simply-laced type. For each positive braid word $\beta$, and for the complete strong duality datum attached to a $Q$-datum, we construct an explicit based monoidal categorification of the quantum cluster algebra of the braid variety $X(\beta )$. We identify this algebra with both a localized quantum Grothendieck ring and a localized level-$\ge 1$ subalgebra of the bosonic extension algebra. Under these identifications, quantum cluster monomials correspond simultaneously to real simple modules and normalized global basis elements. We construct the specialization homomorphism at $q^{1\mathrm{/}2}=1$ and prove that it recovers $\text{CC}[X(\beta )]$; in particular, the resulting integral form is a flat quantum deformation. We also give intrinsic Lusztig parameters for cluster variables attached to double strings, identify the corresponding quantum grid minors, and establish generalized quantum $T$-systems.