Monoidal Categorification and Quantization of Braid Varieties

Abstract: Let (G) be a simple and simply connected algebraic group of simply-laced type. For each positive braid word (β), 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(β)). We identify this algebra with both a localized quantum Grothendieck ring and a localized level-(\geq1) 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^{1/2}=1) and prove that it recovers (\CC[X(β)]); 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.