Quantum cluster structures on bosonic extensions

We construct a quantum cluster algebra structure on the standard Z[q±1/2]-form of the bosonic extension algebra A(b) for every positive braid b and every symmetrizable generalized Cartan matrix. The proof realizes this algebra as the degree-zero part of a localization of a larger quantum unipotent coordinate ring, obtained by thickening a word for b. For symmetric Cartan matrices, this realization also gives a monoidal categorification of the quantum cluster algebra.