Renormalization Group flow in Schur quantization

We develop a general formalism to describe the Renormalization Group Flow of Schur indices and fusion algebras of BPS line defects in four-dimensional $\mathcal{N}=2$ Supersymmetric Quantum Field Theories. The formalism includes and extends known results about the Seiberg-Witten description of these structures. Another application of the formalism is to describe the spectrum of BPS partices of $\mathcal{N}=2$ gauge theories with matter in terms of the spectrum of pure $\mathcal{N}=2$ gauge theories. Applications to the theory of quantum groups and to the quantization of cluster varieties are also discussed.