Double-bosonization and Majid's conjecture (V): Grafting of Quantum Qroups

This paper aims to develop a grafting method to address Majid's conjecture. This method enables the construction of a larger target quantum group by grafting two given smaller quantum groups, and advances the study of the generation, classification, and construction of (quasi-)Hopf algebras. As a foundation for this construction, we establish a multi-tensor-product theory of generalized double-bosonization and use it to extract the crucial data for the associated braiding $R$-matrix. Beyond the braided monoidal category perspective arising from quantum subgroup representations, the grafting procedure needs to incorporate structural information from Lie-theoretic root systems. The resulting theoretic framework provides a one-stop strategy for resolving the generation problem concerning the quantum-groups tree in Majid's conjecture.