Double-bosonization and Majid's conjecture (V): grafting of quantum groups

This paper aims to develop a grafting method to address Majid's conjecture, which enables the construction of a larger target quantum group by grafting two given smaller ones. This method is significant for advancing the understanding of the generation, classification, and construction of (quasi-)Hopf algebras. To pave the way for the grafting method, we first set up a multi-tensor product theory for generalized double-bosonization to acquire the necessary information on the braiding $R$-matrices (see \cite{HH2}). Beyond the perspective of braided monoidal categories arising from the representations of quantum subgroups, the grafting procedure necessitates incorporating structural information from root systems in Lie theory. This approach provides a one-stop strategy for resolving the generation problem in Majid's conjecture on quantum trees.