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

TL;DR

This paper introduces a grafting method for quantum groups, advancing the understanding of their construction and classification by integrating double-bosonization and Lie theory root systems.

Contribution

It develops a novel grafting technique for quantum groups, combining multi-tensor product theory with root system structures to address Majid's conjecture.

Findings

Established a multi-tensor product framework for double-bosonization

Integrated root system data into the grafting process

Provided a new approach to quantum group generation and classification

Abstract

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.