Double-bosonization and Majid's Conjecture, (III): type-crossing and inductions of $E_6$ and $E_7$, $E_8$

TL;DR

This paper advances the theory of quantum groups by demonstrating type-crossing and inductive constructions of exceptional quantum groups $E_6$, $E_7$, and $E_8$ using double-bosonization, confirming Majid's conjecture.

Contribution

It provides a complete inductive construction of all finite-dimensional complex simple Lie algebra quantum groups via double-bosonization.

Findings

Constructed $E_6$, $E_7$, $E_8$ quantum groups through type-crossing methods.

Confirmed Majid's conjecture on inductive quantum group construction.

Extended the double-bosonization framework to exceptional Lie types.

Abstract

Double-bosonization construction in Majid \cite{majid1} is expectedly allowed to generate a tree of quantum groups. Some main branches of the tree in \cite{HH1, HH2} have been depicted how to grow up. This paper continues to elucidate the type-crossing and inductive constructions of exceptional quantum groups of types $E_6$ and $E_7$, $E_8$, respectively, based on the generalized double-bosonization Theorem established in \cite{HH2}. Thus the Majid's expectation for the inductive constructions of $U_q(\mathfrak g)$'s for all finite-dimensional complex simple Lie algebras is completely achieved.