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

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.