Double-bosonization and Majid's conjecture(IV): Type-crossings from A to BCD

Both in Majid’s double-bosonization theory and in Rosso’s quantum shuffle theory, the rankinductive and type-crossing construction for U;（g）’s is still a remaining open question. In this paper, working in Majid’s framework, based on the generalized double-bosonization theorem we proved before, we further describe explicitly the type-crossing construction of U;（g）’s for（BCD）;series directly from type An-1via adding a pair of dual braided groups determined by a pair of（R, R′）-matrices of type A derived from the respective suitably chosen representations. Combining with our results of the first three papers of this series, this solves Majid’s conjecture, i.e., any quantum group U;（g） associated to a simple Lie algebra g can be grown out of U;（sl;）recursively by a series of suitably chosen double-bosonization procedures.