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 Uq(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 Uq(g)’s for (BCD)n series directly from type An−1 via 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 Uq(g) associated to a simple Lie algebra g can be grown out of Uq(sl2) recursively by a series of suitably chosen double-bosonization procedures.