Double-bosonization and majid's conjecture, (I): Rank-inductions of ABCD

Majid developed in [S. Majid, Math. Proc. Cambridge Philos. Soc. 125, 151-192 (1999)] the double-bosonization theory to construct U_q(g) and expected to generate inductively not just a line but a tree of quantum groups starting from a node. In this paper, the authors confirm Majid's first expectation (see p. 178 [S. Majid, Math. Proc. Cambridge Philos. Soc. 125, 151-192 (1999)]) through giving and verifying the full details of the inductive constructions ofU_q(g) for the classical types, i.e., the ABCD series. Some examples in low ranks are given to elucidate that any quantum group of classical type can be constructed from the node corresponding to U_q(sl2).