Double-bosonization and Majid’s Conjecture (II): irregular R-matrices and type-crossing to G ₂

The purpose of the paper is to build up the related theory of weakly quasitriangular dual pairs suitably for non-standard R-matrices (irregular), and establish the generalized double-bosonization construction for irregular R, which generalizes Majid’s results for regular R’s in [M1]. As an application, the type-crossing construction for the exceptional quantum group of type G₂ is given explicitly. This affirms Majid’s expectation that the tree structure of nodes diagram associated with quantum groups can be grown out of the node corresponding to U_q(sl₂) by the (generalized) double-bosonization procedures.