Multiparameter quantum supergroups, deformations and specializations

In this paper, we introduce a multiparameter version of the quantum universal enveloping superalgebras introduced by Yamane in [Quantized enveloping algebras associated to simple Lie superalgebras and their universal R-matrices, Publ. Res. Inst. Math. Sci. 30(1) (1994) 15-87]. For these objects we consider: (1) their deformations by twist and by 2-cocycle (both of "toral type"); in particular, we prove that this family is stable under both types of deformations; (2) their semiclassical limits, which are multiparameter Lie superbialgebras; (3) the deformations by twist and by 2-cocycle (of "toral type") of these multiparameter Lie superbialgebras: in particular, we prove that this family is stable under these deformations, and that "quantization commutes with deformation".