PBW-deformations of quantum groups

In this paper, we investigate certain deformations Bq(g)Bq(g) of the negative part Uq−(g) of quantized enveloping algebras Uq(g)Uq(g). An algorithm is established to determine when a given Bq(g)Bq(g) is a PBW-deformation of Uq−(g). For gg of type A2A2 and B2B2, we classify PBW-deformations of Uq−(g). Moreover, we explicitly construct some PBW bases for a class of PBW-deformations Bq(g)Bq(g) of Uq−(g). As an application, Iorgov–Klimyk's PBW bases for the non-standard quantum deformation Uq′(so(n,C)) of the universal enveloping algebra U(so(n,C))U(so(n,C)) are recovered.