Multiparameter quantum groups at roots of unity

We address the study of multiparameter quantum groups (MpQGs) at roots unity, namely universal enveloping algebras $U\_\mathbf{q}(\mathfrak{g})$ depending on a matrix parameters $\mathbf{q}=(q\_{ij}){i,j\in I}$. This is performed via construction root vectors and suitable “integral forms” $U\mathbf{q}(\mathfrak{g})$, restricted one—generated by divided powers binomial coefficients—and an unrestricted one—where are suitably renormalized. The specializations unity either form “MpQGs unity” we look for. In particular, special subalgebras quotients our MpQGs unity—namely, version small groups—and associated Frobenius morphisms, that link 1 with 1, latter being classical Hopf bearing well precise Poisson-geometrical content. A key point in discussion, often core strategy, every MpQG actually $2$-cocycle deformation algebra structure (a lift of) “canonical” one-parameter group Jimbo–Lusztig, so can rely already established results available for latter. On other hand, chosen $\mathbf{q}$, yield (through choice integral forms their specializations) different semiclassical structures, Lie coalgebra structures Poisson algebraic underlying canonical group.