Multiparameter quantum groups at roots of unity

Abstract

We address the study of multiparameter quantum groups (MpQGs) at roots of unity, namely quantum universal enveloping algebras $U_{\mathbf{q}}(\mathfrak{g})$ depending on a matrix of parameters $\mathbf{q}=(q_{ij}{)}_{i,j\in I}$. This is performed via the construction of quantum root vectors and suitable “integral forms” of $U_{\mathbf{q}}(\mathfrak{g})$, a restricted one—generated by quantum divided powers and quantum binomial coefficients—and an unrestricted one—where quantum root vectors are suitably renormalized. The specializations at roots of unity of either form are the “MpQGs at roots of unity” we look for. In particular, we study special subalgebras and quotients of our MpQGs at roots of unity—namely, the multiparameter version of small quantum groups—and suitable associated quantum Frobenius morphisms, that link the MpQGs at roots of 1 with MpQGs at 1, the latter being classical Hopf algebras bearing a well precise Poisson-geometrical content.

A key point in the discussion, often at the core of our strategy, is that every MpQG is actually a $2$-cocycle deformation of the algebra structure of (a lift of) the “canonical” one-parameter quantum group by Jimbo–Lusztig, so that we can often rely on already established results available for the latter. On the other hand, depending on the chosen multiparameter $\mathbf{q}$, our quantum groups yield (through the choice of integral forms and their specializations) different semiclassical structures, namely different Lie coalgebra structures and Poisson structures on the Lie algebra and algebraic group underlying the canonical one-parameter quantum group.