Multiparameter quantum groups at roots of unity
Gastón Andrés García, Fabio Gavarini
TL;DR
The paper develops a comprehensive framework for multiparameter quantum groups at roots of unity, showing that MpQG's arise as 2-cocycle deformations of the canonical uniparameter quantum group and can be studied via restricted, unrestricted, and mixed integral forms. It constructs quantum root vectors and PBW-type bases, establishes dualities between Borel subalgebras, and extends the theory to MpQG's with larger tori. Specialization at $q=1$ yields semiclassical structures (Lie bialgebras and Poisson groups) dependent on the multiparameter, while specialization at roots of unity is connected by quantum Frobenius morphisms to classical objects and yields small MpQG's as cleft extensions of the specialized ordinary MpQG's. The results unify and extend known uniparameter phenomena to the multiparameter setting, providing a robust platform for further arithmetic, geometric, and topological applications. These advances pave the way for subsequent work on explicit realizations, dualities, and invariants in the multiparameter quantum realm.
Abstract
We address the study of multiparameter quamtum groups (=MpQG's) at roots of unity, namely quantum universal enveloping algebras $ U_{\boldsymbol{\rm q}}(\mathfrak{g}) $ depending on a matrix of parameters $ \boldsymbol{\rm q} = {\big( q_{ij} \big)}_{i, j \in I} \, $. This is performed via the construction of quantum root vectors and suitable "integral forms" of $ U_{\boldsymbol{\rm q}}(\mathfrak{g}) \, $, a \textsl{restricted one} - generated by quantum divided powers and quantum binomial coefficients - and an \textsl{unrestricted\/} one - where quantum root vectors are suitably renormalized. The specializations at roots of unity of either forms are the "MpQG's at roots of unity" we look for. In particular, we study special subalgebras and quotients of our MpQG's at roots of unity - namely, the multiparameter version of small quantum groups - and suitable associated quantum Frobenius morphisms, that link the MpQG's at roots of 1 with MpQG's 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 $ \boldsymbol{\rm q} $ our quantum groups yield (through the choice of integral forms and their specialization) 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.