Poisson orders on large quantum groups

We bring forward the notions of large quantum groups and their relatives. The starting point is the concept of distinguished pre-Nichols algebra arXiv:1405.6681 belonging to a one-parameter family; we call such an object a \emph{large} quantum unipotent subalgebra. By standard constructions we introduce \emph{large} quantum groups and \emph{large} quantum Borel subalgebras. We first show that each of these three large quantum algebras has a central Hopf subalgebra giving rise to a Poisson order in the sense of arXiv:math/0201042. We describe explicitly the underlying Poisson algebraic groups and Poisson homogeneous spaces in terms of Borel subgroups of complex semisimple algebraic groups of adjoint type. The geometry of the Poisson algebraic groups and Poisson homogeneous spaces that are involved and its applications to the irreducible representations of the algebras $U_{\mathfrak{q}} \supset U_{\mathfrak{q}}^{\geqslant} \supset U_{\mathfrak{q}}^+$ are also described. Multiparameter quantum super groups at roots of unity fit in ou context as well as quantizations in characteristic 0 of the 34-dimensional Kac-Weisfeler Lie algebras in characteristic 2 and the 10-dimensional Brown Lie algebras in characteristic 3. All steps of our approach are applicable in wider generality and are carried out using general constructions with restricted and non-restricted integral forms and Weyl groupoid actions. Our approach provides new proofs to results in the literature without reductions to rank two cases.