Poisson orders on large quantum groups

TL;DR

This paper introduces a Poisson geometric framework for the representation theory of large quantum groups at roots of unity, unifying various classes including super groups and characteristic zero quantizations.

Contribution

It develops a uniform Poisson geometric approach for large quantum groups, extending previous methods to super cases using perfect pairings instead of rank-two reductions.

Findings

Explicit description of Poisson algebraic groups and homogeneous spaces.

Application to irreducible representations of quantum algebras.

Framework includes quantum groups, super groups, and characteristic zero quantizations.

Abstract

We develop a Poisson geometric framework for studying the representation theory of all contragredient quantum super groups at roots of unity. This is done in a uniform fashion by treating the larger class of quantum doubles of bozonizations of all distinguished pre-Nichols algebras arXiv:1405.6681 belonging to a one-parameter family; we call these algebras \emph{large} quantum groups. We prove that each of these 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. Besides all (multiparameter) big quantum groups of De Concini--Kac--Procesi and big quantum super groups at roots of unity, our framework also contains the 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. The previous approaches to the above problems relied on reductions to rank two cases and direct calculations of Poisson brackets, which is not possible in the super case since there are 13 kinds of additional Serre relations on up to 4 generators. We use a new approach that relies on perfect pairings between restricted and non-restricted integral forms.