Polynomial families of quantum semisimple coadjoint orbits via deformed quantum enveloping algebras

TL;DR

This paper constructs polynomial families of semisimple module categories over quantum groups, using deformed quantum enveloping algebras, and establishes a comparison theorem in the integral Lusztig setting.

Contribution

It introduces a novel construction of module categories via generalized parabolic induction with deformed quantum enveloping algebras, linking to sheaves on toric varieties.

Findings

Construction of polynomial families of module categories

Definition of sheaves of algebras on toric varieties

Comparison theorem between module categories

Abstract

We construct a polynomial family of semisimple left module categories over the representation category of the Drinfeld-Jimbo deformation, with the fusion rule of the representation category of each Levi subalgebra. In this construction we perform a kind of generalized parabolic induction using a deformed quantum enveloping algebra, whose definition depends on an arbitrary choice of a positive system and corresponds to De Commer's definition for the standard positive system. These algebras define a sheaf of algebras on the toric variety associated to the root system, which contains the moduli of equivariant Poisson brackets. This fact finally produces the family of 2-cocycle. We also obtain a comparison theorem between our module categories and module categories induced from our construction for intermediate Levi subalgebras. The construction of deformed quantum enveloping algebras and the comparison theorem are discussed in the integral setting of Lusztig's sense.