Remarks on the derived center of small quantum groups

TL;DR

This paper investigates the structure of the center of small quantum groups associated with semisimple Lie algebras, establishing equivalences of actions, conjecturing about central elements in type A, and analyzing the Higman ideal.

Contribution

It introduces new equivalences of actions on the center, conjectures about the origin of central elements in type A, and computes the Higman ideal's dimension in this setting.

Findings

Proves equivalence of three actions on the center of small quantum groups.

Conjectures that central elements in type A arise from restrictions of big quantum group elements.

Computes the dimension of the Higman ideal in type A.

Abstract

Let $\mathsf{u}_q(\mathfrak{g})$ be the small quantum group associated with a complex semisimple Lie algebra $\mathfrak{g}$ and a primitive root of unity q, satisfying certain restrictions. We establish the equivalence between three different actions of $\mathfrak{g}$ on the center of $\mathsf{u}_q(\mathfrak{g})$ and on the higher derived center of $\mathsf{u}_q(\mathfrak{g})$. Based on the triviality of this action for $\mathfrak{g} = \mathfrak{sl}_2, \mathfrak{sl}_3, \mathfrak{sl}_4$, we conjecture that, in finite type A, central elements of the small quantum group $\mathsf{u}_q(\mathfrak{sl}_n)$ arise as the restriction of central elements in the big quantum group $\mathsf{U}_q(\mathfrak{sl}_n)$. We also study the role of an ideal $\mathsf{z}_{\mathrm{Hig}}$ known as the Higman ideal in the center of $\mathsf{u}_q(\mathfrak{g})$. We show that it coincides with the intersection of the Harish-Chandra center and its Fourier transform, and compute the dimension of $\mathsf{z}_{\mathrm{Hig}}$ in type A. As an illustration we provide a detailed explicit description of the derived center of $\mathsf{u}_q(\mathfrak{sl}_2)$ and its various symmetries.