Remarks on the derived center of small quantum groups

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.