  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.
