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