Remarks on the derived center of small quantum groups

Abstract

Let $u_q(g)$ be the small quantum group associated with a complex semisimple Lie algebra $g$ and a primitive root of unity q, satisfying certain restrictions. We establish the equivalence between three different actions of $g$ on the center of $u_q(g)$ and on the higher derived center of $u_q(g)$. Based on the triviality of this action for $g = sl_2, sl_3, sl_4$, we conjecture that, in finite type A, central elements of the small quantum group $u_q(sl_n)$ arise as the restriction of central elements in the big quantum group $U_q(sl_n)$. \nWe also study the role of an ideal $z_{Hig}$ known as the Higman ideal in the center of $u_q(g)$. We show that it coincides with the intersection of the Harish-Chandra center and its Fourier transform, and compute the dimension of zHig in type A. As an illustration we provide a detailed explicit description of the derived center of $u_q(sl_2)$ and its various symmetries.