Resumen de Remarks on the derived center of small quantum groups

Anna Lachowska, You Qi

• Let uq(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 uq(g) and on the higher derived center of uq(g). Based on the triviality of this action for g=sl2,sl3,sl4, we conjecture that, in finite type A, central elements of the small quantum group uq(sln) arise as the restriction of central elements in the big quantum group Uq(sln). We also study the role of an ideal zHig known as the Higman ideal in the center of uq(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 uq(sl2) and its various symmetries.