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.

设\({\mathsf {u}}_q(\mathfrak {g})\)是一个与复半单李代数\(\mathfrak {g}\)和一个单位q的原始根相关联的小量子群，满足某些限制。我们建立三种不同的操作之间的等价\（\ mathfrak {G} \）上的中心\（{\ mathsf {蓝}} _ Q（\ mathfrak {G}）\）和较高派生中心\（ {\mathsf {u}}_q(\mathfrak {g})\)。基于\(\mathfrak {g}= \mathfrak {sl}_2, \mathfrak {sl}_3, \mathfrak {sl}_4\)的这个动作的琐碎性，我们推测，在有限类型A 中，中心元素小量子群\({\mathsf {u}}_q(\mathfrak {sl}_n)\)作为大量子群中中心元素的限制出现\({\mathsf {U}}_q(\mathfrak {sl}_n) \)。我们还研究了理想\({\mathsf {z}}_\mathrm{Hig}\)在\({\mathsf {u}}_q(\mathfrak {g} )\)。我们表明，它与哈什·钱德拉中心的交叉点一致，并且它的傅立叶变换，并且计算的尺寸（{\ mathsf {Z}} _ \ mathrm {几点措施} \）\型甲。作为说明，我们提供了\({\mathsf {u}}_q(\mathfrak {sl}_2)\)的导出中心及其各种对称性的详细明确描述。