An upper central series is an ascending sequence obtained by starting with the trivial substructure and successively taking the inverse image of the center of the quotient by the preceding term. The construction applies to several kinds of algebraic structures, with different notions of center.
The group upper central series consists of normal subgroups of a group. Its first term after the trivial group is the group center, and subsequent terms arise from the centers of quotient groups. The term "upper central sequence" is also used for this construction (Wuthrich 2016-2017).
The Lie algebra upper central series consists of ideals in a Lie algebra. Its first term after 0 is the center, whose elements have vanishing Lie bracket with every element of the Lie algebra (de Ruiter 1972).
The loop upper central series consists of normal subloops of an algebraic loop. The center here requires both commutation and association with all other elements. For groups, it reduces to the group upper central series (Semanišinová and Stanovský 2023).
In these settings, reaching the whole structure after finitely many steps characterizes nilpotence, called central nilpotence in the algebraic loop setting.
