Twisted Post-Groups and Skew Trusses

Abstract

To understand the origin of post-groups introduced by Bai, Guo, Sheng, and Tang from the perspective of rings, we introduce the notion of (weak) twisted post-groups. First, we show that every element of a twisted post-group belongs to a unique group, and that a twisted post-group can be viewed as the disjoint union of such groups. Next, we prove that the category of weak twisted post-groups is isomorphic to the category of skew trusses, and that every two-sided twisted post-group admits the structure of a two-sided skew brace. It follows that every abelian two-sided twisted post-group gives rise to a radical ring. We then introduce twisted post-Lie algebras and investigate their algebraic properties, showing that differentiating a twisted post-Lie group yields a twisted post-Lie algebra. Finally, we consider the linearization of (weak) twisted post-groups and propose the notion of (weak) twisted post-Hopf algebras. We show that every twisted post-Hopf algebra gives rise to another Hopf algebra, called the sub-adjacent Hopf algebra.