Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 22 (2026), 099, 30 pages      arXiv:2307.10535      https://doi.org/10.3842/SIGMA.2026.099

Twisted Post-Groups and Skew Trusses

Shukun Wang ab
a) School of Mathematics and Big Data, Anhui University of Science and Technology, Huainan 232001, P.R. China
b) Anhui Province Engineering Laboratory for Big Data Analysis and Early Warning Technology of Coal Mine Safety, Huainan 232001, P.R. China

Received July 13, 2026, in final form October 09, 2026; Published online October 19, 2026

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.

Key words: (weak) twisted post-group; skew truss; Rota-Baxter system of groups; twisted post-Lie algebra.

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