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.

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Cite

BibTeX RIS

BibTeX

@article{wang2026twisted,
  title = {{Twisted Post-Groups and Skew Trusses}},
  author = {Shukun Wang},
  journal = {Symmetry, Integrability and Geometry: Methods and Applications},
  year = {2026},
  month = oct,
  doi = {10.3842/sigma.2026.099},
  url = {https://sigma-journal.com/2026/099/},
  note = {CC BY-SA 4.0},
}

RIS

TY  - JOUR
TI  - Twisted Post-Groups and Skew Trusses
AU  - Shukun Wang
PY  - 2026
DA  - 2026/10/19
JO  - Symmetry, Integrability and Geometry: Methods and Applications
DO  - 10.3842/sigma.2026.099
UR  - https://sigma-journal.com/2026/099/
AB  - 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.
SN  - 1815-0659
ER  -

Abstract

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