(Weak) Twisted post-groups, skew trusses and rings

TL;DR

This paper introduces and explores (weak) twisted post groups, establishing their connections with rings, skew trusses, Lie algebras, and Hopf algebras, revealing new algebraic structures and their interrelations.

Contribution

It defines (weak) twisted post groups, proves their structural properties, and links them to rings, skew braces, Lie algebras, and Hopf algebras, expanding the algebraic framework.

Findings

Category of weak twisted post groups is isomorphic to skew trusses

Every abelian two-sided twisted post group relates to a radical ring

Twisted post Hopf algebras lead to sub-adjacent Hopf algebras

Abstract

In an attempt to understand the origin of post groups introduced by C. Bai, L. Guo, Y. Sheng, R. Tang, from the perspective of rings, we introduce the notion of (weak) twisted post groups. First, we show that every element in a twisted post group is attached to a unique group and the twisted post group can be decomposed as the disjoint union of such groups. Next, we show the category of weak twisted post groups and the category of skew trusses are isomorphic, every two-sided twisted post group has the structure of two-sided skew braces. Furthermore, we prove that every abelian two-sided twisted post group is associated with a radical ring. Then we introduce the notion of twisted post Lie algebras, and study their algebraic properties. Indeed, we show that the differentiation of every twisted post Lie group is a twisted post Lie algebra. Finally, we linearize (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 sub-adjacent Hopf algebra.