Hopf q-braces structures on rank one pointed Hopf algebras

TL;DR

This paper classifies all Hopf q-brace structures on rank one pointed Hopf algebras, analyzes their socles, identifies Hopf skew-braces, and explores isomorphisms and weak braiding operators, advancing the understanding of algebraic structures in Hopf algebra theory.

Contribution

It provides a complete classification of Hopf q-brace structures on rank one pointed Hopf algebras and characterizes their properties and interrelations.

Findings

All Hopf q-brace structures on rank one pointed Hopf algebras are determined.

The socle of each Hopf q-brace is computed.

Conditions for isomorphism and identification of Hopf skew-braces are established.

Abstract

In this paper we determine all the Hopf q-brace structures on rank one pointed Hopf algebras and compute the socle of each one of them. We also identify which among them are Hopf skew-braces. Then we determine when two Hopf q-brace structures on rank one pointed Hopf algebras are isomorphic, and, finally, we compute all the weak braiding operators on these Hopf algebras.