Hopf brace, braid equation and bicrossed coproduct

TL;DR

This paper explores the structure of Hopf braces, establishing their equivalence with certain categories and constructing new examples on various Hopf algebras, advancing the understanding of their algebraic properties.

Contribution

It provides new characterizations of Hopf braces, proves categorical equivalences, and constructs numerous examples on different classes of Hopf algebras.

Findings

Category of Hopf braces is equivalent to categories of bijective 1-cocycles and Hopf matched pairs.

Constructed many Hopf braces on polynomial, Long copaired, and Drinfel'd double Hopf algebras.

Established conditions for bicrossed coproducts to be Hopf braces.

Abstract

In this paper, we mainly give some equivalent characterisations of Hopf braces, show that the category $\mathcal{CB}(A)$ of Hopf braces is equivalent to the category $\mathcal{C}(A)$ of bijective 1-cocycles, and prove that the category $\mathcal{CB}(A)$ of Hopf braces is also equivalent to the category $\mathcal{M}(A)$ of Hopf matched pairs. Moreover, we construct many more Hopf braces on polynomial Hopf algebras, Long copaired Hopf algebras and Drinfel'd doubles of finite dimensional Hopf algebras, and give a sufficient and necessary condition for a given bicrossed coproduct $A\bowtie H$ to be a Hopf brace if $A$ or $H$ is a Hopf brace.