About Hopf braces and crossed products

TL;DR

This paper investigates conditions under which certain constructions of Hopf braces, involving matched pairs and crossed products, result in new Hopf braces, with applications to Drinfeld's Double.

Contribution

It provides new criteria for when crossed product constructions of Hopf braces form valid Hopf braces, extending the understanding of their algebraic structure.

Findings

Established conditions for Hopf braces from crossed products.

Applied results to analyze when Drinfeld's Double forms a Hopf brace.

Extended the theory of Hopf braces in symmetric monoidal categories.

Abstract

The present article represents a step forward in the study of the following problem: If $\mathbb{A}=(A_{1},A_{2})$ and $\mathbb{H}=(H_{1},H_{2})$ are Hopf braces in a symmetric monoidal category C such that $(A_{1},H_{1})$ and $(A_{2},H_{2})$ are matched pairs of Hopf algebras, then we want to know under what conditions the pair $(A_{1}\bowtie H_{1},A_{2}\bowtie H_{2})$ constitutes a new Hopf brace. We find such conditions for the pairs $(A_{1}\otimes H_{1},A_{2}\bowtie H_{2})$ and $(A_{1}\bowtie H_{1},A_{2}\sharp H_{2})$ to be Hopf braces, which are particular situations of the general problem described above, and we apply these results to study when the Drinfeld's Double gives rise to a Hopf brace.