About Hopf braces and crossed products
Ramón González Rodríguez, Brais Ramos Pérez
TL;DR
This work investigates how to construct new Hopf braces from matched pairs and crossed-product constructions within a strict symmetric monoidal category, introducing explicit compatibility criteria that ensure bicrossed and smash-type products yield Hopf braces. It presents two main theorems (mainth2 and mainth) detailing necessary and sufficient conditions, expressed via equations (E1)-(E2) and (C1)-(C3), that extend Agore’s earlier crossed-product results to mixtures of two Hopf braces. The authors apply these criteria to Drinfeld’s Double $D(H)$, showing that $(T(H),D(H))$ is a Hopf brace when $H$ is cocommutative and giving conditions for $(D(H),T(H))$ in other cases, including commutative and non-cocommuttative settings. The results illuminate the interplay between cocommutativity classes and Hopf-brace structures, and establish categorical links between cocommutative Hopf braces and matched-pair data, with implications for factorization problems and quantum Yang–Baxter equation solutions.
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.