About Hopf braces and crossed products

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 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, analyzing when these are cocommutative, leading to solutions to the Quantum Yang-Baxter equation. These results are applied to study when the Drinfeld’s Double gives rise to a Hopf brace.