Brais Ramos Pérez
When is the Drinfeld Double Hopf algebra part of a Hopf brace structure?
Given two Hopf braces in a symmetric monoidal category \({\sf C}\), namely \(\mathbb{A}=(A_{1},A_{2})\) and \(\mathbb{H}=(H_{1},H_{2})\), such that \((A_{1},H_{1})\) and \((A_{2},H_{2})\) form matched pairs of Hopf algebras, we investigate the conditions under which the pair
\[(A_{1}\bowtie H_{1},\, A_{2}\bowtie H_{2})\]
defines a new Hopf brace. In particular, we establish necessary and sufficient conditions for the pairs
\[(A_{1}\otimes H_{1},\, A_{2}\bowtie H_{2})\quad\text{and}\quad(A_{1}\bowtie H_{1},\, A_{2}\sharp H_{2})\]
to be Hopf brace structures, viewing them as special instances of the general problem above.
The Drinfeld Double is one of the most prominent examples of a quasitriangular Hopf algebra, typically neither commutative nor cocommutative, and therefore provides solutions to the Quantum Yang-Baxter equation. As an application of the previous constructions, we study conditions under which the Drinfeld Double can be realized as part of a Hopf brace.
