Matched pairs and Yetter–Drinfeld braces

Abstract It is proven that a matched pair of actions on a Hopf algebra H is equivalent to the datum of a Yetter–Drinfeld brace, which is a novel structure generalizing Hopf braces. This [-30pt] improves a theorem by Angiono, Galindo, and Vendramin, originally stated for cocommutative Hopf braces. These Yetter–Drinfeld braces produce Hopf algebras in the category of Yetter–Drinfeld modules over H , through an operation that generalizes Majid’s transmutation. A characterization of Yetter–Drinfeld braces via 1-cocycles, in analogy to the one for Hopf braces, is given. Every coquasitriangular Hopf algebra H will be seen to yield a Yetter–Drinfeld brace, where the additional structure on H is given by the transmutation. We compute explicit examples of Yetter–Drinfeld braces on the Sweedler’s Hopf algebra, on the algebras $E(n)$ , on $\mathrm {SL}_{q}(2)$ , and an example in the class of Suzuki algebras.