Matched pairs and Yetter-Drinfeld braces

TL;DR

This paper establishes a new equivalence between matched pairs of actions on Hopf algebras and Yetter-Drinfeld braces, generalizing Hopf braces and producing new Hopf algebra structures in Yetter-Drinfeld categories.

Contribution

It introduces Yetter-Drinfeld braces as a generalization of Hopf braces and proves their equivalence to matched pairs of actions on Hopf algebras, extending previous theorems.

Findings

Yetter-Drinfeld braces produce Hopf algebras via a generalized transmutation.

Explicit examples of Yetter-Drinfeld braces are computed for several Hopf algebras.

A characterization of Yetter-Drinfeld braces via 1-cocycles is provided.

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 generalising Hopf braces. This 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 generalises Majid's transmutation. A characterisation 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.