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Matched pairs and Yetter-Drinfeld braces

Davide Ferri, Andrea Sciandra

TL;DR

The paper removes the cocommutativity constraint in the Hopf-brace framework and proves a tight correspondence between matched pairs of actions on a Hopf algebra $H$ and Yetter--Drinfeld braces, realized via Majid's transmutation in the braided category $^{H^ullet}_{H^ullet} ext{YD}$. It provides a comprehensive 1-cocycle characterisation and shows how coquasitriangular Hopf algebras yield broad classes of Yetter--Drinfeld braces, expanding the landscape beyond cocommutative examples. A converse equivalence between Yetter--Drinfeld braces and matched pairs is established, giving an isomorphism of categories, and the theory is illustrated with explicit braces on the Sweedler algebra $H_4$, the algebras $E(n)$, $ ext{SL}_q(2)$, and Suzuki algebras. The work unifies transmutation with the Yetter--Drinfeld-brace framework and provides concrete constructions of Hopf algebras in $ ext{YD}$-categories, enriching the toolkit for studying solutions to the Yang--Baxter equation in noncocommutative settings.

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.

Matched pairs and Yetter-Drinfeld braces

TL;DR

The paper removes the cocommutativity constraint in the Hopf-brace framework and proves a tight correspondence between matched pairs of actions on a Hopf algebra and Yetter--Drinfeld braces, realized via Majid's transmutation in the braided category . It provides a comprehensive 1-cocycle characterisation and shows how coquasitriangular Hopf algebras yield broad classes of Yetter--Drinfeld braces, expanding the landscape beyond cocommutative examples. A converse equivalence between Yetter--Drinfeld braces and matched pairs is established, giving an isomorphism of categories, and the theory is illustrated with explicit braces on the Sweedler algebra , the algebras , , and Suzuki algebras. The work unifies transmutation with the Yetter--Drinfeld-brace framework and provides concrete constructions of Hopf algebras in -categories, enriching the toolkit for studying solutions to the Yang--Baxter equation in noncocommutative settings.

Abstract

It is proven that a matched pair of actions on a Hopf algebra 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 , 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 will be seen to yield a Yetter-Drinfeld brace, where the additional structure on is given by the transmutation. We compute explicit examples of Yetter-Drinfeld braces on the Sweedler's Hopf algebra, on the algebras , on , and an example in the class of Suzuki algebras.
Paper Structure (17 sections, 20 theorems, 139 equations, 8 tables)

This paper contains 17 sections, 20 theorems, 139 equations, 8 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Matched pairs of actions
  4. Hopf braces
  5. Matched pairs of actions and Yetter--Drinfeld braces
  6. Yetter--Drinfeld braces
  7. A general example
  8. A converse connection
  9. Yetter--Drinfeld braces as 1-cocycles
  10. Coquasitriangular Hopf algebras and Yetter--Drinfeld braces
  11. Coquasitriangular bialgebras
  12. Coquasitriangular Hopf algebras and matched pairs of actions
  13. Examples
  14. The Sweedler's Hopf algebra
  15. The Hopf algebras E(n)
  16. ...and 2 more sections

Key Result

Lemma 2.3

Let $\mathcal{M}$ be a (strict) braided monoidal category with braiding $\sigma$, let $H$ be an object in $\mathcal{M}$, and $m\colon H\otimes H\to H$, $u\colon \mathbbm{1}\to H$ any morphisms in $\mathcal{M}$. Then, the following hold: Moreover, if $m$ is a monomorphism in $\mathcal{M}$ and $\varsigma\colon H\otimes H\to H\otimes H$ is a morphism satisfying braided1--braided4, then $\varsigma$ s

Theorems & Definitions (63)

  • Definition 2.1
  • Remark 2.2
  • Lemma 2.3
  • Lemma 2.4: cf. Tambara tambara1990coendomorphism
  • proof
  • Remark 2.5
  • Lemma 2.7: cf. Majid Majid-book
  • proof
  • Definition 2.9
  • Proposition 2.10: Angiono, Galindo and Vendramin angiono2017hopf
  • ...and 53 more