Matched pairs and Yang-Baxter operators

TL;DR

This paper characterizes when certain algebraic structures related to Hopf algebras produce involutive Yang-Baxter operators, solving an open problem and classifying examples in a specific non-semisimple Hopf algebra.

Contribution

It provides necessary and sufficient conditions for matched pairs of actions on Hopf algebras to induce involutive Yang-Baxter operators, addressing an open problem and exploring structural properties.

Findings

Involutive Yang-Baxter operators correspond to braided commutative Hopf algebras.

The double cross product $H\bowtie H$ forms a Hopf algebra with a projection.

All Yang-Baxter operators on the 8-dimensional non-semisimple Hopf algebra $A_{C_2\times C_2}$ are involutive.

Abstract

Recently, Ferri and Sciandra introduced two equivalent algebraic structures, matched pair of actions on an arbitrary Hopf algebra and Yetter-Drinfeld brace. In fact, they equivalently produce braiding operators on Hopf algebras satisfying the braid equation, thus generalize the construction of Yang-Baxter operators by Lu, Yan and Zhu from braiding operators on groups, and also by Angiono, Galindo and Vendramin from cocommutative Hopf braces. In this paper, we provide equivalence conditions for such kind of Yang-Baxter operators to be involutive. Particularly, we give a positive answer for an open problem raised by Ferri and Sciandra, namely, a matched pair of actions on a Hopf algebra $H$ induces an involutive Yang-Baxter operator if and only if its intrinsic Hopf algebra $H_\rightharpoonup$ in the category of Yetter-Drinfeld modules over $H$ is braided commutative. Also, we show that the double cross product $H\bowtie H$ is a Hopf algebra with a projection and $H_\rightharpoonup$ serves as its subalgebra of coinvariants. As an illustration, we use a simplified characterization to classify matched pairs of actions on the 8-dimensional non-semisimple Hopf algebra $A_{C_2\times C_2}$ and analyze the associated Yang-Baxter operators to find that they are all involutive.