Matched pairs of actions on the Kac-Paljutkin algebra $H_8$

TL;DR

This work classifies all matched pairs of actions on the Kac–Paljutkin Hopf algebra $H_8$ and analyzes their connection to coquasitriangular structures and Yang–Baxter operators. It identifies six distinct matched pairs on $H_8$, four arising from coquasitriangular data and two not; the latter yield Yang–Baxter operators that are involutive. The authors develop a systematic strategy to classify these pairs by examining interactions among basis elements, obtaining two explicit Situation patterns that lead to the two non-coquasitriangular pairs. Consequently, exactly these two non-derived pairs give rise to involutive Yang–Baxter operators, highlighting a precise link between non-coquasitriangular matched pairs and involutivity in the associated braid solutions.

Abstract

The notion of matched pair of actions on a Hopf algebra generalizes the braided group construction of Lu, Yan and Zhu, and efficiently provides Yang-Baxter operators. In this paper, we classify matched pairs of actions on the Kac-Paljutkin Hopf algebra $H_8$. Through calculations, we obtain 6 matched pairs of actions on $H_8$. Based on such a classification result, we find that four of them can be derived from the coquasitriangular structures of $H_8$, while the other two can not. Furthermore, we discover that the Yang-Baxter operators associated to exactly these two distinguished matched pairs of actions are involutive.

Theorems & Definitions (8)

• Definition 1.1: Ma
• Definition 1.2: FS
• Theorem 2.1
• Remark 2.2
• Theorem 2.3
• Theorem 3.1
• Theorem 4.1: Li
• Theorem 4.2