Associative Rota--Baxter operators on the Sweedler algebra $H_4$

TL;DR

This work provides a complete classification of nontrivial Rota--Baxter operators on the Sweedler algebra $H_4$ with weight $\lambda\neq 0$, up to conjugation and dualization. Through a systematic kernel-dimension analysis (dimensions 3, 2, 1, 0) and careful use of $H_4$ subalgebras and automorphisms, the authors derive explicit operator families and establish conjugacy relations that collapse many apparent variations into a small set of orbits. The dualization analysis reveals how dual operators populate and relate these orbits, yielding a full list of RB-operators modulo the stated equivalences. The results extend the classification program for RB-operators to a concrete finite-dimensional Hopf-like algebra, and connect with existing classifications in related algebras.

Abstract

In this paper, we classify all Rota--Baxter operators on the Sweedler algebra $H_4$ up to conjugation and dualization. Modulo algebra (anti)automorphisms of $H_4$, we first describe its subalgebras and then analyse the kernel of a Rota--Baxter operator. The classification is carried out according to the dimension of this kernel, yielding a complete description of such operators. A complete list of operators is given in the theorem of the final section.

Theorems & Definitions (32)

• Definition
• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• Theorem 1
• proof
• Theorem 2
• proof