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

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.

💡 Research Summary

The paper undertakes a complete classification of associative Rota–Baxter operators on the Sweedler Hopf algebra (H_{4}), a four‑dimensional non‑commutative, non‑cocommutative algebra over a field (F) of characteristic different from two. A Rota–Baxter operator of weight (\lambda\neq0) is a linear map (R:A\to A) satisfying the identity
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