Constructions of Rota–Baxter operators by L-R smash products
Colloquium Mathematicum
MSC: Primary 16T05; Secondary 17B38
DOI: 10.4064/cm9730-3-2026
Published online: 3 August 2026
Abstract
Let $A$ and $H$ be two cocommutative Hopf algebras such that $A$ is an $H$-bimodule Hopf algebra. Suppose that $R:A\rightarrow A$ is a linear map and $B$ is a Rota–Baxter operator of $H$. We characterize the Rota–Baxter operators on the L-R smash product $A\mathbin\natural H$ and give necessary and sufficient conditions for $\overline{B}(a\mathbin\natural h)=B(h_1)\triangleright R(a)\triangleleft B(h_2)\mathbin\natural B(h_3)$ to be a Rota–Baxter operator of $A\mathbin\natural H$ for $a\in A$ and $h\in H$. Then we consider the dual case, and construct a Rota–Baxter co-operator on the L-R smash coproduct $C\ltimes H$, where $C$ and $H$ are commutative Hopf algebras and $C$ is an $H$-bicomodule Hopf algebra.
