Narrow Results

If A is an associative algebra, then we can define the adjoint Lie algebra $A^{(-)}$ and Jordan algebra $A^{(+)}$. It is easy to see that any associative Rota–Baxter (RB) operator on A induces a Lie and Jordan RB operator on $A^{(-)}$ and $A^{(+)}$, respectively. Are there Lie (Jordan) RB operators, which are not associative RB operators? In this paper, we explore these questions for the Sweedler algebra $H_4$, which is a 4-dimensional non-commutative Hopf algebra. More precisely, we describe the RB operators on the adjoint Lie algebra $H_4^{(-)}$.