Rota-Baxter operators on a six-dimensional simple Leibniz algebra

We study Rota-Baxter operators of weight zero on a six-dimensional simple Leibniz
algebra whose associated Lie algebra is sl 2 (C) and whose Leibniz kernel is the three-dimensional
irreducible sl 2 -module V (2). Using a fixed module isomorphism between the Lie part and the
Leibniz kernel, we represent an arbitrary linear operator in block form and reduce the Rota-Baxter
identity to a system of operator equations on sl 2 . We completely describe the case in which the block
mapping the Leibniz kernel to the Lie part vanishes. When the Lie component is zero, we obtain
an intrinsic description of all admissible operators in terms of the Killing form. For a nonzero Lie
component, we use the known classification of weight-zero Rota–Baxter operators on sl 2 to determine
the compatible remaining blocks and the dimensions of the corresponding solution spaces. Several
exceptional parameter values at which these dimensions increase are also identified. The cases in
which the block from the Leibniz kernel to the Lie part is nonzero are left for further investigation.