Classification of Nijenhuis operators on three-dimensional multiplicative simple Bihom-Lie algebras

We provide a complete classification of Nijenhuis operators on three-dimensional multiplicative simple BiHom-Lie algebras over the complex field C. Based on the classification of such algebras by Saadaoui into three families L1, L2, and L3, we derive the complete system of equations arising from the equivariance conditions and the Nijenhuis identity. We provide explicit matrix representations of all solutions, including all degenerate parameter cases. We construct the BiHom-Frölicher-Nijenhuis bracket on the cochain complex of a BiHom-Lie algebra, showing that Maurer-Cartan elements of this graded Lie algebra are precisely Nijenhuis operators. This enables us to define the cohomology of Nijenhuis operators and study formal deformations. We introduce BiHom-NS-Lie algebras as the algebraic structure underlying Nijenhuis operators. We provide explicit deformed BiHom-Lie algebra structures and prove the non-isomorphism of different families. We also study the moduli space of Nijenhuis operators and provide computational verification of all results.