Cohomology and Extensions of BiHom-LieDer Pairs

We introduce BiHom–LieDer pairs, consisting of a BiHom–Lie algebra endowed with a compatible (α, β)-derivation. Representations of such pairs are defined, and an associated cohomology complex is constructed by combining the BiHom–Lie algebra cohomology with an operator induced by the derivation. We prove that the second cohomology group classifies central extensions of BiHom–LieDer pairs. Furthermore, given a central extension of the underlying BiHom–Lie algebras, we determine necessary and sufficient conditions for derivations on the kernel and the quotient to lift to a derivation on the whole extension. The obstruction to this lifting is shown to be a cohomology class in the second cohomology group of the quotient. These results extend the corresponding theories for Lie and Hom–Lie algebras with derivations.