Biderivations and Commuting Linear Maps on Hom-Lie Algebras

Our purpose is to determine skew-symmetric biderivations [Formula: see text] and commuting linear maps [Formula: see text] on a multiplicative Hom-Lie algebra [Formula: see text] having their ranges in an [Formula: see text]-module [Formula: see text], which are both closely related to [Formula: see text], the centroid of [Formula: see text] on [Formula: see text]. We give the relationship between biderivations and commuting linear maps on a regular Hom-Lie algebra and those on the related Lie algebra. Under appropriate assumptions, we also prove that every [Formula: see text] in [Formula: see text] is of the form [Formula: see text] for some [Formula: see text], and [Formula: see text] coincides with [Formula: see text]. Besides, we give the algorithms for describing [Formula: see text] and [Formula: see text].