We prove that the universal central extension of a direct limit of perfect Hom- Lie algebras (Li, Li ) is (isomorphic to) the direct limit of universal central extensions of (Li, Li ). As an application we provide the universal central extensions of some multiplicative Hom-Lie algebras. More precisely, we consider a family of multiplicative Hom-Lie algebras {(slk(A), k)}k∈I and describe the universal central extension of its direct limit.
