LAUSR

We introduce the notion of virtual endomorphisms of Lie algebras and use it as an approach for constructing self-similarity of Lie algebras. This is done in particular for a class of metabelian Lie algebras having homological type $$FP_n$$FPn, which are Lie algebra analogues of lamplighter groups. We establish several criteria when the existence of virtual endomorphism implies a self-similar Lie structure. Furthermore, we prove that the classical Lie algebra $$sl_n(k)$$sln(k), where char(k) does not divide n affords non-trivial faithful self-similarity.