On Self-Similar Lie Algebras and Virtual Endomorphisms

TL;DR

This paper introduces virtual endomorphisms for Lie algebras to construct and analyze self-similar structures, especially in metabelian Lie algebras and classical Lie algebras like sln(k).

Contribution

It develops a new framework using virtual endomorphisms to identify self-similarity in Lie algebras, including classical and metabelian types.

Findings

Criteria established for virtual endomorphisms implying self-similarity.

Classical Lie algebra sln(k) admits non-trivial faithful self-similarity.

Application to metabelian Lie algebras with homological type FPn.

Abstract

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 F Pn, which are variants of lamp-lighter 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 sln(k), where char(k) does not divide n affords non-trivial faithful self-similarity.