Self-similar Lie algebras

TL;DR

This paper introduces a broad definition of self-similar Lie algebras, demonstrating that key examples are nil and establishing conditions under which this property holds, with implications for well-known algebraic structures.

Contribution

It provides a general framework for self-similar Lie algebras and proves their nilpotency under certain conditions, linking them to important algebraic examples.

Findings

Self-similar Lie algebras encompass key examples like those related to Grigorchuk's and Gupta-Sidki's groups.

Sufficient conditions for nilpotency of self-similar Lie algebras are established.

The results apply to classes of algebras constructed by Petrogradsky, Shestakov, and Zelmanov.

Abstract

We give a general definition of self-similar Lie algebras, and show that important examples of Lie algebras fall into that class. We give sufficient conditions for a self-similar Lie algebra to be nil, and prove in this manner that the self-similar algebras associated with Grigorchuk's and Gupta-Sidki's torsion groups are nil as well as self-similar. We derive the same results for a class of examples constructed by Petrogradsky, Shestakov and Zelmanov.