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On self-similar Lie algebras and virtual endomorphisms

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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 \(FP_n\), 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 \(sl_n(k)\), where char(k) does not divide n affords non-trivial faithful self-similarity.

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Acknowledgements

The authors thank the referee for the suggestions that improved the paper. The first author was partially supported by the CNPq grant (200783/2018-1) and by the Fapesp grant (2014/09310-5), the second author was partially suported by FAPESP grant “projeto regular” 2016/05678-3 and CNPq grant “bolsa de produtividade em pesquisa” 301779/2017-1. The third author was suported by a FAPESP grant 2016/05271-0 for a visit to UNICAMP in September 2016.

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Authors and Affiliations

  1. University of São Paulo (USP), São Paulo, Brazil

    Vyacheslav Futorny

  2. State University of Campinas (UNICAMP), Campinas, Brazil

    Dessislava H. Kochloukova

  3. Universidade de Brasília (UnB), Brasília, Brazil

    Said N. Sidki

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  1. Vyacheslav Futorny
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  2. Dessislava H. Kochloukova
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  3. Said N. Sidki
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Correspondence to Vyacheslav Futorny.

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Futorny, V., Kochloukova, D.H. & Sidki, S.N. On self-similar Lie algebras and virtual endomorphisms. Math. Z. 292, 1123–1156 (2019). https://doi.org/10.1007/s00209-018-2146-6

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