Tangent Lie algebras of automorphism groups of free algebras
Shestakov I. Umirbaev U.
1 April 2026Elsevier Inc.
Linear Algebra and Its Applications
2026#734193 - 223 pp.
We study an analogue of the Andreadakis–Johnson filtration for automorphism groups of free algebras and introduce the notion of tangent Lie algebras for certain automorphism groups, defined as subalgebras of the Lie algebra of derivations. We show that, for many classical varieties of algebras, the tangent Lie algebra is contained in the Lie algebra of derivations with constant divergence. We also introduce the concepts of approximately tame and absolutely wild automorphisms of free algebras in arbitrary varieties and employ tangent Lie algebras to investigate their properties. It is shown that nearly all known examples of wild automorphisms of free algebras are absolutely wild, with the notable exceptions of the Nagata and Anick automorphisms. We show that the Bergman automorphism of free matrix algebras of order two is absolutely wild. Furthermore, we prove that free algebras in any variety of polynilpotent Lie algebras–except for the abelian and metabelian varieties–also possess absolutely wild automorphisms.
Automorphism , Derivation , Divergence , Filtration , Free algebra
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Instituto de Matemática e Estatística, Universidade de São Paulo, Caixa Postal 66281, SP, São Paulo, 05315-970, Brazil
Department of Mathematics, Wayne State University, Detroit, 48202, MI, United States
Institute of Mathematics and Mathematical Modeling, Almaty, 050010, Kazakhstan
Instituto de Matemática e Estatística
Department of Mathematics
Institute of Mathematics and Mathematical Modeling
10 лет помогаем публиковать статьи Международный издатель
Книга Публикация научной статьи Волощук 2026 Book Publication of a scientific article 2026