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Tangent Lie Algebras of Automorphism Groups of Free Algebras

Ivan Shestakov, Ualbai Umirbaev

TL;DR

This work develops tangent Lie algebras $T(H)$ for automorphism groups of free algebras in characteristic zero, providing a unified framework to study tame versus wild automorphisms through derivations and divergences. It shows that, for several varieties, $T(H)$ sits inside the derivations of constant/divergence, enabling a correspondence between automorphism dynamics and tangent derivations, and uses this to analyze tameness and wildness. The authors prove that most known wild automorphisms are absolutely wild, except for classic exceptional cases like the Nagata and Anick automorphisms, and they establish that the Bergman endomorphism is absolutely wild in rank two. They also demonstrate that free algebras in a broad class of polynilpotent Lie varieties (beyond the abelian and metabelian cases) possess absolutely wild automorphisms, illustrating the wide reach of the tangent-algebra approach to detecting wild behavior.

Abstract

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.

Tangent Lie Algebras of Automorphism Groups of Free Algebras

TL;DR

This work develops tangent Lie algebras for automorphism groups of free algebras in characteristic zero, providing a unified framework to study tame versus wild automorphisms through derivations and divergences. It shows that, for several varieties, sits inside the derivations of constant/divergence, enabling a correspondence between automorphism dynamics and tangent derivations, and uses this to analyze tameness and wildness. The authors prove that most known wild automorphisms are absolutely wild, except for classic exceptional cases like the Nagata and Anick automorphisms, and they establish that the Bergman endomorphism is absolutely wild in rank two. They also demonstrate that free algebras in a broad class of polynilpotent Lie varieties (beyond the abelian and metabelian cases) possess absolutely wild automorphisms, illustrating the wide reach of the tangent-algebra approach to detecting wild behavior.

Abstract

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.

Paper Structure

This paper contains 5 sections, 21 theorems, 139 equations.

Table of Contents

  1. Introduction
  2. Tangent Lie algebras
  3. Universal enveloping algebras, Jacobian matrices, and divergence
  4. Approximately tame automorphisms and density
  5. Absolutely wild automorphisms and their detection

Key Result

Proposition 1

Let $R$ be a subalgebra of $L$ containing $L_0$. Then $(a)$$R$ is homogeneous with respect to each variable $x_i$. In particular, $R$ is homogeneous with respect to the degree function $\mathrm{deg}$. $(b)$$R$ is invariant under the action of the group of linear automorphisms $GL_n(K)$.

Theorems & Definitions (22)

  • Proposition 1
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Theorem 1
  • Lemma 5
  • Definition 1
  • Corollary 1
  • Lemma 6
  • ...and 12 more