[Paper Review] The group of automorphisms of the category of free associative algebras

Aivars Berzins|ArXiv.org|Feb 20, 2005

Advanced Topics in Algebra5 references11 citations

TL;DR

This paper solves the long-standing problem of characterizing the automorphism group of the category of free associative algebras over a field. It proves that this group is generated exclusively by semi-inner automorphisms—compositions of field automorphisms and algebra automorphisms—and mirror automorphisms, which reverse the order of non-commutative monomials. The result establishes a complete structural description of the automorphism group in the associative case, analogous to known results in commutative and Lie algebra settings.

ABSTRACT

In this paper, the problem formulated in [8] is solved. We prove, that the group of automorphisms of the category of free associative algebras is generated by semi-inner and mirror automorphisms

Motivation & Objective

• To determine the structure of the automorphism group of the category of free associative algebras over a field, a problem posed in earlier work by Plotkin.
• To resolve the open question of whether automorphisms of this category are generated by semi-inner and mirror automorphisms, as conjectured in the literature.
• To provide a method independent of the structure of Aut(W(x₁,…,xₙ)) for n > 2, which remains unknown and is known to differ significantly from the n=2 case.
• To extend the framework of universal algebraic geometry by clarifying the role of automorphisms in the category of free algebras in the associative variety.
• To establish that every automorphism of the category arises from a family of bijections on free algebras that are either algebra automorphisms or compositions with the mirror antiautomorphism.

Proposed method

• The proof uses the category-theoretic definition of automorphisms of the category of free associative algebras, where each automorphism τ is induced by a family of bijections μ = {μᵢ} on free algebras Wᵢ with i generators.
• It leverages the fact that τ acts on morphisms via conjugation: sᵀ = μⱼ ∘ s ∘ μᵢ⁻¹ for s ∈ Hom(Wᵢ, Wⱼ).
• The analysis begins with the 1-generated case W(x) = P[x], where the automorphism group is known to be generated by field automorphisms and algebra automorphisms, so μ₁ must be of the form αη₁.
• By composing with the inverse of the field automorphism, the problem reduces to analyzing a normalized automorphism τ′ for which μ₁ is an algebra automorphism.
• The key step involves analyzing the behavior of μ on the free 2-generated algebra W(x,y), using endomorphisms s ∈ Hom(W₁, W₂) and Hom(W₂, W₁) to derive constraints on μ.
• It proves that μ must satisfy μ(x+y) = μ(x) + μ(y) and μ(xy) = αμ(x)μ(y) + βμ(y)μ(x) with α, β ∈ P, and then shows that either α=1, β=0 (automorphism) or α=0, β=1 (antiautomorphism), leading to the conclusion that μ is either an automorphism or mirror antiautomorphism on W(x,y).

Experimental results

Research questions

• RQ1 What is the complete structure of the automorphism group of the category of free associative algebras over a field?
• RQ2 Can the automorphism group be generated by semi-inner and mirror automorphisms, as conjectured in universal algebraic geometry?
• RQ3 Is it possible to describe this group without relying on the unknown structure of Aut(W(x₁,…,xₙ)) for n > 2?
• RQ4 How do endomorphisms in Hom(Wᵢ, Wⱼ) constrain the possible forms of the bijections μᵢ that induce automorphisms of the category?
• RQ5 What conditions force a bijection μ on a free algebra to be an automorphism or antiautomorphism?

Key findings

• The automorphism group of the category of free associative algebras over a field P is generated by semi-inner automorphisms and mirror automorphisms.
• Semi-inner automorphisms are those induced by compositions of field automorphisms and algebra automorphisms on each free algebra.
• Mirror automorphisms arise from the antiautomorphism that reverses the order of monomials, and they are not inner or semi-inner.
• The proof shows that any automorphism of the category must act via bijections that are either algebra automorphisms or compositions with the mirror antiautomorphism on each free algebra.
• The analysis of the 2-generated free algebra W(x,y) leads to a key identity: μ(xy) = αμ(x)μ(y) + βμ(y)μ(x), with α, β ∈ P, and the only consistent solutions are α=1, β=0 or α=0, β=1.
• This implies that the induced bijection μ on W(x,y) is either an automorphism or an antiautomorphism, and this property extends to all higher-generated free algebras via endomorphism lifting.