Defining relations for automorphism groups of free algebras

Abstract

We describe a set of defining relations for automorphism groups of finitely generated free algebras of Nielsen-Schreier varieties. In particular, this gives a representation of the automorphism groups of free Lie algebras by generators and defining relations.

Free Steiner triple systems and their automorphism groups

“…These conjectures fit the context of the work [13] on linear Nielsen-Schreier varieties of algebras.…”

“…Contrary to the case of the automorphism group of free Steiner loop with n > 3 generators, we prove that the group Aut (S(x 1 , x 2 , x 3 )) is generated by three involutions (12), (13) and ϕ = e 1 (x 2 ).…”

“…The group Aut(S(x 1 , x 2 , x 3 )) is generated by three involutions (12), (13) and ϕ = e 1 (x 2 ) with relations (12)(13)(12) = (13)(12)(13), (ϕ(12)) 3 = (ϕ(13)) 4 = 1.…”

“…In paper [13] it has been proved that the automorphism group of a free algebra of an arbitrary linear Nielsen-Schreier variety is generated by elementary automorphisms with some specific relations (2)-(4) ( [13, pp. 210-211]).…”

The automorphism group of the rank 2 Lie algebra free in the variety $$\mathfrak{A}\mathfrak{N}_2 $$

“…Many articles are devoted to studying the automorphism groups of relatively free algebras. A list of references on this topic (without any claim to completeness) can be found for instance in [3,4].…”