Automorphisms of a free associative algebra of rank 2. II

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Abstract
Let R R be a commutative domain with 1. R ⟨ x , y ⟩ R\langle x,y\rangle stands for the free associative algebra of rank 2 over R ; R [ x ~ , y ~ ] R;R[\tilde x,\tilde y] is the polynomial algebra over R R in the commuting indeterminates x ~ \tilde x and y ~ \tilde y . We prove that the map Ab : Aut ⁡ ( R ⟨ x , y ⟩ ) → Aut ⁡ ( R [ x ~ , y ~ ] ) \text {Ab}: \operatorname {Aut} (R\langle x,y\rangle ) \to \operatorname {Aut} (R[\tilde x,\tilde y]) induced by the abelianization functor is a monomorphism. As a corollary to this statement and a theorem of Jung [5], Nagata [7] and van der Kulk [8]* that describes the automorphisms of F [ x ~ , y ~ ] F[\tilde x,\tilde y] ( F F a field) we are able to conclude that every automorphism of F ⟨ x , y ⟩ F\langle x,y\rangle is tame (i.e. a product of elementary automorphisms).

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