Automorphisms of a free associative algebra of rank $2$. I

Abstract:

Let ${R_{\left \langle 2 \right \rangle }} = R\left \langle {x,y} \right \rangle$ be the free associative algebra of rank 2, on the free generators $x$ and $y$, over $R$ ($R$ a field, a Euclidean domain, etc.). We prove that if $\varphi$ is an automorphism of ${R_{\left \langle 2 \right \rangle }}$ that keeps $(xy - yx)$ fixed (up to multiplication by an element of $R$), then $\varphi$ is tame, i.e. it is a product of elementary automorphisms of ${R_{\left \langle 2 \right \rangle }}$. This follows from a more general result about endomorphisms of ${R_{\left \langle 2 \right \rangle }}$ via a theorem due to H. Jung [6] concerning automorphisms of a commutative and associative algebra of rank 2.