Automorphisms of a Free Associative Algebra of Rank 2. I

Let Ä<2> = R that keeps (xy — yx) fixed (up to multiplication by an element of R), then . This follows from a more general result about endomorphisms of Ä<2> via a theorem due to H. Jung [6] concerning automorphisms of a commutative and associative algebra of rank 2. Introduction. Let F be a commutative domain with 1. We will further assume that every invertible matrix with coefficients in F is a product of elementary matrices (e.g. F a field, F a Euclidean domain). Let F be the free associative algebra of rank n over F. It has been conjectured (see P. M. Cohn [1, p. 33]) that every F automorphism of F is tame (i.e. a product of elementary automorphisms of F). The conjecture has been proved true for a free Lie algebra by P. M. Cohn [3], and for R(x,y), the free commutative and associative algebra of rank 2 (i.e. the polynomial ring in 2 commuting indeterminates x and y) by Jung [6]. We shall prove here that a certain class of automorphisms of F<2> satisfies the conjecture, namely that if F so that [A, B] = X[x, y], X a unit of F, is it true that A and B generate F<2>? We prove that the answer is "yes" if A=x + P, B=y+ Q, where F and Q satisfy condition (iii) below. The following is an outline of our proof. Given <p e AutB (R(.x, y}) so that <p keeps (xy—yx) fixed, Jung's theorem allows us to reduce the problem of proving that <p is tame to the case in which the canonical projection of <p to R(x, y) is the identity. More generally we prove _ Received by the editors October 27, 1970. AMS 1970 subject classifications. Primary 16A06, 16A72; Secondary 20F55, 16A02.