A Commutator Test for Two Elements to Generate the Free Algebra of Rank Two

Abstract

Nielsen [8] showed that the group of automorphisms of the free group F on two generators X, Y is generated by the three automorphisms XHV, Y^X; X\\->X~X, Y\\-^Y; X)->XY, Y H * Y. Each of these automorphisms sends the commutator (X, Y) = X~ { Y'1 XY to a conjugate of {X, Y) or (X, Y)~K Using this, Nielsen showed further that two elements X', Y ' (freely) generate F if and only if (X1, Y') is a conjugate of (X, Y) or (X,Y)~l. In [1, p. 197] Cohn conjectured that there were analogous results for the free associative algebra on two generators X, Y over a (commutative) field k. This was vindicated when Makar-Limanov [7] (cf [6]) and Czerniakiewicz [3] independently showed that the group of/c-algebra automorphisms of this free /c-algebra, k(X, Y>, is generated by the automorphisms of the form X\\-+Y, Yh+X; X n a X, 7 H Y, {<xek,u f 0); Each of these automorphisms sends the commutator [X, Y] = AT — YX to a nonzero scalar multiple of [X, Y~\\. Now if two elements X', Y ' generate k(X, Y> then they freely generate k(X, Y> (see [2, p. 244]), so [X', Y'] is a nonzero scalar multiple of [X, V]. The purpose of this note is to prove the converse and thus complete the analogy with Nielsen's results. We shall need the following important fact (that was crucial also in the arguments of Makar-Limanov and Czerniakiewicz). THE JUNG-VAN DER KULK THEOREM. For a field k the group of k-algebra automorphisms of the polynomial ring in two variables, /c[x, y], is generated by the automorphisms of the form x i— • y, y H-> x; x •- • ax + f(y), y H-> y (aefc,a ^ 0,f{y) e /c[y]). This was proved for fields of characteristic zero by Jung [4] and for arbitrary fields by Van der Kulk [10] (cf also [9, 6.3.1] for a relatively simple proof). We now come to our main result.