ON GENERATORS OF POLYNOMIAL ALGEBRAS IN 2 COMMUTING OR NON-COMMUTING VARIABLES

An element of a free associative algebra A(2) = K[x(1),x(2)] is called primitive if it is an automorphic image of x(1). We address the probl em of detecting primitive elements of A2: We present an algorithm that distinguishes primitive elements, and also give a couple of very hand y necessary conditions for primitivity that allow one to rule out many sorts of non-primitive elements of A(2) just by inspection. We also g ive a structural description of the automorphism groups Aut(A(2)) and Aut(P-2) (Where P-2 = K[x(1),x(2)] is the polynomial algebra in two va riables over the same ground field K) which is different from previous ly known descriptions. (C) 1998 Elsevier Science B.V. All rights reser ved.