On generators of polynomial algebras in two commuting or non-commuting variables
Let \(A_2\) be the free associative algebra on \(x\), \(y\) over a field \(K\) and \(P_2\) the polynomial ring on \(x\), \(y\) over \(K\). The authors study their automorphism groups \(\Aut(A_2)\) and \(\Aut(P_2)\). They observe that these groups are isomorphic, with the Abelianization \(A_2\to P_2\) providing a natural isomorphism. They now define two subgroups of \(\Aut\). \(\Aut^0(A_2)\) is the group of augmentation-preserving automorphisms of \(A_2\) and \(\Aut_{1L}(A_2)\) generated by elementary automorphisms (i.e. of form \(x\to x+f(y)\), \(y\to y\) or \(x\to x\), \(y\to y+f(x)\)). Similar definitions apply for \(P_2\). They prove the following theorem: For any ground field \(K\), \(\Aut^0(A_2)\) can be expressed as a semidirect product of \(\Aut_{1L}(A_2)\) and \(\text{GL}_2(K)\). When the polynomials in the generators of \(\Aut_{1L}(A_2)\) have no constant or linear terms, the resultant group is a free product \(L*L\), where \(L\) is the direct sum of countably many copies of the additive group of \(K\). -- The rest of the paper is concerned with primitive elements. An algorithm is presented to distinguish primitive elements (in characteristic zero) and some necessary conditions for primitivity are given, e.g. every primitive element is reversible (admits the antiautomorphism fixing \(x\) and \(y\)).
scientific article; zbMATH DE number 1308255