On the extension of automorphisms of polynomial rings
Let \(F\) be a field and let \(A=F\langle t_1,\dots,t_n\rangle\) be a free algebra with generators \(t_1,\dots,t_n\). Let us denote by \(T\) the commutator ideal of \(A\). Then \(C=A/T\) is a ring of polynomials and \(B=A/T^2\) is a free metabelian algebra. Let \(C'\) be the algebra which is anti-isomorphic to \(C\). The author introduces a ``gradient'' from \(B\) to \((C'\otimes_FC)^n\) and with the help of this gradient connects with every endomorphism of \(B\) a matrix with elements from \(C'\otimes_FC\) which is analogous to the Jacobi matrix. He shows that an endomorphism of \(B\) is an automorphism if and only if the corresponding matrix is invertible over \(C'\otimes_FC\). He also proves that any automorphism of the polynomial ring is induced by an automorphism of the corresponding free metabelian algebra.
scientific article; zbMATH DE number 923547