The group of automorphisms of a commutative algebra
We study the automorphism group \(\text{Aut}(A)\) of a local commutative algebra \(A\) and its identity component \(G_ A\). The monomial algebras are then characterized as those for which \(G_ A\) has rank \(n = \dim J/J^ 2\) and, in particular, the algebras of the form \(K[X_ 1,\dots, X_ n]/ (X_ 1, \dots, X_ n)^ \ell\) are characterized as those for which \(\text{Aut} (A)\) is a semidirect product of a unipotent normal subgroup and the linear group \(\text{GL}(n,K)\). A general strategy to study \(\text{Aut} (A)\) via \(\text{Aut} (G(A))\), where \(G(A)\) is the associated graded algebra, is also developed. That strategy allows us to prove that, when \(n = 2\) and \(A\) is not isomorphic to \(K[X,Y] / (X,Y)^ \ell\), \(G_ A\) is a solvable group. If \(U_ A\) is the (unipotent) kernel of the canonical morphism \(\text{Aut} (A) \to \text{GL} (J/J^ 2)\), then a precise description of the quotient \(G_ A/U_ A\) is given, when \(A\) is gradable by the radical. In the final section it is seen that, again with \(n = 2\), no algebra with Loewy length \(L(A) \leq 4\) has unipotent \(G_ A\) and we identify, up to isomorphism, all those with \(L(A) = 5\) for which \(G_ A\) is unipotent or \(\text{Aut}(A)\) is nilpotent.
scientific article; zbMATH DE number 784098