The structure of automorphism groups of zero-dimensional monomial algebras

Source abstract

Let $A$ be a zero-dimensional monomial algebra over an algebraically closed field of characteristic zero, that is, a finite-dimensional quotient of a polynomial ring by a monomial ideal. Its automorphism group $G$ is a linear algebraic group, described through the homogeneous nilpotent derivations of $A$. We analyze the structure of $G$ in detail. Its identity component $G^0$ is a semidirect product of its unipotent radical and a reductive subgroup isomorphic to a product of general linear groups, and for each root degree we characterize when the associated derivations give rise to an additive root subgroup, and determine its dimension. Using the Lie brackets of these derivations, we then give an explicit algorithm that produces, out of the minimal monomial generators of the ideal, a family of root subgroups generating $G^0$ together with a maximal torus. Such a family is minimal in the generic case. We also show that the component group $G/G^0$ can be arbitrary: every finite group arises as the component group of the automorphism group of some zero-dimensional monomial algebra. Finally, we apply these results to the algebras $\mathbf{k}[\mathbf{x}]/\mathfrak{m}^d$, showing that the subgroup generated by a maximal torus and the outer root subgroups is exactly the subgroup of automorphisms with constant Jacobian determinant, and we deduce from this a new proof of Anick's theorem on the density of the tame automorphisms of $\mathbf{k}[\mathbf{x}]$.