On Solvability of Automorphism Groups of Commutative Algebras

TL;DR

This paper investigates conditions under which the identity component of the automorphism group of a finite-dimensional commutative algebra is solvable, extending previous work by removing restrictions on the radical quotient dimension.

Contribution

It provides new sufficient conditions for the solvability of automorphism groups of commutative algebras, generalizing prior results that required specific radical quotient dimensions.

Findings

Established new criteria for solvability of automorphism groups

Extended previous results to broader classes of algebras

Removed restrictions on the dimension of the radical quotient

Abstract

Let $A$ be a finite-dimensional commutative associative algebra with unity over an algebraically closed field $\mathbb{K}$. The purpose of the paper is to study the solvability of $G_A$, where $G_A$ is the identity component of $\text{Aut}_\mathbb{K}(A)$. Inspired by Pollack's work, Saor\'in and Asensio have started this study for a commutative associative algebra $A$ when $\text{dim}(R/R^2)=2$, where $R$ is the Jacobson radical of $A$. In this paper, we give new sufficient conditions on $A$ so that $G_A$ is solvable without any restriction on $\text{dim}(R/R^2)$.