One generator algebras

TL;DR

This paper classifies algebras generated by one element over a field, describing their automorphism groups and structure as products of simpler components, with applications to finite field extensions and algebraic automorphisms.

Contribution

It provides a comprehensive classification of one-generator algebras over perfect fields, detailing their structure, automorphism groups, and isomorphism conditions, including explicit descriptions of automorphism groups as wreath products.

Findings

Classification of algebras as products of quotients of polynomial rings

Automorphism groups split into wreath products of automorphisms and symmetric groups

Explicit description of automorphism groups involving semi-direct products and algebraic subgroups

Abstract

For $R_1,R_2,R_3,\dots$ a family of non isomorphic rings (or algebras) having each only 2 idempotents ($1$ and $0$), we classify up to isomorphism the rings (or algebras) obtained by taking products of powers of the different $R_i$. We show that the automorphism groups of such rings (or algebras) split naturally into the product of wreath products $Aut( R_n)\wr \mathfrak{S}_{m_n} $ for different $n$. These results are applied to algebras generated by one element over a perfect field $\mathbb{K}$. Such algebra is either $\mathbb{K}[X]$ or a quotient of $\mathbb{K}[X]$. We show that in the later case the algebra is isomorphic to a finite product of the form $A=\prod (\mathbb{L}_i[X]/(X^j))^{n_{i,j}}$, where the $\mathbb{L}_i$ are non isomomorphic finite field extensions of $\mathbb{K}$ $($not isomophic as $\mathbb{K}$-algebras$)$, with restrictions on the numbers $n_{i,j}$ if $\mathbb{K}$ is finite. We classify these algebras up to isomorphism. We have also that the $\mathbb{K}$-algebra automorphism group of $A=\prod (\mathbb{L}_i[X]/(X^j))^{n_{i,j}}$ splits naturally into the product of wreat products $Aut_\mathbb{K}(\mathbb{L}_i[X]/(X^j) )\wr \mathfrak{S}_{n_{i,j}}$ ($Aut_\mathbb{K}(-)$ is for $\mathbb{K}$-algebra automorphism group). Finally, we prove that $Aut_\mathbb{K}(\mathbb{L}_i[X]/(X^n) )$ is isomorphic to the semi-direct product $G_n(\mathbb{L}_i)\rtimes Aut_\mathbb{K}(\mathbb{L}_i)$ ($Aut_\mathbb{K}(-)$ is for $\mathbb{K}$-algebra automorphism group), where $G_n(\mathbb{L}_i)\simeq Aut_{\mathbb{L}_i}(\mathbb{L}_i[X]/(X^n) )$ ($\mathbb{L}_i$ algebra automorphism group) is an algebraic subgroup of invertible lower triangular matrices of dimension $(n-1)\times (n-1)$ with coefficients in $\mathbb{L}_i$; the conjugate of a matrix $M\in G_n(\mathbb{L}_i)$ by $\sigma \in Aut_\mathbb{K}(\mathbb{L}_i)$ is the matrix obtained from $M$ by applying $\sigma$ to its coefficients.