Finite-dimensional monomial algebras are determined by their automorphism group

TL;DR

This paper proves that finite-dimensional monomial algebras can be uniquely identified by their automorphism groups among certain local algebras, enabling reconstruction of the monomial ideal from the automorphism group.

Contribution

It establishes a characterization of finite-dimensional monomial algebras via their automorphism groups and provides a method to recover the monomial ideal from this group.

Findings

Finite-dimensional monomial algebras are uniquely determined by their automorphism groups.

Automorphism groups can be used to recover the original monomial ideal.

The result applies to local algebras with fixed cotangent space dimension.

Abstract

A monomial algebra is the quotient of a polynomial algebra by an ideal generated by monomials. We prove that finite-dimensional monomial algebras are characterized by their automorphism group among finite-dimensional, local algebras with cotangent space of fixed dimension. In particular, we show how to recover a monomial ideal given the automorphism group of the corresponding monomial algebra.