Classifying local Artinian Gorenstein algebras

TL;DR

This paper provides a detailed classification of local Artinian Gorenstein algebras by analyzing group actions on polynomial rings, offering explicit formulas for orbits and tangent spaces, and classifying specific Hilbert function cases.

Contribution

It introduces an explicit formula for orbits and tangent spaces in the classification of local Artinian Gorenstein algebras, and classifies algebras with specific Hilbert functions.

Findings

Classified algebras with Hilbert function (1, 3, 3, 3, 1)

Classified algebras with Hilbert function (1, 2, 2, 2, 1, 1, 1)

Provided explicit formulas for orbits and tangent spaces

Abstract

The classification of local Artinian Gorenstein algebras is equivalent to the study of orbits of a certain non-reductive group action on a polynomial ring. We give an explicit formula for the orbits and their tangent spaces. We apply our technique to analyse when an algebra is isomorphic to its associated graded algebra. We classify algebras with Hilbert function (1, 3, 3, 3, 1), obtaining finitely many isomorphism types, and those with Hilbert function (1, 2, 2, 2, 1, 1, 1). We consider fields of arbitrary, large enough, characteristic.