Two-generated algebras and standard-form congruence

TL;DR

The paper introduces standard-form congruence to classify algebras with two generators and one quadratic relation, providing canonical forms and applications to homogenizations, extending matrix congruence concepts.

Contribution

It develops a new classification method using standard-form congruence for two-generated algebras with quadratic relations, including canonical forms and homogenization classifications.

Findings

Canonical forms for 3x3 matrices under standard-form congruence

Classification of two-generated algebras with quadratic relations

Application to homogenizations of these algebras

Abstract

Matrix congruence can be used to mimic linear maps between homogeneous quadratic polynomials in $n$ variables. We introduce a generalization, called standard-form congruence, which mimics affine maps between non-homogeneous quadratic polynomials. Canonical forms under standard-form congruence for three-by-three matrices are derived. This is then used to give a classification of algebras defined by two generators and one degree two relation. We also apply standard-form congruence to classify homogenizations of these algebras.