Two-generated algebras and standard-form congruence

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.

Final version

References in corpus (3)

• Canonical forms for complex matrix congruence and *congruence
• Isomorphisms of some quantum spaces
• PBW deformations of Artin-Schelter regular algebras

Cited by in corpus (7)

• On the discriminant of twisted tensor products
• PBW deformations of Artin-Schelter regular algebras
• Cancellation and skew cancellation for Poisson algebras
• Meta Algebras and Biorthogonal Rational Functions: The Hahn Case
• Congenial Algebras: Extensions and Examples
• Some algebras similar to the 2x2 Jordanian matrix algebras
• Orthogonal polynomials and the deformed Jordan plane