Classification of Frobenius algebra structures on two-dimensional vector space over any base field

TL;DR

This paper classifies all Frobenius algebra structures on two-dimensional vector spaces over any field, providing canonical forms and considering different field characteristics.

Contribution

It offers a complete classification of two-dimensional Frobenius algebras over arbitrary fields, including explicit canonical representatives for various characteristics.

Findings

Classification of all associative algebra structures with non-degenerate bilinear forms

Identification of Frobenius algebras among these structures

Canonical representatives for different field characteristics

Abstract

Classifying Frobenius algebras is a key question that has been addressed in various contexts. The structure of finite-dimensional Frobenius algebras depends on the base field and the dimension of the algebra, leading to different classification results depending on whether the base field has characteristic zero, characteristic $p$, or other properties. Frobenius algebras over fields of characteristic zero have been well-studied, often related to semisimple algebra theory. The behavior of Frobenius algebras over fields of positive characteristic presents new challenges, with connections to modular representation theory. In the paper, we first classify all associative algebra structures on a two-dimensional vector space over any base field equipped with a non-degenerate bilinear form. We then identify which of these are Frobenius algebras. Lists of canonical representatives of the isomorphism classes of these algebras are provided for a base field with characteristics not equal to two or three, as well as for characteristics two and three.