Classification of bilinear maps with radical of codimension 2

TL;DR

This paper classifies all bilinear maps on an n-dimensional space with a radical of codimension 2, effectively classifying certain n-dimensional algebras with large annihilators, providing a comprehensive algebraic structure overview.

Contribution

It offers a complete classification of bilinear maps with radical of codimension 2, extending to the classification of n-dimensional algebras with large annihilators.

Findings

Classification of bilinear maps with radical of codimension 2

Complete description of n-dimensional algebras with annihilator of dimension n-2

Framework for understanding annihilator extensions of 2-dimensional algebras

Abstract

Let ${\mathbb V}$ be an $n$-dimensional linear space over an algebraically closed base field. We provide a classification, up to equivalence, of all of the bilinear maps $f:{\mathbb V} \times {\mathbb V} \to {\mathbb V}$ such that $dim(rad(f)) =n-2$. This is equivalent to give a complete classification (up to isomorphism) of all $n$-dimensional algebras with annihilator of dimension $n-2$ or, in other words, a classification of the annihilator extensions of all $2$-dimensional algebras.