On the classification of bilinear maps with radical of a fixed codimension

ABSTRACT

Let $\mathbb{V}$ be an n-dimensional linear space over an algebraically closed base field. We provide a general method for classifying, up to equivalence, all bilinear maps $f:\mathbb{V}\times \mathbb{V}\to \mathbb{V}$ such that $\mathrm{d}\mathrm{i}\mathrm{m}\left(\mathrm{r}\mathrm{a}\mathrm{d}\right(f\left)\right)=n-m$, in case a classification of bilinear maps $f:\mathbb{W}\times \mathbb{W}\to \mathbb{W}$ when $\mathrm{d}\mathrm{i}\mathrm{m}\left(\mathbb{W}\right)=m$ is known. This is equivalent to giving a procedure to classify (up to isomorphism) all n-dimensional algebras with annihilator of dimension n−m from a given classification of m-dimensional algebras. As an example of application, we consider the case in which m = 2.

Keywords:

• Bilinear maps
• algebras
• extension by radical
• classification theorem

2000 MSC:

• 15A03
• 15A63
• 17A01

Acknowledgments

We would like to thank the referee for the detailed reading of this work and for the valuable suggestions to improve the same.

Table 2. Classification of 3-dimensional bilinear pairs with 1-dimensional radicals.

Table 3. Classification of 4-dimensional bilinear pairs with 2-dimensional radicals.

Table 4. Classification of 5-dimensional bilinear pairs with 3-dimensional radicals.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The work is Supported by the PCI of the UCA ‘Teoria de Lie y Teoria de Espacios de Banach’, by the PAI with project number FQM298 and by the 2014-2020 ERDF Operational Programme and by the Department of Economy, Knowledge, Business and University of the Regional Government of Andalusia [Project reference: FEDER-UCA18-107643].