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We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

MiS Preprint
1/2022

The Geometries of Jordan nets and Jordan webs

Arthur Bik and Henrik Eisenmann

Abstract

A Jordan net (resp. web) is an embedding of a unital Jordan algebra of dimension $3$ (resp. $4$) into the space $\mathbb S^n$ of symmetric $n\times n$ matrices. We study the geometries of Jordan nets and webs: we classify the congruence-orbits of Jordan nets (resp. webs) in $\mathbb S^n$ for $n\leq 7$ (resp. $n\leq 5$), we find degenerations between these orbits and list obstructions to the existence of such degenerations. For Jordan nets in $\mathbb S^n$ for $n\leq5$, these obstructions show that our list of degenerations is complete. For $n=6$, the existence of one degeneration is still undetermined.

To explore further, we used an algorithm that indicates numerically whether a degeneration between two orbits exists. We verified this algorithm using all known degenerations and obstructions, and then used it to compute the degenerations between Jordan nets in $\mathbb S^7$ and Jordan webs in $\mathbb S^n$ for $n=4,5$.

Received:
12.01.22
Published:
12.01.22
MSC Codes:
17C50, 14M15, 14L30, 65K10

Related publications

inJournal
2022 Journal Open Access
Arthur Bik and Henrik Eisenmann

The geometries of Jordan nets and Jordan webs

In: Annali di matematica pura ed applicata, 201 (2022) 5, pp. 2413-2464