Search

MiS Preprint Repository

Delve into the future of research at MiS with our preprint repository. Our scientists are making groundbreaking discoveries and sharing their latest findings before they are published. Explore repository to stay up-to-date on the newest developments and breakthroughs.

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:
Jan 12, 2022
Published:
Jan 12, 2022
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