Generalized sharped cubic form and split spin factor algebra

Abstract

There is a well-known construction of a Jordan algebra via a sharped cubic form. We introduce a generalized sharped cubic form and prove that the split spin factor algebra is induced by this construction, and it satisfies the identity $((a,b,c),d,b)+((c,b,d),a,b)+((d,b,a),c,b)=0$. The split spin factor algebras have recently appeared in the classification of 2-generated axial algebras of Monster type fulfilled by T. Yabe; their properties were studied by J. McInroy and S. Shpectorov.

Keywords:

• Lie triple system
• sharped cubic form
• split spin factor algebra

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 17A30

Acknowledgments

The authors are grateful to the anonymous referee for useful remarks.

Additional information

Funding

V. Gubarev is supported by Mathematical Center in Akademgorodok under agreement No. 075-15-2022-281 with the Ministry of Science and Higher Education of the Russian Federation. A.S. Panasenko is supported by the Program of fundamental scientific researches of Russian Academy of Sciences, project FWNF-2022-0002. F. Mashurov is supported by the Postdoctoral program at Shenzhen International Center for Mathematics (SICM), SUSTech. The results of Section 2 are supported by the Program of fundamental scientific researches of Russian Academy of Sciences, project FWNF-2022-0002. The results of Section 4–6 are supported by Mathematical Center in Akademgorodok under agreement No. 075-15-2022-281 with the Ministry of Science and Higher Education of the Russian Federation.