Contractions of Low-Dimensional Nilpotent Jordan Algebras

Abstract

In this article, we classify the laws of three-dimensional and four-dimensional nilpotent Jordan algebras over the field of complex numbers. We describe the irreducible components of their algebraic varieties and extend contractions and deformations among their isomorphism classes. In particular, we prove that 𝒥² and 𝒥³ are irreducible and that 𝒥⁴ is the union of the Zariski closures of the orbits of two rigid Jordan algebras.

Key Words:

• Contraction
• Jordan algebras
• Nilpotent
• Rigidity

2000 Mathematics Subject Classification:

• 17C10
• 17C55

ACKNOWLEDGMENTS

The authors express their gratitude to the referee for correcting several mistakes and pointing out references [5] and [6].

The first author was supported in part by MTM 2006-09152 and CC607-UCM/ESP-2922.

Notes

Communicated by I. Shestakov.