Deformations of Jordan Algebras via the Jordan Defect: An Explicit Low--Degree Deformation Complex

TL;DR

This paper provides a concrete, explicit description of low-degree deformation theory for Jordan algebras over characteristic zero fields, focusing on the Jordan defect and an associated deformation complex.

Contribution

It introduces an explicit low-degree deformation complex for Jordan algebras using the Jordan defect, facilitating computation of infinitesimal deformations and obstructions.

Findings

Explicit low-degree deformation complex constructed

Second cohomology classifies infinitesimal deformations

Obstruction space identified for extending deformations

Abstract

Over a field of characteristic $0$ we give a concrete, computation--ready description of Jordan algebra structures and their low--order deformation theory. The Jordan identity is quartic in the elements and cubic in the multiplication, and in characteristic $0$ it is equivalent to its standard four--variable polarization. We encode this polarization as a cubic map in the product~$\mu$, called the \emph{Jordan defect} $J(\mu)$. Linearizing this defect yields an explicit low--degree deformation complex \[ C^1(J)\xrightarrow{\;\delta_\mu\;} C^2(J)\xrightarrow{\;d_\mu\;} C^3(J), \] whose second cohomology classifies infinitesimal deformations modulo equivalence and whose obstruction space \[ \mathrm{Obs}^3_\mu := C^3(J)/\operatorname{im}(d_\mu) \] contains the primary obstruction to extending such deformations. We emphasize that this construction captures only the low--degree part of the operadic deformation theory and does not claim to produce the full governing $L_\infty$ structure.