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

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 \[ C^1(J)\xrightarrow{\;δ_μ\;} C^2(J)\xrightarrow{\;d_μ\;} C^3(J), \] whose second cohomology classifies infinitesimal deformations modulo equivalence and whose obstruction space \[ \mathrm{Obs}^3_μ:= C^3(J)/\operatorname{im}(d_μ) \] 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.

Revised version. Corrected and streamlined presentation; clarified the low-degree deformation complex and its scope; added explicit examples and improved exposition