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

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~$μ$, called the \emph{Jordan defect} $J(μ)$. 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.

💡 Research Summary

The paper presents a concrete, computation‑ready framework for studying low‑order deformations of Jordan algebras over a field of characteristic zero. A Jordan algebra is defined by a commutative bilinear product μ satisfying the quartic Jordan identity ((x^2\circ y)\circ x = x^2\circ (y\circ x)). By polarizing this identity the author introduces a cubic multilinear map, the Jordan defect (J(\mu)), which vanishes exactly when μ is a Jordan product.

The central construction is the linearization of this defect. For a linear endomorphism (D\in\operatorname{End}(J)) the infinitesimal change of μ induced by the one‑parameter automorphism (\exp(tD)) yields the equivalence differential
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