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

Vincent E. Coll

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~$μ$, 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.

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

Abstract

Over a field of characteristic 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 it is equivalent to its standard four--variable polarization. We encode this polarization as a cubic map in the product~, called the \emph{Jordan defect} . Linearizing this defect yields an explicit low--degree deformation complex whose second cohomology classifies infinitesimal deformations modulo equivalence and whose obstruction space 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 structure.
Paper Structure (7 sections, 3 theorems, 17 equations)

This paper contains 7 sections, 3 theorems, 17 equations.

Table of Contents

  1. Jordan algebras and the Jordan defect
  2. The low--degree deformation complex
  3. Examples
  4. Dimension one
  5. Dual numbers (explicit calculation)
  6. Symmetric 3 by 3 matrices
  7. Comparison with the classical linearization

Key Result

Proposition 1.3

A commutative bilinear map $\mu$ is a Jordan product if and only if $J(\mu)=0$.

Theorems & Definitions (11)

  • Remark 1: Relation to the full operadic picture
  • Remark 2: What this viewpoint adds
  • Definition 1.1: Jordan defect
  • Remark 1.2
  • Proposition 1.3
  • proof
  • Lemma 2.1
  • proof
  • Remark 2.2: Canonical scope of the deformation complex
  • Proposition 4.1
  • ...and 1 more