2025/12/23 by Vincent E. Coll, Coll, Vincent E.
Mathematics · Physics and Astronomy · #Advanced Topics in Algebra #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Nonlinear Waves and Solitons #Rings and Algebras (math.RA)
paper · doi:10.48550/arxiv.2512.19975
openalex publication_date 2025/12/23 · openalex created_date 2025/12/25 · openalex updated_date 2026/07/28
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 Jordan defect J(μ). Linearizing this defect yields an explicit low--degree deformation complex C1(J)\xrightarrow δμ C2(J)\xrightarrow dμ C3(J), whose second cohomology classifies infinitesimal deformations modulo equivalence and whose obstruction space Obs3μ:= C3(J)/im(dμ) 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_∞ structure.