Deformations of Jordan Algebras via the Jordan Defect: An Explicit Low--Degree Deformation Complex
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908818247516160 |
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| author | Coll Jr, Vincent E. |
| author_facet | Coll Jr, Vincent E. |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_19975 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Deformations of Jordan Algebras via the Jordan Defect: An Explicit Low--Degree Deformation Complex Coll Jr, Vincent E. Rings and Algebras 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. |
| title | Deformations of Jordan Algebras via the Jordan Defect: An Explicit Low--Degree Deformation Complex |
| topic | Rings and Algebras |
| url | https://arxiv.org/abs/2512.19975 |