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

Technical Summary: Deformations of Jordan Algebras via the Jordan Defect

Author: Vincent E. Coll, Jr.
Subject Area: Algebraic Deformation Theory, Jordan Algebras, Operad Theory.

1. The Problem: Complexity of Jordan Deformation Theory

The deformation theory of Jordan algebras is inherently more complex than that of associative algebras. While associative algebras are governed by a quadratic operad (leading to the standard Hochschild cohomology), the Jordan identity is quartic in its elements and cubic in its multiplication.

In the language of operads, this means the Jordan operad is not quadratic. Consequently, the governing $L_\infty$-algebra for Jordan deformations requires higher-order brackets ($\ell_n$ for $n \geq 3$). Traditional operadic methods provide a theoretical framework for these deformations but often fail to provide a "computation-ready" description. They typically require complex resolutions that make it difficult to perform explicit, low-degree calculations for specific algebras without invoking heavy homotopical machinery.

2. Methodology: The "Jordan Defect" Approach

The author proposes a shift from abstract operadic resolutions to a concrete, polynomial-based approach. The methodology centers on the following steps:

• The Jordan Defect $J(\mu)$: Instead of working with the non-linear Jordan identity, the author defines a multilinear map $J(\mu): J^{\otimes 4} \to J$. This "defect" is a cubic map in the multiplication $\mu$ that represents the fully polarized version of the Jordan identity. A commutative bilinear map $\mu$ is a Jordan product if and only if $J(\mu) = 0$.
• Linearization: By linearizing this cubic defect, the author derives an explicit differential. If $\mu$ is a Jordan product and $\phi$ is a symmetric bilinear map, the differential $d_\mu \phi$ is defined as the first-order variation of the defect: $d_\mu \phi = \frac{d}{dt} \big|_{t=0} J(\mu + t\phi)$.
• Construction of a Truncated Complex: The author constructs a low-degree deformation complex:
  $$C^1(J) \xrightarrow{\delta_\mu} C^2(J) \xrightarrow{d_\mu} C^3(J)$$
  where:
  • $C^1(J) = \text{End}(J)$ (representing infinitesimal automorphisms/changes of coordinates).
  • $C^2(J) = \text{Sym}^2(J^*) \otimes J$ (representing infinitesimal deformations of the product).
  • $C^3(J) = \text{Sym}^3(J^*) \otimes J^* \otimes J$ (representing the space of potential obstructions).

3. Key Contributions and Results

• Explicit Differentials: The paper provides closed-form expressions for the equivalence differential $\delta_\mu$ (which handles gauge equivalence/coordinate changes) and the linearized Jordan differential $d_\mu$ (which handles the cocycle condition).
• Classification of Infinitesimal Deformations: The author proves that the second cohomology group $H^2_\mu$ of this complex classifies infinitesimal deformations modulo equivalence.
• Primary Obstruction Space: The author defines the obstruction space $\text{Obs}^3_\mu := C^3(J) / \text{im}(d_\mu)$. This space contains the primary obstruction to extending an infinitesimal deformation to a higher-order formal deformation.
• Verification via Examples:
  • Dimension 1: Confirms the complex correctly identifies the rigidity of the trivial unital case ($H^2 = 0$).
  • Dual Numbers: Provides a manual calculation showing that a specific deformation $\phi$ is a cocycle but not a coboundary, yielding $\dim H^2_\mu = 1$. This demonstrates the complex's ability to detect non-trivial deformations.
  • Symmetric $3 \times 3$ Matrices: Confirms compatibility with classical results by showing the complex recovers the known formal rigidity of $\text{Sym}_3(k)$.

4. Significance and Scope

The significance of this work lies in its computational utility. By reducing the deformation problem to the linearization of a single polynomial identity, the author transforms a high-level homotopical problem into a problem of finite linear algebra.

Limitations/Scope: The author explicitly notes that this is a "low-degree" theory. It does not attempt to construct the full $L_\infty$-structure or the higher-order obstruction theory (which would require defining a differential on $C^3$ and beyond). Instead, it provides the necessary tools to solve the most common practical questions in deformation theory: Is this algebra rigid? and What are its infinitesimal deformations?