Deformation Theory of Jordan Algebras

• Deformation theory of Jordan algebras is framed by dg-Lie algebra models that satisfy the Jordan identity via Maurer–Cartan equations.
• The cohomological approach classifies infinitesimal deformations (H²) and obstructions (H³) using symmetric cochain complexes.
• Matched pair and deformation map techniques extend classification to Jordan algebra complements and explicit factorization scenarios.

The deformation theory of Jordan algebras investigates families of algebra structures on a given vector space near a fixed Jordan product, focusing on describing, classifying, and controlling deformations via cohomological and algebraic models. Recent developments have provided explicit differential graded Lie algebra (dg-Lie) frameworks and new matched pair–deformation approaches, yielding computational tools and new perspectives on extension theory, classification of complements, and local algebraic invariants.

1. Symmetric Cohain Complex and the dg-Lie Algebra Model

Let $(J, \circ)$ be a Jordan algebra over a field $k$ of characteristic zero. The deformation theory is constructed using the symmetric cochain complex: $C^n(J) = \Sym^n(J^\vee) \otimes J, \quad n \geq 0,$ where $\Sym^n(J^\vee)$ denotes the symmetric power of the dual space. Cochains $f \in C^n(J)$ are totally symmetric multilinear maps $f: J^n \to J$, with grading $|f| = n-1$.

A graded Lie algebra structure is established on the shifted complex $C^\bullet(J)[1]$ via an unshuffle-averaged insertion operation: $$(f \circ g)(x_1, \ldots, x_{p+q-1}) = \frac{1}{(p-1)! q!} \sum_{\sigma \in \mathrm{Sh}(p-1, q)} f\big(x_{\sigma(1)}, \ldots, x_{\sigma(p-1)},\, g(x_{\sigma(p)}, \ldots, x_{\sigma(p+q-1)})\big),$$ for $f\in C^p(J)$, $g\in C^q(J)$, and $\mathrm{Sh}(a,b)$ the set of $(a,b)$-unshuffles. The graded commutator

$[f, g] = f \circ g - (-1)^{|f||g|} g \circ f$

satisfies the graded Jacobi identity, rendering the complex a genuine dg-Lie algebra when equipped with a suitable differential.

2. The Deformation Differential and the Maurer–Cartan Formalism

Let $\mu \in C^2(J)$ represent the Jordan multiplication: $\mu(x, y) = x \circ y$. The differential is defined by

$d_\mu := [\mu,\,\cdot\,] : C^n(J) \to C^{n+1}(J).$

A direct computation yields

$[\mu, \mu] = 2 (\mu \circ \mu),$

where

$$(\mu\circ\mu)(x, y, z) = \tfrac{1}{2} \sum_{\mathrm{cyc}} \left(\mu(\mu(x, y), z) - \mu(x, \mu(y, z))\right).$$

The vanishing $[\mu, \mu]=0$ is shown to be equivalent, by polarization, to the Jordan identity

$$(x\circ x)\circ(y\circ x) = x\circ (x\circ(y\circ x)), \quad \forall x, y \in J.$$

Thus, $(C^\bullet(J), [\,\cdot,\,\cdot\,], d_\mu)$ forms a dg-Lie algebra controller for deformations of $\mu$ (Coll, 23 Dec 2025).

3. Recursion, Gauge, and Cohomological Interpretation

A formal 1-parameter deformation

$$\mu_t = \mu + t \varphi_1 + t^2 \varphi_2 + \cdots,$$

with $\varphi_i \in C^2(J)$, satisfies the Maurer–Cartan equation $[\mu_t, \mu_t] = 0$. Expanding in powers of $t$ leads to the recursive system: $$\begin{cases} t^1: & d_\mu \varphi_1 = 0 \ t^2: & d_\mu \varphi_2 = -\frac{1}{2}[\varphi_1, \varphi_1] \ t^n: & d_\mu \varphi_n = -\frac{1}{2}\sum_{i+j=n} [\varphi_i, \varphi_j],\quad n \ge 3 \end{cases}$$ where gauge equivalence is induced by the formal adjoint action of $\exp(t f_1 + t^2 f_2 + \cdots)$, with $f_i \in C^1(J)$, so $\varphi_1$ is defined modulo coboundaries. The cohomology $H^2$ classifies infinitesimal deformations, obstructions to their integration lie in $H^3$, and formal equivalence classes correspond to adjoint orbits in the dg-Lie algebra (Coll, 23 Dec 2025).

Explicit low-degree formulas for the differential $d_\mu$ retrieve the classical McCrimmon differentials:

• $C^1(J) = \mathrm{End}(J)$, with $$(d_\mu f)(x, y) = f(x \circ y) - f(x) \circ y - x \circ f(y)$$
• $C^2(J)$ as symmetric bilinear maps, with

$$(d_\mu \varphi)(x, y, z) = \sum_{\mathrm{cyc}} \Big( \mu(\varphi(x, y), z) - \varphi(\mu(x, y), z)\Big)$$

4. Matched Pairs, Bicrossed Products, and Deformation Maps

A distinct but complementary deformation theory arises from extension and factorization problems. Given Jordan algebras $A$ and $V$, a matched pair consists of bilinear maps (right action $\triangleleft: V \times A \to V$, left action $\triangleright: V \times A \to A$) obeying six compatibility axioms, guaranteeing that the bicrossed product

$$A \bowtie V = A \oplus V,\quad (a, x)\circ(b, y) = (ab + x\triangleright b + y\triangleright a,\ x\triangleleft b + y\triangleleft a + xy)$$

inherits a Jordan structure (Agore et al., 2022).

The "deformation map" formalism enables the classification of complements inside split extensions. For a fixed embedding $A \hookrightarrow E \cong A \bowtie B$, a deformation map $r: B \to A$ satisfies the equation

$$r(xy) - r(x) r(y) = x \triangleright r(y) + y \triangleright r(x) - r(x \triangleleft r(y) + y \triangleleft r(x)),$$

defining new Jordan algebra structures $(B, *_r)$ with

$$x *_r y = xy + x \triangleleft r(y) + y \triangleleft r(x).$$

Isomorphism classes of $A$-complements correspond to $$\mathcal{H}^2(B, A \mid \triangleleft, \triangleright)$$, the quotient of deformation maps by a natural equivalence. This approach provides a cohomological classification for the complements problem and reveals a local analog of the second cohomology set (Agore et al., 2022).

5. Examples and Explicit Computations

Concrete instances illuminate the structure theory and practical computation:

• For $J = k e \oplus k u$, $e \circ x = x$, $u \circ u = a e + b u$, the explicit calculation of $d_\mu$ on $C^1(J)$ and $C^2(J)$ yields the constraints for infinitesimal deformations and derivations.
• In the setting of matched pairs, consider $$A = \langle a, b \mid a^2 = a, b^2 = b, ab = 0\rangle$$, $$V = \langle u, v \mid u^2 = u, v^2 = 0, uv = 0\rangle$$ with a trivial left action and a specified right action. The deformation map equations are solved, yielding a finite number of non-isomorphic 2-dimensional Jordan algebra structures as deformations of a single split extension and establishing the factorization index (Coll, 23 Dec 2025, Agore et al., 2022).

6. Significance, Implications, and Directions

The dg-Lie model provides a computation-friendly, cohomological framework paralleling the established deformation theories of associative and Lie algebras. It enables direct realization of the Jordan identity as a Maurer–Cartan condition and organizes deformation obstructions and equivalence systematically (Coll, 23 Dec 2025). The matched pair and deformation map technique extends the reach of deformation theory to new classes of problems, including the explicit classification of algebra complements in extensions, and leverages combinatorial and bicrossed product data to define and understand algebraic deformations beyond the traditional tangent-cohomology viewpoint (Agore et al., 2022).

These structures suggest a rich interplay between cohomological invariants, extension theory, and explicit classification in the landscape of Jordan algebras. Further exploration is expected to yield a deeper “deformation cohomology” with parallels to Hopf and group algebra extension theory, systematic obstruction classes, and new insights into the analytic and algebraic deformation spaces of finite and infinite-dimensional Jordan systems.