Axial Algebra: Jordan Type 1/2

Updated 25 August 2025

Axial algebras of Jordan type ½ are nonassociative, commutative algebras generated by primitive idempotents (axes) with a Peirce decomposition into eigenvalues 1, 0, and ½.
Their fusion laws and Miyamoto involutions impose a Z/2-grading that connects the algebraic structure to 3-transposition groups and vertex operator algebras such as the Griess algebra.
Classification results for 2- and 3-generated algebras reveal distinct types and criteria based on solid subalgebras, advancing our understanding of both Jordan and non-Jordan systems.

An axial algebra of Jordan type ½ is a commutative (in all current prototypes), generally nonassociative algebra over a field of characteristic ≠2, generated by a set of primitive idempotents—called axes—whose adjoint map (left multiplication) is diagonalizable with minimal polynomial dividing $(x-1)x(x-\frac{1}{2})$ . The associated Peirce decomposition and fusion laws mirror those of Jordan algebras generated by their idempotents, but allow for direct generalization to important structures arising in vertex operator algebras and finite group theory, such as the Griess algebra and Matsuo algebras. The Jordan type ½ case is structurally richer than the classical (associative) situation and is rigidly characterized by fusion rules, Miyamoto involutions, and the presence—or absence—of solid subalgebras.

1. Foundational Definitions: Primitive Axes, Peirce Decomposition, and Fusion Law

For an axis $a$ in an axial algebra of Jordan type ½, the adjoint map ${\rm ad}_a$ acts on $A$ with minimal polynomial dividing $(x-1)x(x-\frac{1}{2})$ , yielding the eigenspace (or Peirce) decomposition: $A = A_1(a) \oplus A_0(a) \oplus A_{½}(a),$ where $A_1(a) = \mathbb{F} a$ is 1-dimensional—this is the primitivity constraint. The fusion law prescribes allowed products between eigenvectors, encoded by the multiplication table:

$\star$	1	0	½
1	{1}	∅	{½}
0	∅	{0}	{½}
½	{½}	{½}	{1,0}

Notable consequences:

$A_1(a) \cdot A_0(a) = 0$
$A_{½}(a) \cdot A_{½}(a) \subseteq A_0(a) \oplus A_1(a)$
$A_1(a) \cdot A_1(a) = A_1(a)$ (since $a$ is idempotent)

These fusion rules enforce a $Z/2$ -grading: $A_+(a) = A_1(a) \oplus A_0(a), \qquad A_-(a) = A_{½}(a),$ and guarantee existence of the Miyamoto involution $T(a)$ : $T(a): x = x_+ + x_- \mapsto x_+ - x_-,$ an automorphism of order 2.

2. Miyamoto Involutions, 3-Transposition Groups, and Automorphism Structure

The involutive automorphisms $T(a)$ associated to each axis generate a normal set of 3-transpositions within $\operatorname{Aut}(A)$ : for any two such involutions $t,s$ , their product $ts$ has order in $\{1,2,3\}$ (Hall et al., 2014). The classification of finite 3-transposition groups (Fischer's theory) underpins characterization of many axial algebras of Jordan type ½.

In classical examples (Griess algebra, Monster group context), the Miyamoto involutions correspond to symmetries generated by specific involutions, and their algebraic fusion rules dictate structural features of the corresponding algebra (Hall et al., 2016).

3. Classification Results: 2- and 3-Generated Algebras, Solid Subalgebras

For axial algebras of Jordan type ½ generated by two primitive axes, the algebra falls into one of three types ("1A", "2B", and "3C(½)"), the latter characterized by basis $\{c_0, c_1, c_2\}$ and multiplication: $c_i \cdot c_j = \tfrac{1}{2} (c_i + c_j - c_k), \quad \{i,j,k\} = \{0,1,2\}.$ Any $n$ -generated algebra is spanned by its axes, and the associated Miyamoto group is a quotient of a 3-transposition group (Hall et al., 2014, Hall et al., 2017).

For 3-generated primitive axial algebras (with axes $a,b,c$ ), dimension is at most 9; universal examples depend on Frobenius form parameters $(\alpha, \beta, \gamma, \psi)$ , and, where the radical of the form vanishes, yield classical Jordan matrix algebras or Hermitian Jordan algebras (Bildanov et al., 2023). Solid subalgebras—those in which all primitive idempotents generated are axes—play a central role in discerning whether an algebra is Jordan or not (Gorshkov et al., 29 Jan 2024, Desmet, 27 Mar 2024, Rowen, 22 Aug 2025).

Solid Subalgebra Table (Characteristic Zero):

Frobenius Value $(a,b)$	Solid?	Structure
$\notin \{0, \tfrac{1}{4}, 1\}$	Yes	Jordan or Matsuo
$0$, $1$	Yes	Jordan or Matsuo
$\tfrac{1}{4}$	No	Exactly 3 axes

The property that every 2-generated subalgebra is solid is equivalent (in char $\neq 2$ , and under technical spanning requirements) to the entire algebra being Jordan (Desmet, 27 Mar 2024).

4. Universal Constructions, Matsuo Algebras, and Beyond

Matsuo algebras, $M(\Gamma, \frac{1}{2}, \mathbb{F})$ , constructed from Fischer spaces and 3-transposition groups (Hall et al., 2014, Hall et al., 2017, Gorshkov et al., 2023), provide canonical examples: $a_p^2 = a_p; \quad a_p a_q = 0 \text{ if %%%%29%%%% not collinear}; \quad a_p a_q = \tfrac{1}{2}(a_p + a_q - a_r) \text{ if %%%%30%%%% is a line}.$ Miyamoto involutions on Fischer spaces coincide with the algebra's internal involutions, tightly linking automorphism group theory with the fusion structure (Hall et al., 2016, McInroy et al., 2022).

Recent results expose wider landscape: there are infinitely many primitive axial algebras of Jordan type ½ that are not Matsuo algebras nor Jordan algebras (Gorshkov et al., 2023, Rowen, 22 Aug 2025). Universal decomposition algebras, constructed using formal fusion rules, provide generic objects for categorical and classification purposes (Yabe, 2022).

5. Frobenius Forms, Power-Associativity, and Characteristic Constraints

Every primitive axial algebra of Jordan type ½ carries a unique symmetric Frobenius form (bilinear, associative with multiplication, normalized so $(a,a)=1$ for any axis) (Hall et al., 2017, Gorshkov et al., 2022, Gorshkov et al., 29 Jan 2024). This form imposes orthogonality between Peirce components and controls the possible interactions of axes. Unital algebras are guaranteed under quasi-definite and strong axiality; the unit decomposes as a minimal sum of orthogonal axes (Gorshkov et al., 2022).

Power-associativity, though not automatically assured by the fusion rules, is forced when certain polynomial identities ( $x^2x^2=x^4$ , $x^3x^2=xx^4$ ) hold strictly (Segev, 2017). In such cases, the algebra is Jordan. Characteristic restrictions (typically char $\neq2$ , sometimes $\neq3$ ) are required for the full strength of classification and fusion rule arguments (Burde et al., 2014, Segev, 2017).

6. Noncommutative Generalizations and Weak Primitivity

Decomposition algebras and weakly primitive variants allow broader fusion rules and noncommutative multiplication, with axes defined via left or right multiplication decompositions. Noncommutative generalizations maintain Peirce-type gradings, yield flexible (but not always Jordan) algebras, and support solidness criteria and analytic classification in low dimensions (Rowen et al., 2021, Rowen et al., 16 Aug 2024).

7. Broader Impact, Applications, and Open Directions

Axial algebras of Jordan type ½ have unified varied phenomena from nonassociative algebra, group theory (notably 3-transposition groups, Griess algebra, Monster, Majorana theory), and Frobenius geometry (matrix Jordan algebras, Hermitian structures, Albert algebra). The presence—or absence—of solid subalgebras governs the global nature of the algebra (Jordan vs. non-Jordan), with generic constructions showing the possibility of non-Matsuo, non-Jordan examples.

Current classification schemes are complete for 2- and 3-generated algebras; results for four primitive axes include a sharp dimension bound (81) (Medts et al., 2023). Questions remain about the full landscape in higher dimensions, the structure of automorphism groups, and refinement of solidness vs. genericity criteria—especially in the context of affine algebraic geometry and universal constructions (Rowen, 22 Aug 2025).

A plausible implication is that the variety-theoretic and polynomial identity approach will yield further examples, perhaps characterizing structural deformations of standard fusion-rule algebras. The existence of universal axial identities and variety-theoretic moduli spaces (with nondegenerate Frobenius forms) points toward a classification program that is simultaneously algebraic, geometric, and combinatorial.

Summary Table: Core Properties of Axial Algebras of Jordan Type ½

Property	Description	Reference
Fusion Law	Prescribed via Peirce decomposition and explicit multiplication table	(Hall et al., 2014)
Miyamoto Involutions	Automorphisms implementing $Z/2$ -grading; generate 3-transposition group	(Hall et al., 2016)
Frobenius Form	Unique, symmetric, associative bilinear form; normalizes axes, controls orthogonality	(Hall et al., 2017)
Solid Subalgebras	All primitive idempotents in 2-generated subalgebras are axes; equivalent to Jordan property	(Gorshkov et al., 29 Jan 2024, Desmet, 27 Mar 2024)
Universal Constructions	Existence of category-theoretic universal objects for fusion rules; facilitate classification	(Yabe, 2022, Rowen, 22 Aug 2025)
Noncommutative Cases	Decomposition algebras generalize structure/fusion rules to flexible, noncommutative models	(Rowen et al., 2021)

These collective results integrate algebraic, combinatorial, and categorical perspectives to yield a robust and rapidly expanding theory of axial algebras of Jordan type ½.