Weakly Primitive Axial Algebras

• Weakly primitive axial algebras are nonassociative, commutative algebras defined by idempotent generators and a fusion law that governs eigenvector products.
• Their structure relaxes the strict minimality of the 1-eigenspace, allowing broader examples such as band semigroup algebras and noncommutative variants.
• These algebras find applications in vertex operator algebras, group construction, and combinatorial representation theory, supported by Frobenius forms and Miyamoto involutions.

Weakly primitive axial algebras are a generalization of the theory of primitive axial algebras, retaining the core structural features—commutative, nonassociative, idempotent-generated, multiplicity-free adjoint actions, and a fusion law—but relaxing the minimality of the 1-eigenspace attached to each axis. This relaxation enables the inclusion of broader families of examples, such as certain band semigroup algebras and noncommutative variants, while maintaining a coherent Peirce/fusion-rule framework. The paper of weakly primitive axial algebras is fundamental for extending the algebraic and group-theoretic connections originally observed in the context of Jordan and Monster type algebras, underpinning applications to vertex operator algebras, group construction, and combinatorial representation theory.

1. Structural Definition and Fusion Rule Formalism

A weakly primitive axial algebra over a field $\mathbb{F}$ is a commutative, possibly nonassociative, algebra $A$ generated by a distinguished set of idempotents (axes) $\mathcal{X}$ such that, for each axis $a \in \mathcal{X}$, the adjoint action $\mathrm{ad}_a: x \mapsto a x$ is semisimple with minimal polynomial dividing $(x - 1)x(x - \eta)$. This yields a Peirce decomposition: $$A = A_1(a) \oplus A_0(a) \oplus \bigoplus_{(\mu, \nu) \in S(a)} A_{(\mu, \nu)}(a)$$ with $A_1(a) = \mathbb{F} a$ (“weak” primitivity: the 1-eigenspace is spanned by $a$), a possibly larger $A_0(a)$, and additional two-sided eigenspaces $A_{(\mu, \nu)}(a)$. The essential relaxation is that while the “primitive” condition requires $A_1(a)$ to be $\mathbb{F}a$ for both left and right multiplication and for all axes, the weakly primitive notion allows the existence of additional (possibly off-diagonal) eigenspaces, provided the core $A_1(a) = \mathbb{F}a$ is preserved as the unique eigenline for eigenvalue $1$ (Rowen et al., 16 Aug 2024, Rowen et al., 21 Sep 2025).

The product of eigenvectors is governed by a specified fusion law $\ast$, which determines, for each pair of eigenvalues (or generalized eigenpairs), the set of possible target eigenspaces of their product: $$A_{\alpha}(a) \cdot A_{\beta}(a) \subseteq \bigoplus_{\gamma \in \alpha \ast \beta} A_{\gamma}(a)$$ This law extends to paired indices: e.g., for $A_{(\mu,\nu)}(a) \cdot A_{(\rho,\sigma)}(a)$, the product is only nonzero when $(\mu,\nu) = (\rho,\sigma)$, and then lands in either $A_1(a)$ or $A_0(a)$ according to the fusion rule (Rowen et al., 16 Aug 2024).

A crucial additional requirement is that $A_1(a) + A_0(a)$ form a subalgebra for each axis $a$ (a structural echo of the Seress condition), ensuring that trivial eigenparts are closed under multiplication, and that each $A_{(\mu,\nu)}(a)$ is a left and right module for $A_1(a) + A_0(a)$.

2. Fusion Rules, Grading, and Miyamoto Involutions

Weakly primitive axial algebras are organized by a “fusion rule,” which can be encoded as a symmetric relation on an index set (typically the set of eigenvalues, or pairs for two-sided decompositions): $\ast: S \times S \to 2^S$ The Jordan type fusion rule, for example, mandates multiplication tables of the form: $$\begin{aligned} A_1(a)A_1(a) &\subseteq A_1(a)\ A_0(a)A_0(a) &\subseteq A_0(a)\ A_1(a)A_0(a) &\subseteq A_0(a)\ A_1(a) A_{\eta}(a),\ A_0(a) A_{\eta}(a) &\subseteq A_{\eta}(a)\ A_{\eta}(a)A_{\eta}(a) &\subseteq A_1(a) + A_0(a) \end{aligned}$$ and the presence of a $\mathbb{Z}_2$-grading—where $A^+(a) = A_1(a) \oplus A_0(a)$ and $A^-(a) = \bigoplus_{(\mu, \nu)} A_{(\mu,\nu)}(a)$—supports the existence of involutive automorphisms (Miyamoto involutions) acting as $+1$ on $A^+$ and $-1$ on $A^-$ (Hall et al., 2014, Rehren, 2014). Such involutions, when taken across all axes, generate the so-called Miyamoto group, which is a (possibly 3-)transposition group controlling the automorphism structure of the algebra.

This grading and the Miyamoto group action persist in the weakly primitive setting, often with richer combinatorial and group-theoretic structure due to the possibility of nontrivial multiplicities in the decomposition (Rehren, 2014, Rowen et al., 16 Aug 2024).

3. Classification of 2-Generated Examples and Dimension Bounds

A central result, generalizing the Sakuma-type theorems for the primitive case, is that in the strictly primitive Jordan-type setting, any 2-generated algebra falls into a narrow list: $1\mathrm{A}$ (trivial), $2\mathrm{B}$ (two orthogonal idempotents), and $3\mathrm{C}(\eta)$ (3-dimensional, with structure constants parameterized by $\eta$). Weakly primitive axial algebras, by contrast, admit more elaborate behavior:

• Additional eigenvalues, or off-diagonal eigenspaces, may result in 4- or higher-dimensional algebras even for 2 axes (Rowen et al., 16 Aug 2024).
• For each axis $a$, the dimensions of $A_0(a)$ and the extra $A_{(\mu, \nu)}(a)$ directly influence the dimension and structure constants of the algebra.
• Specific algebraic conditions, such as $(ab)a = a(ba)$ and constraints on structure constants for products involving $A_{(\mu, \nu)}(a)$, are necessary for the extension beyond the classical dimension bounds (e.g., Theorems 2.4, 3.4 of (Rowen et al., 16 Aug 2024); see also classification tables in (Rowen et al., 21 Sep 2025)).

A concise summary of dimension characteristics in the various regimes:

Class/Regime	Max Dimension (2-generated)	Typical Fusion Law
Primitive (Jordan type)	3	$(x-1)x(x-\eta)$
Weakly primitive (general $S$)	$>3$ possible	Extended, with $S$
Band semigroup algebra	$\text{rank of band}$	(0,1) Peirce

The inclusion of band algebras and noncommutative examples relies on relaxing strict primitivity and allowing fusion rules to organize a larger spectrum of eigenspaces (Rowen et al., 16 Aug 2024, Rowen et al., 21 Sep 2025).

4. Frobenius Forms and Orthogonality

Weakly primitive axial algebras generated by a homogeneous set of special axes admit an associative symmetric bilinear form—a Frobenius form—satisfying

$(xy, z) = (x, yz)$

for all $x, y, z \in A$ (Rowen et al., 21 Sep 2025). The normalization (for axes $a$) is customarily $(a,a) = 1$, and, crucially, $(a, y)$ equals the projection of $y$ onto $A_1(a)$.

An important orthogonality property emerges: products of eigenvectors in $A_{(\mu, \nu)}(a)$ and $A_{(\mu', \nu')}(a)$ are 0 unless $(\mu, \nu) = (\mu', \nu')$, and even then are often forced to be “trivial” or lie in the radical—giving rise to explicit, degenerate Frobenius forms on certain subspaces. Consequently, weakly primitive axial algebras can support nontrivial radicals for the Frobenius form, detectable via the vanishing of products between off-diagonal eigenspaces (Rowen et al., 21 Sep 2025, Gorshkov et al., 2022).

5. Automorphism Structure, Ideals, and the Miyamoto Group

The automorphism group generated by the Miyamoto involutions reflects the global symmetry induced by the fusion rule and the eigenstructure:

• In Jordan type, the Miyamoto group is a 3-transposition group acting on the axes.
• In Monster-type and more general settings, the fusion law may be more elaborate, but the action is always tightly constrained by the $\mathbb{Z}_2$ (or higher) grading and the positions of axes.
• In the weakly primitive context, every nontrivial ideal of the algebra must (under suitable hypotheses) contain an axis, and the closure of the axes under the Miyamoto group often generates the entire algebra (“axis-spanning” property) (McInroy et al., 2022, Rowen et al., 16 Aug 2024).

These features are critical for classification arguments and for understanding connections to the combinatorics of Fischer spaces, 3-transposition groups, and vertex operator algebras.

6. Extended Family: Examples and Applications

The expansion to weak primitivity allows the inclusion of numerous families not captured in earlier frameworks:

• Band semigroup algebras (associative, generated by idempotents, with only 0,1 eigenvalues) (Rowen et al., 16 Aug 2024).
• Noncommutative generalizations, including left-regular bands and decomposable nonsemisimple algebras.
• Code algebras constructed from binary codes, whose “small idempotents” can be primitive or weakly primitive, and whose fusion laws may be explicitly tabulated in terms of code properties (Castillo-Ramirez et al., 2017, Castillo-Ramirez et al., 2018).
• Section 5.1 of (Rowen et al., 16 Aug 2024) and Theorems 4.10–4.14 detail new classes of 4-dimensional, noncommutative axial algebras absent in the strictly primitive catalog.

The theory further accommodates “decomposition algebras” (Medts et al., 2019), wherein both the fusion law and the decomposition structure are decoupled from individual axes and encoded independently—useful for constructing universal objects and quotient classification (Yabe, 2022).

7. Consequences for Classification and Further Directions

Comprehensive classification of weakly primitive axial algebras, especially in the 2-generated case, is achievable via universal (decomposition) algebras and categorification techniques (Yabe, 2022, Rowen et al., 21 Sep 2025):

• Construction of a universal object in the suitable category allows for all 2-generated weakly primitive axial algebras to be obtained as quotients, with the isomorphism types determined by analysis of the quotient ideals.
• The existence of a Frobenius form, Miyamoto group structure, and the axis-spanning property places strict constraints on possible “exotic” behaviors.
• Future work examines the higher generated case, the impact of weakening primitivity for only some axes, and the full spectrum of noncommutative and multifusion generalizations.

A plausible implication is that the theory of weakly primitive axial algebras will connect to and generalize existing results on VOAs, 3-transposition groups, and combinatorial representation theory, while simultaneously enabling the systematic construction and classification of previously inaccessible families of nonassociative, idempotent-generated algebras.

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Key Formulas/Statements from Cited Works:

• Weakly primitive axis condition: $A_1(a) = \mathbb{F} a$ (Rowen et al., 16 Aug 2024, Rowen et al., 21 Sep 2025)
• Fusion rules for two-sided eigenspaces: $$A_{(\mu, \nu)}(a) \cdot A_{(\rho, \sigma)}(a) \subseteq \delta_{(\mu, \rho)} \delta_{(\nu, \sigma)} (A_1(a) + A_0(a))$$ (Rowen et al., 16 Aug 2024)
• Frobenius form normalization: $(a,y) =$ projection onto $A_1(a)$, $(a,a) = 1$ (Rowen et al., 21 Sep 2025)
• Existence of nontrivial higher-dimensional examples in the weak regime, even with just two axes (Rowen et al., 16 Aug 2024, Rowen et al., 21 Sep 2025)