Axial Algebras of Monster Type

• Axial algebras of Monster type are commutative, nonassociative structures generated by idempotents (axes) with a local structure determined by the Monster fusion law.
• They are classified into families such as 3A, 4A, and 5A, including exceptional and infinite cases, with explicit structural constants and dihedral symmetry actions.
• These algebras are fundamental in connecting Majorana theory, vertex operator algebras, and 3-transposition groups, and they provide a basis for further computational and theoretical research.

Axial algebras of Monster type are a distinguished class of commutative, generally nonassociative algebras generated by idempotents called axes, whose local structure is governed by the Monster fusion law. This concept abstracts and extends the features of the Griess algebra—whose automorphism group is the Monster sporadic simple group—but crucially encompasses an array of mathematical phenomena connected to Majorana theory, vertex operator algebras, and 3-transposition groups. The classification and structure theory of 2-generated Monster-type axial algebras is fundamental, as these algebras serve as the atomic building blocks for larger constructions.

1. Monster-Type Fusion Law and Axial Algebras

An axial algebra of Monster type is a commutative algebra $A$ over a field, generated by a set of axes $X$, each of which is a semisimple idempotent. The key structure is determined by the Monster fusion law $M(\alpha, \beta)$, which specifies for each axis $a$ a Peirce decomposition

$$A = A_1(a) \oplus A_0(a) \oplus A_\alpha(a) \oplus A_\beta(a),$$

with eigenvalues $F = \{1, 0, \alpha, \beta\}$, and the multiplication of eigencomponents is controlled by the table: $\begin{array}{c|cccc} \star & 1 & 0 & \alpha & \beta \ \hline 1 & \{1\} & \emptyset & \{\alpha\} & \{\beta\} \ 0 & \emptyset & \{0\} & \{\alpha\} & \{\beta\} \ \alpha& \{\alpha\}& \{\alpha\}& \{1,0\} & \{\beta\} \ \beta & \{\beta\} & \{\beta\} & \{\beta\} & \{1,0,\alpha\} \end{array}$ This law is $C_2$-graded, and each axis $a$ defines a Miyamoto involution $\tau_a$ acting as $+1$ on $A_{1,0,\alpha}(a)$ and $-1$ on $A_\beta(a)$. Monster-type axial algebras generalize the Griess algebra and encapsulate the fusion behavior observed in Majorana involutions and VOAs (McInroy et al., 31 Dec 2025, Franchi et al., 2021, McInroy et al., 2022).

2. Classification of 2-Generated Monster-Type Axial Algebras

The full classification of symmetric 2-generated axial algebras of Monster type is due to Yabe, Franchi–Mainardis, and McInroy and collaborators. Every such algebra is a quotient of exactly one of the following:

• Jordan-type algebras: Special 2B or 3C-types and spin-factor exceptions; these correspond to the classical Jordan cases with fusion type $J(\eta)$ for $\eta = \alpha$ or $\beta$.
• Twelve infinite one-parameter families: Denoted $nL(\alpha, \beta)$, indexed by $n = 3, 4, 5, 6, \infty$. Each family has an $n$-gon axet structure for the set of axes, with the dihedral Miyamoto group $D_n$ acting transitively. Representative families include 3A($\alpha,\beta$), 4A($\frac14, \beta$), 5A($\alpha, \frac{5-\alpha}8$), 6A($\alpha, -\frac{\alpha^2}{4(2-\alpha)}$), various $Y$, $B$, $J$ families, plus infinite-axet types IY$_3$, IY$_5$ (McInroy et al., 31 Dec 2025).
• Exceptional cases: The (infinite-dimensional) Highwater algebra $H$ of type $M(\alpha, \beta)$ and its characteristic-5 cover $H^5$. These are universal objects for the parameter locus $\alpha=2,\beta=2$ or in characteristic $5$ for type $(2,\frac12)$ (Franchi et al., 2021, Franchi et al., 2022).

The families are determined algebraically by the fusion parameters and admit explicit structural constants and bases.

3. Structure Theory: Ideals, Idempotents, Subalgebras

The structure of each family is characterized by:

• Bases and multiplication: For each $nL$, an explicit basis of $n$ axes plus additional vectors (commonly $w, c, z, u$) is given. Multiplication between basis elements is determined by polynomial functions of $(\alpha, \beta, \mu)$, concrete in Tables 3–8 of (McInroy et al., 31 Dec 2025). The action of the dihedral $D_n$ uniformizes the combinatorics of the products and the orbits of idempotents.
• Idempotents: All idempotents are solutions to $x^2 = x$. The dihedral symmetry groups the idempotents into $2^{\dim A}$ orbits. Each representative idempotent can itself have Monster, Jordan, or related fusion type, with explicit eigenvalue multiplicities recorded (McInroy et al., 31 Dec 2025).
• Ideals and quotients: All nontrivial ideals are contained in the radical $R$, that is invariant under all Miyamoto involutions. The structure of $R$ and its $D_n$-module decomposition is determined via explicit calculation of degeneracy loci in the parameter space (vanishing Gelfand–Kirillov form determinants); closed $D_n$-submodules of $R$ yield the ideals, and quotients reflect subalgebra or collapsed-axet structures.
• Subalgebra lattices: Any subaxet $X(k) \subset X(n)$ yields a $k$-generated subalgebra, again of Monster type. Special cases arise when $k\,|\,n$ (preserving family structure) or when forming Jordan-type subalgebras otherwise.
• Prime characteristic: Features such as block fusion in the $D_n$-module structure and degeneration of the fusion law occur in characteristics dividing $|D_n|$, with detailed stratification provided for parameters with coinciding structures or collapsed axes.

4. Exceptional Isomorphisms and Algebraic Identifications

Several exceptional isomorphisms between families occur at special parameter values, determined up to axial equivalence. Notably:

• $4A(\frac14, \frac18) \cong 4J(\frac14, \frac18)$,
• $4J(\frac12, \frac14) \cong 4Y(\frac12, \frac14)$,
• $6A(\frac25) \cong 6J(\frac25)$,
• $$IY_3(\alpha, \frac12, -\frac12) \cong 3A(\alpha, \frac12)$$,
• $IY_5(\alpha, \frac12)$ in char $5$ $\cong 5A(\alpha, \frac12)$.

Additionally, there are non-axial isomorphisms (e.g., $$4Y(\alpha, \frac12(1-\alpha^2)) \cong 4B(1-\alpha, \frac{(1-\alpha)^2}2)$$ via $a_i \mapsto 1-b_i$). These isomorphisms allow the unification of previously distinct families under specializations and connect the new constructions to classical Norton–Sakuma algebras (McInroy et al., 31 Dec 2025).

5. Skew Axial Algebras and Non-Symmetric Cases

The theory also encompasses the classification of skew axets, where the Miyamoto involution structure is non-regular. The paradigmatic example is the skew axet $X'(1+2)$, with axes subject to non-transitive involution action. Up to isomorphism, every 2-generated primitive skew Monster-type algebra falls into one of:

• $C(\alpha, 1-\alpha)$ for $\alpha \neq \frac12$,
• $Q_2(\frac13, \frac23)$ for $\alpha = \frac13,\, \beta = \frac23$ in $\operatorname{char} \neq 5$,
• $Q_2(\frac13)^\times \oplus 1$ for $\operatorname{char}=5$ (Turner, 2023, Turner, 2023).

This fundamentally restricts the landscape of skew 2-generated Monster-type examples, in contrast to the richer symmetric family hierarchy.

6. Applications, Computational Tools, and Further Directions

These 2-generated Monster-type algebras serve as the building blocks for more complex axial algebras via "shape and construction" algorithms, encoding all larger combinatorial or group-theoretic configurations (McInroy et al., 31 Dec 2025, McInroy et al., 2021). Magma code and explicit change-of-basis matrices for several families support computational work in this area (see ancillary files of (McInroy et al., 31 Dec 2025)).

The explicit classification underpins the construction of higher-generator, non-collapsing Monster-type algebras, informs the study of forbidden configurations via axet/shape methods, and facilitates the investigation of automorphism groups (notably, all finite-dimensional Monster-type axial algebras have finite automorphism group if $1/2$ is not invertible) (Gorshkov et al., 2023). Key open directions include the structure of higher-rank (e.g., 3-generated) Monster-type algebras and the exploration of new exceptional behavior in positive characteristic or for other fusion parameters.

7. Summary Table: Families of 2-Generated Axial Algebras of Monster Type

Family Notation	Axet Size	Parameterization	Key Features
3A($\alpha,\beta$)	3	$\alpha,\,\beta$	Bicyclic; simplest Monster-type case
4A, 4J, 4B, 4Y	4	Varied, e.g., $4A(\frac14,\beta)$	Rich isomorphisms; exceptional cases
5A($\alpha,\cdot$)	5	$\alpha,\,\frac{5-\alpha}8$	Higher dihedral symmetry
6A, 6J, 6Y	6	e.g., $6A(\alpha, -\alpha^2/(4(2-\alpha)))$	Largest finite-axet families
IY$_3$, IY$_5$	$\infty$	Parameters incl. $\mu$	Infinite-axet cases, fibered by $\mu$
Highwater $H$	$\infty$	$(\alpha,\beta)$ or $H^5$	Universal in char 5

These families, together with the exceptional isomorphisms and skew cases, exhaust all possibilities for 2-generated Monster-type axial algebras (McInroy et al., 31 Dec 2025). The explicit description of their structure—multiplication, ideal theory, subalgebra lattices, and idempotent classification—constitutes the foundational database for Monster-type axial algebra theory.