Axial Identities in Axial Algebras

• The paper introduces axial identities that formalize fusion laws and eigenspace decompositions essential for primitive axial algebras.
• It employs a polynomial identity framework to verify conditions for solid subalgebras and to ensure radical-free structures.
• The work generalizes classical Jordan and Matsuo algebras, constructing universal examples beyond established parameter spaces.

Axial identities in axial algebras encapsulate the algebraic relationships imposed by the fusion law and the eigenspace structure associated with idempotent generators known as axes. These identities provide a unifying formalism to understand structural properties of axial algebras, facilitate generalizations beyond classical settings (such as Jordan or Matsuo algebras), and clarify existing results on “solid” subalgebras. Recent developments include the introduction of idempotental and axial polynomial identities as formal languages to encode these structures, enabling the systematic paper of the algebraic consequences of fusion rules, as well as the construction of new generic examples outside the established classes.

1. Polynomial, Idempotental, and Axial Identities

Idempotental and axial identities are generalizations of classical polynomial identities to the context of nonassociative algebras generated by idempotents (axes) with specified eigenvalue decompositions. In this framework, a “generalized monomial” is a nonassociative word

$f(y_1,\dots,y_m; X_1,\dots,X_n)$

where some arguments are reserved to be specialized to idempotents or axes, while others are arbitrary. An idempotental identity vanishes for all substitutions where distinguished positions are specialized to idempotents from a distinguished set; an axial identity further restricts specialization to axes, typically of a given fusion type.

For instance, a primary identity in the context of primitive axis is: $$E_1 X_1 + (1-\lambda)(E_1; X_1) E_1 - E_1(E_1 X_1) = 0$$ where $E_1$ is to be an idempotent, $\lambda$ is the eigenvalue arising in the fusion rule, and $(E_1; X_1)$ indicates the operation acting in a formal linearization. Such identities are designed so that, upon specializing $E_1$ to an axis $a$, the decomposition of any $y$ as $y = y_1 + y_0 + y_\lambda$ is encoded via corresponding projection identities, forcing, for example, the primitivity condition that the 1-eigenspace is one-dimensional.

By systematically formulating the algebraic implications of the Peirce decomposition and fusion rules as families of such identities, one obtains a constructive and universal way to describe the structure of all axial algebras subject to a given type.

2. Encapsulation of Fusion Rules and Eigenstructure

The fusion rule in an axial algebra determines which products of eigenvectors (with respect to an axis' adjoint operator) can belong to which eigenspaces. For an axis $a$, the algebra decomposes as

$A = A_1(a) \oplus A_0(a) \oplus A_\lambda(a)$

and the multiplication must satisfy, for $x \in A_p(a)$ and $y \in A_q(a)$,

$x y \in \bigoplus_{r \in p \star q} A_r(a)$

with $\star$ given by the fusion law.

These constraints can be encoded in generalized polynomial identities. For the Jordan case, for example, the identities may assert $A_\lambda(a)^2 \subseteq A_1(a) \oplus A_0(a)$ and $A_0(a)^2 \subseteq A_0(a)$. These are expressed as identities involving the corresponding projections, which, after appropriate specialization, must vanish in the algebra.

The general approach is to derive all such consequences (often in linearized form) at the level of the free nonassociative algebra with distinguished idempotent positions, before quotienting out by the ideal generated by these axial identities to construct universal or generic objects.

3. Solid Subalgebras and the Role of Axial Identities

A “solid” subalgebra, as defined by J. Desmet, I. Gorshkov, S. Shpectorov, and A. Staroletov, is a subalgebra in which every idempotent is a primitive axis. The language of axial identities precisely packages the necessary and sufficient conditions: if all the idempotental identities (that is, the polynomial identities encoding primitivity, the Peirce decomposition, and the relevant fusion rules) hold for every idempotent in a subalgebra, then that subalgebra is solid.

The verification of solidness thus becomes the verification of a finite collection of identities, often reducible (via multilinearization and Vandermonde arguments) to checking that certain key two- and three-variable identities vanish. This formal framework allows for a conceptual understanding and efficient proof of the results of Desmet, Gorshkov, Shpectorov, and Staroletov regarding conditions under which all idempotents in a 2-generated subalgebra are necessarily axes and when the subalgebra is solid (Rowen, 22 Aug 2025).

4. Construction of Universal and Generic Axial Algebras

The language of axial identities enables the construction of universal objects, notably generic primitive axial algebras of Jordan type (PAJ-1), as quotients of the free algebra modded out by the ideal generated by the appropriate identities. By varying the set of imposed identities—specifically, by omitting those that characterize the Matsuo class (for instance, the identities expressing the product of axes via the Miyamoto involution)—one realises generic axial algebras that lie outside the Jordan or Matsuo classes.

For example, the product formula characterizing Matsuo algebras,

$$ab = \frac{1-\lambda}{2} \left( a + b - b^{\tau_a} \right)$$

may be omitted in the universal construction. Consequently, the resulting algebra need not be power-associative (it may fail to be 4-power associative) and is neither Jordan nor a homomorphic image of a Matsuo algebra, yet, by virtue of the imposed axial identities, still possesses the essential characteristics of a primitive axial algebra of Jordan type.

Furthermore, identities for the Frobenius form—such as those enforcing that for every axis $a$ the projection of $y$ onto $A_1(a)$ is given by $(a, y) a$—are included to ensure nondegeneracy (a radical zero condition) in the generic Frobenius structure.

5. Formal Comparison and Explanatory Power

This axiomatic and language-theoretic approach provides a conceptual unification of various phenomena previously established via ad hoc or case-specific computations. It clarifies why certain reductions (such as restriction to 2- or 3-generated subalgebras or cases where the bilinear form assumes exceptional values) are sufficient for structural results. Moreover, it enables systematic exploration of new parameter spaces and types, giving rise to explicit counterexamples to previous conjectures—such as the existence of radical-free primitive axial algebras of Jordan type that are neither Jordan nor Matsuo.

The approach also facilitates understanding of which properties—like solidness or Frobenius form degeneracy—are consequences of underlying identities and which are additional, potentially independent, constraints.

6. Key Identities and Algebraic Formulations

Some of the principal identities highlighted in this framework include:

Role	Identity	Context
Projection to 1-eigenspace	$$E_1 X_1 + (1-\lambda)(E_1;X_1)E_1 - E_1(E_1X_1) = 0$$	Characterizes primitivity in axes
Fusion rule for Jordan type	$A_\lambda(a)^2 \subseteq A_1(a) \oplus A_0(a)$	Expressed as vanishing of specialized polynomials
Matsuo product formula	$ab = \frac{1-\lambda}{2}(a + b - b^{\tau_a})$	Characterizes Matsuo algebras
Frobenius identity	$(a,y) a =$ (projection of $y$ onto $A_1(a)$)	Enforces orthogonality and nondegeneracy of the form

7. Implications and Generality

The systematic use of idempotental and axial identities extends the theory of axial algebras beyond traditional confines—yielding a larger, richer class of radical-free primitive axial algebras, some of which are not Jordan or images of Matsuo algebras (Rowen, 22 Aug 2025). This formalism facilitates the construction and explicit classification of new examples, provides a lens through which the solidness of subalgebras and other properties become algebraic rather than case-specific, and equips researchers with a universal toolkit for the paper of fusion rules and eigenstructure in nonassociative, commutative algebraic systems.

The resulting theory reveals that the universe of axial algebras of Jordan type is strictly larger than previously recognized, with the axiomatization via identities forming a bridge between structural theory, explicit computation, and new algebraic constructions.