Pre-Jacobi–Jordan Algebras

• Pre-Jacobi–Jordan algebras are nonassociative systems defined on a vector space where the symmetrized product yields a Jacobi–Jordan structure.
• They employ operad-theoretic techniques to split associative operations, revealing high-degree identities and facilitating classification via Koszul duality.
• Recent research links their representation, cohomology, and deformation theories to applications in physics and advanced higher algebra frameworks.

Pre-Jacobi–Jordan algebras (pre-JJ algebras) are nonassociative algebraic systems whose defining binary product “precedes” and encodes much of the Jordan-type symmetry present in Jacobi–Jordan algebras. These structures arise naturally through the symmetrization of a nonassociative product and are intimately related to identities governing representations, cohomology, and deformation theory, as well as to the operad-theoretic frameworks connecting dendriform algebras, Jordan trialgebras, and dialgebraic systems. Recent advances systematically clarify their algebraic properties, their role in the landscape of nonassociative algebras, and their connections to physical applications and higher algebraic structures.

1. Structural Foundations and Defining Identities

A pre-Jacobi–Jordan algebra is a vector space $A$ with a bilinear multiplication $x \cdot y$ such that the anticommutator $x \circ y = x \cdot y + y \cdot x$ endows $A$ with the structure of a Jacobi–Jordan algebra, and left multiplication $L_x(y) = x \cdot y$ defines a representation of the resulting Jordan algebra on $A$ (Bremner et al., 2012). The key governing identities are:

• For pre-Jordan algebras (as introduced by Hou–Ni–Bai), the defining multilinear identities (see explicit forms PJ₁ and PJ₂ in (Bremner et al., 2012)) ensure compatibility with the Jordan algebra structure and constrain the possible nonassociative behaviors.
• In left pre-JJ algebras, the associator is antisymmetric in the first two arguments: $$(x, y, z)_1 := (x \cdot y) \cdot z + x \cdot (y \cdot z)$$ obeys $(x, y, z)_1 = - (y, x, z)_1$. This guarantees a controlled “pre-associativity” condition (Haliya et al., 2020, Attan et al., 5 Aug 2025).

The symmetrization operation is central: the sub-adjacent Jacobi–Jordan algebra is $(A, \circ)$ where $x \circ y := x \cdot y + y \cdot x$ satisfies commutativity and a Jacobi-type identity, often written as

$$(x \circ y) \circ z + (y \circ z) \circ x + (z \circ x) \circ y = 0,$$

implying strong nilpotency conditions ($x^3 = 0$) and, in many cases, equivalence between Jacobi and Jordan identities for the symmetrized product (Zusmanovich, 2016).

2. Operad-Theoretic and Splitting Constructions

The role of operads is crucial in understanding pre-JJ algebras as splittings of associative or Jordan structures:

• In the context of dendriform algebras, if $A$ is endowed with operations $\succ$, $\prec$ satisfying splitting identities so that $x * y = x \prec y + x \succ y$ is associative, the pre-Jordan product is defined by $x \cdot y = x \succ y + y \prec x$. Its symmetrization always yields a Jordan algebra (Bremner et al., 2012, Bagherzadeh et al., 2016).
• The algebraic identities of pre-Jordan products in free dendriform algebras have been exhaustively computed: all multilinear polynomial identities up to degree 7 are consequences of two degree-4 generators (PJ₁, PJ₂). In degree 8, new identities emerge, generating S₈-modules isomorphic to those of Jordan diproducts in associative dialgebras, a phenomenon governed by Koszul duality for operads (Bremner et al., 2012).
• Jordan trialgebras and post-Jordan algebras are more elaborate operadic splittings: they combine commutative “Jordan” products with nonsymmetric “diproducts” and encode their identities via explicit operad morphisms and computer algebra computations (Bagherzadeh et al., 2016). The resulting structures often have all degree ≤6 identities generated by arity-4 ones, with novel independent identities conjectured at arity 8.

The commutative square of operad morphisms postulated in (Bagherzadeh et al., 2016) organizes these varieties and suggests a general method for constructing further nonassociative systems, including pre–Jacobi–Jordan algebras, from splitting and dualization procedures.

3. Representation Theory and Specialness

The representation theory of pre-JJ algebras is distinctly nontrivial:

• Embedding pre-JJ algebras into associative algebras (called “being special”) is possible only in select cases: the abelian examples always admit faithful representations, but many “exceptional” mock-Lie (pre-JJ) algebras do not—markedly failing analogs of the Ado or PBW theorems characteristic of Lie and Jordan theory (Zusmanovich, 2016).
• For a mock-Lie algebra (commutative, Jacobi-identity, hence $x^3=0$), the representation is given by a linear map $p: L \rightarrow \mathrm{End}(V)$ such that $p(x \circ y) = -p(x)p(y) - p(y)p(x)$; after rescaling, this is a Jordan algebra homomorphism. The failure of faithful representation is diagnostic of “exceptional” structures; classification into special vs exceptional guides the deeper paper of representation theory, module extensions, and cohomology (Zusmanovich, 2016, Haliya et al., 2020).
• The notion of bimodules and matched pairs in pre-JJ algebra is essential for constructing semi-direct sums, double constructions, and for categorically describing algebraic extensions (Haliya et al., 2020, Attan, 2021).

Antiderivations (maps $D$ such that $D(x \circ y) = -p(x)D(y) - p(y)D(x)$) replace derivations from Lie theory and deeply inform the cohomology and deformation theory for both mock-Lie and pre-JJ algebras.

4. Cohomology and Deformation Theory

Recent work has produced an explicit cohomology theory for pre-JJ algebras:

• The “zigzag” cohomology complex is constructed using two families of operators (differentials and zigzag differentials) acting on multilinear cochains, with compatibility conditions arising from pre-JJ identities (Baklouti et al., 2021, Attan et al., 5 Aug 2025).
• In low degrees, the cohomology has concrete algebraic significance: $H^0$ captures invariants, $H^1$ classifies outer antiderivations, and $H^2$ encodes infinitesimal deformations of the algebra structure. Linear deformations $x \cdot_t y = x \cdot y + t\,\omega(x, y)$ are governed by 2-cocycles; a deformation is trivial if $\omega$ is a coboundary (e.g., if there exists a Nijenhuis operator $N$ such that $$\omega(x, y) = N(x) \cdot y + x \cdot N(y) - N(x \cdot y)$$) (Attan et al., 5 Aug 2025).
• Nijenhuis operators are characterized by the torsion-free condition $$N(x) \cdot N(y) = N(N(x) \cdot y + x \cdot N(y) - N(x \cdot y))$$ and are shown to always produce trivial deformations; in some cases, they coincide with Rota–Baxter operators of suitable weights (Attan et al., 5 Aug 2025, Attan, 2021).

Cohomology, extensions, and formal deformation theory can be extended to Hom-pre-JJ structures (including twisting by an endomorphism), matched pairs, and O-operator frameworks, with applications in integrable system theory and operadic deformation (Attan, 2021).

5. Classification, Examples, and Generalizations

Pre-Jacobi–Jordan algebras have been classified in low dimensions and constructed from operator-theoretic principles:

Dimension	Classification Result	Remarks
1	Two types: associative and null algebra	Complete
2	Four nonisomorphic classes (e.g. $e_1 \cdot e_1 = e_2$, trivial, etc.)	Classification via structure constants and symmetry (Haliya et al., 2020)
3	20 classes of Jordan algebras, each admitting several pre-Jordan structures	Only subset realized via Rota–Baxter operators (Sun et al., 2021)

Rota–Baxter operators on Jordan algebras yield pre-Jordan algebra structures via $x \cdot y = R(x) \circ y$ but generate only a small fraction of all pre-Jordan algebras in low dimensions (Sun et al., 2021).

Double constructions (inspired by Frobenius and Drinfeld doubles) endow the direct sum $A \oplus A^*$ of a pre-JJ algebra and its dual with invariant symmetric bilinear forms, extending the class of “doubled” pre-JJ and Jacobi–Jordan structures (Haliya et al., 2020).

Generalizations to higher arity (ternary and $n$-ary versions) have been formulated: totally symmetric $n$-ary products satisfying derivation-type commutator identities, constructing ternary Jordan algebras and their noncommutative D-derivation variants via Cayley–Dickson algebras (Kaygorodov et al., 2017).

Hom-pre-Jacobi–Jordan algebras, endowed with a twisting map $\alpha$, further broaden the context and allow richer hierarchies of compatible algebraic systems, introduced via O-operators and Nijenhuis elements (Attan, 2021).

6. Operadic Dualities, Koszul Connections, and Dual Algebraic Structures

The paper of special identities at high degree, the isomorphism of S₈-modules for pre-Jordan product identities and Jordan diproducts, and the conjectural commutative square of operad morphisms underline deep Koszul dualities:

• The duality between dendriform and associative dialgebra operads manifests as mutually isomorphic modules of new polynomial identities (in degree 8 and above) in both settings, signaling a universal pattern for “splittings” of associative and Jordan algebras (Bremner et al., 2012, Bagherzadeh et al., 2016).
• These dualities reinforce the viewpoint that pre-Jacobi–Jordan structures are a natural consequence of canonical splittings of associative and Jordan algebraic operads.

Such operadic perspectives unify pre-JJ, dendriform, diassociative, and trialgebra frameworks, suggesting methods for constructing new classes of nonassociative algebras exhibiting prescribed symmetry and fusion law properties.

7. Applications and Interconnections with Physics and Higher Structures

Pre-Jacobi–Jordan algebras and their generalizations are relevant in several contexts:

• In the theory of symmetric matrices, the Jordan locus in the Grassmannian is classified by subspaces closed under the Jordan product and inversion, providing a geometric model for “almost” Jordan algebraic structures encountered in optimization and covariance analysis (Bik et al., 2020).
• Axial algebras of Jordan type and their noncommutative generalizations (“decomposition algebras”) have concrete links to vertex operator algebras, 3-transposition groups, and symmetry algebras used in mathematical physics (Hall et al., 2014, Rowen et al., 2021).
• Recent results demonstrate that every Jacobi–Jordan algebra admits a nontrivial post-Jacobi–Jordan structure via skew-symmetric “anti-biderivation” products, bridging pre-, post-, and admissible algebra frameworks (Benayadi et al., 9 May 2025).

Linear deformations, cohomology, and extension theory for pre-JJ algebras mirror the situation for Lie and Jordan algebras, with applications in quantization, deformation theory in physics, and advanced operadic algebra.

Summary Table: Core Properties and Connections

Algebra Type	Key Identities / Properties	Structural Invariants	Operadic Context
Pre-Jacobi–Jordan	$x \cdot y$ antisymmetric associator	Anticommutator is a JJ algebra	Dendriform / Triassociative
Jacobi–Jordan	$x \circ y$ symmetric, Jacobi identity	$x^3=0$ nilpotency	Polynomial operads
Post-Jacobi–Jordan	Skew-symmetric “anti-biderivation” product	Present in all JJ algebras	Dual to pre-JJ via symmetry
Hom-pre-JJ	Twisting by $\alpha$, compatibility conditions	O-operators, Nijenhuis operators	Hom-algebraic operads
Ternary $n$-ary variants	Total symmetry; commutator derivation	Cayley–Dickson examples, so(n) derivs	Multilinear operad theory

Pre-Jacobi–Jordan algebras thus provide a foundational bridge from classical associative and Lie structures to a panorama of splittings, deformations, and representations in nonassociative algebraic systems, with operadic perspectives furnishing both classification schemes and methods for constructing new varieties. Their structural roles and cohomological frameworks continue to inform both pure algebraic theory and diverse applications in geometry, physics, and higher algebra.