Left Pre-Jacobi–Jordan Algebras

• Left pre-Jacobi–Jordan algebras are non-associative structures defined by a left-sided derivational identity and a symmetrized product yielding Jacobi-like properties.
• They integrate operator techniques such as Rota–Baxter and Nijenhuis operators, enabling double constructions, matched pairs, and clear classification methods.
• Recent advances in cohomology and deformation theory reveal higher-degree special identities that are essential for distinguishing algebraic varieties.

A left pre-Jacobi–Jordan algebra is a non-associative algebraic structure uniquely characterized by identities that intertwine left-sided derivational behavior and a symmetrized Jordan–like product. This variety sits at the intersection of concepts arising from Jordan algebras, dialgebra theory, and the module-based framework endemic to operator approaches such as Rota–Baxter and Nijenhuis operators. Recent work provides rigorous cohomological foundations, detailed classification methods in low dimensions, and an understanding of the critical role of higher-degree identities within the identity theory of such algebras.

1. Algebraic Definition and Structural Identities

A left pre-Jacobi–Jordan algebra (A, ·) is built upon a bilinear product “·” satisfying a left-sided pre-Jacobi–Jordan identity, typically involving symmetric and anti-symmetric behaviors of the associator. This is formalized by the cyclic symmetry condition: $$(x, y, z)_{-1} := (x · y) · z + x · (y · z) = - (y · x) · z - y · (x · z)\quad \forall x,y,z \in A$$ A standard construction passes from (A, ·) to its sub-adjacent Jacobi–Jordan algebra by taking the anticommutator: $[x, y] := x · y + y · x$ This induces a commutative product on A which, under the defining pre-Jacobi–Jordan relations, satisfies the Jacobi identity distinguishing these structures from generic Jordan algebras. Specific incarnations such as the quasi-Jordan algebra arise from dialgebra theory, where the product is symmetrized from two “almost associative” operations satisfying compatibility conditions, leading to identities such as:

• Right-commutativity: $a(bc) = a(cb)$
• Quasi-Jordan: $(ba)a^2 = (ba^2)a$
• Associator-derivation: $(b,a^2,c) = 2(b,a,c)a$ with $(a,b,c) = (ab)c - a(bc)$

Every multilinear identity in degrees ≤ 7 for the quasi-Jordan product is a consequence of these three degree ≤ 4 identities (Bremner et al., 2010). This demonstrates the foundational strength and minimality of such systems in the low-degree regime.

2. Double Constructions, Bimodules, and Matched Pairs

Extending the structure, double constructions provide a canonical method to glue a left pre-Jacobi–Jordan algebra (A, ·) to its dual (A*, ∘), yielding a richer algebra on $A \oplus A^*$. Here, a nondegenerate symmetric bilinear form

$$B(x + a^*,\, y + b^*) = \langle x, b^* \rangle + \langle a^*, y \rangle$$

is required to be invariant with respect to the new product (Haliya et al., 2020). Matched pairs of pre-Jacobi–Jordan algebras, specified via representation maps $(l_A, r_A), (l_B, r_B)$ subject to compatibility conditions, furnish the algebraic background for these double constructions. Bimodules are defined through linear maps $l, r : A \to \text{gl}(V)$ satisfying compatibility relations such as $[l(x), r(y)] = - [l(y), r(x)]$, ensuring the semidirect sum $A \oplus V$ inherits a natural pre-JJ structure.

Hom-analogues generalize these constructions using twisting maps $\alpha$, allowing further flexibility. Importantly, matched pairs in the Hom-(pre)-Jacobi–Jordan setting provide direct sum algebras with induced (pre-)Jacobi–Jordan products, maintaining the essential cyclic and symmetry properties (Attan, 2021).

3. Classification and Operator Techniques

Explicit classification of pre-Jacobi–Jordan algebras in low dimensions is achieved via formal characteristic matrices encoding the structure constants and solving the resultant system of cubic polynomial equations arising from the compatibility identities (Sun et al., 2021). The approach fixes a Jordan structure and solves for all bilinear operations “·” compatible via $x · y + y · x = x \circ y$, using automorphism group actions to identify isomorphism classes.

A critical aspect is the role of operator-induced products, notably Rota–Baxter operators. For a Jordan algebra $(J, \circ)$ and a Rota–Baxter operator $R$, the induced “pre-Jordan” product is

$x \cdot y = R(x) \circ y$

Pre-Jacobi–Jordan algebras generated by such operators form only a subset of all possible structures, highlighting the limited but important class accessible by operator methods. Analogous constructions for left pre-Jacobi–Jordan algebras via operator identities suggest avenues for further investigation and classification.

4. Cohomology, Linear and Simultaneous Deformation Theory

A cohomology theory for left pre-Jacobi–Jordan algebras is now available (Attan et al., 5 Aug 2025), using a zigzag-type cochain complex. The n-cochain space is $C^n(A, V)$, and the differential operators $d^n$ and $\delta^n$ are constructed using the representation maps and the algebra product. The compatibility $d^n \circ \delta^{n-1} = 0$ for $n \geq 2$ enables rigorous definitions of cocycles, coboundaries, and cohomology groups $\mathcal{H}^k(A, V)$.

Linear deformations are defined by introducing a bilinear cocycle $\omega$,

$x \cdot_t y = x \cdot y + t\,\omega(x, y)$

with conditions ensuring that $(A, \omega)$ is a pre-Jacobi–Jordan algebra and $\omega$ is a 2-cocycle. Two deformations are equivalent if their cocycles differ by a coboundary: $\omega - \omega' = \delta N$.

Simultaneous deformation theory (as developed by Frégier and Zambon) extends to the deformation of both the algebra structure and a distinguished operator, employing a differential graded Lie algebra via derived brackets to control the deformations (Djibril et al., 5 Aug 2025). In this context, trivial deformations correspond to cocycles cohomologous to zero via Nijenhuis elements.

5. Nijenhuis Operators and Triviality of Certain Deformations

A Nijenhuis operator $N : A \to A$ satisfies $$N(x) \cdot N(y) = N(N(x) \cdot y + x \cdot N(y) - N(x \cdot y))$$, intertwining twisted and original products (Attan et al., 5 Aug 2025, Attan, 2021). Deformations generated via $\omega = \delta N$ yield

$$x \cdot_t y = x \cdot y + t\,(N(x) \cdot y + x \cdot N(y) - N(x \cdot y))$$

and the map $Id + tN$ is an algebra isomorphism up to $O(t^2)$, rendering such deformations trivial. This result connects to Gerstenhaber’s theory for associative algebras and informs the interpretation of cohomology classes in deformation theory. Further, it establishes an equivalence between certain Nijenhuis and Rota–Baxter operators, unifying approaches across Jordan-type structures.

6. Higher-Degree Special Identities and Structural Richness

Computer algebra computations detect new degree 8 identities in quasi-Jordan algebras, not derivable from the lower-degree standard relations (Bremner et al., 2010). These identities, some of which serve as noncommutative preimages of the Glennie identity, indicate that the identity theory for left pre-Jacobi–Jordan algebras is substantially richer at high degrees. They suggest that structurally distinct varieties cannot be captured solely by low-degree constraints. Explicitly, the emergence of such identities—irreducible under the action of the symmetric group on the variables—enables finer distinction between algebras arising directly from associative dialgebras and those with genuinely noncommutative or non-special features.

7. Implications, Representation Theory, and Open Problems

The paper of left pre-Jacobi–Jordan algebras is tightly coupled to questions of representability, module structure, and classification of exceptional versus special cases (Zusmanovich, 2016). Faithful representation (embedding into a symmetrized algebra), the structure of universal enveloping algebras, and the impact of nilpotency enforced by the Jacobi identity are central. While most small-dimensional examples are special, exceptional algebras appear in higher dimensions and are often linked to failure of certain higher-degree identities.

Cohomological and operator-theoretic techniques, classification protocols, and recognition of trivial versus essential deformations form the methodological foundation for current research. The framework is adaptable to related structures, such as mock-Lie algebras and Hom-type generalizations, and informs applications in vertex operator algebra theory, combinatorial algebra, and the representation theory of non-associative systems.

Key Formulas:

• Pre-Jacobi–Jordan associator symmetry: $(x, y, z)_{-1} = - (y, x, z)_{-1}$
• Sub-adjacent Jacobi–Jordan product: $[x,y] = x \cdot y + y \cdot x$
• Rota–Baxter induced product: $x \cdot y = R(x) \circ y$
• Nijenhuis product: $$x \cdot_N y = N(x) \cdot y + x \cdot N(y) - N(x \cdot y)$$

This synthesis establishes left pre-Jacobi–Jordan algebras as a robust algebraic family possessing rich identity theory, explicit cohomological and deformation frameworks, deep connections to operator approaches, and active classification efforts.