Cohomology of Left Pre-Jacobi–Jordan Algebras

• The paper introduces a zigzag cochain complex that categorizes derivations, deformations, and extensions in left pre-Jacobi–Jordan algebras.
• It employs dual operator sequences and matched pair constructions to bridge classical Jordan identities with nonassociative structures.
• The approach integrates Nijenhuis and relative Rota–Baxter operators to identify trivial deformations and establish cohomological classifications.

A left pre-Jacobi–Jordan algebra is a nonassociative algebraic structure characterized by bilinear operations whose failure of associativity satisfies a left skew-symmetry condition related to the classical Jordan and Jacobi identities. The algebraic cohomology theory of left pre-Jacobi–Jordan algebras provides the formal framework to classify and analyze derivations, extensions, and deformations of these algebras. The contemporary theory uses a zigzag cochain complex constructed from dual operator sequences and is closely linked to deformation theory, Nijenhuis and relative Rota–Baxter operators, and broader Jordan-type algebraic structures.

1. Pre-Jacobi–Jordan Algebras: Structure and Defining Identities

A left pre-Jacobi–Jordan algebra $(A, \cdot)$ over a field (char $K\ne2,3$) is defined by a bilinear product $\cdot$ satisfying the antiassociator identity

$$(x\cdot y)\cdot z + x\cdot(y\cdot z) = -\Big((y\cdot x)\cdot z + y\cdot(x\cdot z)\Big),\quad\forall\, x, y, z\in A$$

or equivalently,

$$(x,y,z)_1 := (x\cdot y)\cdot z + x\cdot(y\cdot z) = - (y, x, z)_1.$$

This left symmetry condition distinguishes pre-JJ algebras from both associative and fully Jordan structures. The symmetrized product $x\ast y := x\cdot y + y\cdot x$ defines the so-called “sub-adjacent” Jacobi–Jordan algebra, which satisfies the commutative Jordan product and Jacobi identity. The antiassociative case (i.e., $(x\cdot y)\cdot z = - x\cdot(y\cdot z)$) is a special subclass.

Key structural features:

• Nonassociativity is strictly controlled by the antiassociator skew-symmetry.
• When symmetrized, the product satisfies Jacobi–Jordan (Jordan plus Jacobi) identities, ensuring compatibility with classical Jordan algebraic cohomological techniques.
• Bimodules, matched pairs, and invariant bilinear forms are natural modules of paper in the representation and cohomological analysis (Haliya et al., 2020).

2. Algebraic Cohomology Complex: Zigzag Operators and Graded Structure

The algebraic cohomology theory for left pre-Jacobi–Jordan algebras is realized through a zigzag complex constructed from two operator sequences acting on cochains:

• For a representation $(V; \rho, \mu)$ of $A$, the cochain spaces $C^n(A, V)$ are the $n$-multilinear maps $A^n \to V$.
• There is a distinguished subspace $$A^n(A,V)\subset \operatorname{Hom}(\wedge^{n-1}A \otimes A, V)$$ where the cochains obey a specific skew-symmetry (see condition (cA) in (Attan et al., 5 Aug 2025)).

The two key operator sequences are:

1. The differential $d^n: C^n(A, V) \to C^{n+1}(A, V)$, defined by

$$\begin{aligned} d^n f(x_1, ..., x_{n+1}) &= \sum_{i=1}^{n} \rho(x_i)f(x_1,...,\hat{x}_i,...,x_{n+1}) \ &\quad + \sum_{i=1}^{n} \mu(x_{n+1})f(x_1,...,\hat{x}_i,...,x_n, x_i) \ &\quad + \sum_{i=1}^{n} f(x_1,...,\hat{x}_i,...,x_n, x_i\cdot x_{n+1}) \ &\quad + \sum_{1\leq i<j\leq n} f(x_i * x_j, x_1,...,\hat{x}_i,...,\hat{x}_j,...,x_{n+1}), \end{aligned}$$

where $*$ denotes the symmetrized product.

1. The secondary operator $\delta^n : A^n(A,V) \to C^{n+1}(A,V)$ with appropriate sign conventions.

A crucial property is $d^n \circ \delta^{n-1} = 0$ for all $n \geq 2$, ensuring the integrity of the zigzag complex: $$\dots \xrightarrow{\,\delta^{n-1}\,} C^n(A, V) \xrightarrow{\,d^n\,} C^{n+1}(A, V) \xrightarrow{\,\delta^n\,} \dots$$ The cohomology groups are then defined as

$$H^n(A,V) = \ker(d^n)/ \operatorname{im}(\delta^{n-1}).$$

These groups measure obstructions to extensions, deformations, and the existence of invariant structures.

3. Classification and Extension Problems: Bimodules, Matched Pairs, and Double Structures

The construction of bimodules and matched pairs is foundational for extension and cohomology theory in pre-JJ algebras (Haliya et al., 2020).

• Bimodules: A bimodule $V$ comes equipped with left and right actions $l, r: A \rightarrow \mathfrak{gl}(V)$ satisfying compatibility relations ensuring that the extended action respects the pre-JJ algebra structure. These modules are used as coefficients in the cochain complexes.
• Matched pairs: Two pre-JJ algebras $A$ and $B$ can be assembled as a direct sum $A\oplus B$, with cross-actions providing a unified pre-JJ product under compatibility conditions (see Theorem 3.5 in (Haliya et al., 2020)).
• Double constructions: The direct sum $A \oplus A^*$, with $A^*$ carrying a dual product and module action, equipped with a nondegenerate invariant symmetric bilinear form $B$, yields new symmetric pre-JJ algebras. This mirrors Drinfeld’s double in bialgebra theory and is parametrized via matched pair data (Haliya et al., 2020).

Such double constructions and bilinear invariants are essential for defining and classifying symmetric extension classes and for establishing cohomological dualities.

4. Deformation Theory: Linear Deformations, Cocycle Conditions, and Nijenhuis Operators

Deformation theory in the pre-JJ context follows the Gerstenhaber paradigm, emphasizing linear deformations governed by cohomology:

• Given $(A, \cdot)$, a formal deformation is defined as $x \cdot_t y = x\cdot y + t\,\omega(x,y)$ with $\omega$ a bilinear map (the infinitesimal). For associativity of the deformed product to persist at first order, $\omega$ must satisfy the cocycle condition: it is a 2-cocycle in the cohomology of the regular representation (Attan et al., 5 Aug 2025).
• Quadratic constraints must also be fulfilled for the higher-order deformation to remain pre-JJ, typically restricting possible $\omega$ beyond cocycle conditions alone.

Nijenhuis operators provide a systematic source of trivial (integrable) deformations:

• A linear operator $N:A\rightarrow A$ is called a Nijenhuis operator if

$$N(x) \cdot N(y) = N\big(N(x) \cdot y + x \cdot N(y) - N(x \cdot y)\big), \quad\forall\,x,y\in A,$$

where the $N$-deformed product $$x \cdot_N y := N(x)\cdot y + x\cdot N(y) - N(x\cdot y)$$.

• Deformations generated via $N$—specifically, choosing $\omega(x,y) = x\cdot_N y$—are always trivial in cohomology, with the isomorphism given by the map $\mathrm{id}+tN$ (Attan et al., 5 Aug 2025).
• Rota–Baxter operators of weight $-1$ are shown to be equivalent to Nijenhuis operators in this setting, further connecting these operational approaches with deformation rigidity.

5. Connections to Related Algebraic and Cohomological Frameworks

The theory of pre-Jacobi–Jordan algebraic cohomology both extends and absorbs established cohomological machinery:

• Analogies to the cohomology of (pre-)Lie, Jordan, and Jacobi–Jordan algebras are direct: e.g., symmetrization to Jordan-type operations, and nilpotency properties leading to vanishing or simplified higher-degree cohomologies (Burde et al., 2014).
• Embedded cohomological frameworks: The adaptation of cohomology via enveloping Lie algebras (cf. the Tits–Koecher–Kantor construction for Jordan triples) provides structural insight, though the asymmetric pre-JJ identities require specific modification of the complex's symmetry and module structure (Chu et al., 2015).
• Hom-type generalizations: Recent developments have extended structure and cohomology to Hom-pre-Jacobi–Jordan algebras (involving twisting maps), wherein the zigzag complex and operator-based deformation theory remain structurally parallel, with suitable compatibility imposed for associativity and representation (Attan, 2021, Anitheou et al., 2023).

A table summarizing the key algebraic correspondences:

Structure	Product Symmetry	Key Cohomology Feature
Lie algebra	Skew (antisymmetric)	Chevalley–Eilenberg complex
Jordan algebra	Commutative	Jordan cohomology, deformation from H²
Jacobi–Jordan algebra	Commutative + Jacobi	Zigzag cohomology, H² classifies deformations
Pre-Jacobi–Jordan algebra	Weak left sym. (pre-JJ)	Zigzag complex: cocycle + quadratic constraints

6. Low-Degree Cohomology, Extensions, and Examples

Low-degree cohomology spaces admit classical algebraic interpretations:

• $H^0(A,V)$ identifies invariants in $V$ under the module action: those $v\in V$ such that $\rho(x)v + \mu(x)v = 0$ for all $x\in A$.
• $H^1(A,V)$ consists of outer antiderivations modulo inner ones—these measure derivational symmetries not realized by the algebra’s own multiplication (Attan et al., 5 Aug 2025, Baklouti et al., 2021).
• $H^2(A, A)$ classifies equivalence classes of abelian extensions and, crucially, infinitesimal linear deformations of the product. Nontriviality in $H^2$ reveals the presence of nonrigid deformations, while the vanishing of $H^2$ signifies rigidity.

Explicit examples worked out in (Haliya et al., 2020, Baklouti et al., 2021), and (Yang, 2022) include double constructions, abelian extension classes, and deformation computations for low-dimensional pre-JJ and Jacobi–Jordan algebras.

7. Triviality and Simultaneous Deformation: Relative Rota–Baxter Operators and Nijenhuis Elements

The theory of relative Rota–Baxter operators on (pre-)Jacobi–Jordan algebras provides a mechanism for constructing new structures and studying their deformations (Djibril et al., 5 Aug 2025):

• The cohomology theory for such operators parallels that of algebra deformations, using an adapted zigzag complex.
• Linear deformations of relative Rota–Baxter operators are controlled by 1-cocycles in the corresponding cohomology; triviality of the deformation relates to the existence of a Nijenhuis element—an operator whose associated deformation is cohomologically trivial.
• These techniques fit into a broader theme of simultaneous deformation studied in the derived bracket framework for other algebra–morphism pairs, as seen in the work of Frégier and Zambon and adapted to the pre-JJ context (Djibril et al., 5 Aug 2025).

Key foundational and recent works include:

• (Burde et al., 2014) for the structure and classification of Jacobi–Jordan algebras in low dimensions and their role in cohomology.
• (Haliya et al., 2020) for bimodules, matched pairs, and double constructions.
• (Attan et al., 5 Aug 2025) for the formal development of cohomology and deformation theory specific to left pre–Jacobi–Jordan algebras.
• (Djibril et al., 5 Aug 2025) for cohomologies and linear deformations of relative Rota–Baxter operators and simultaneous deformation analogies.

The culmination of these developments is a systematic, algebraically robust, and functorially rich cohomology theory for left pre-Jacobi–Jordan algebras, serving as a foundation for the paper of their deformations, extensions, and higher algebraic properties, with broad implications for nonassociative algebra, algebraic deformation theory, and mathematical physics.