Weighted Rota-Baxter Jacobi-Jordan Algebras

• Weighted Rota–Baxter Jacobi–Jordan algebras are algebraic systems combining commutative, nonassociative Jacobi–Jordan structures with a weighted Rota–Baxter operator.
• They generate a canonical twisted product and feature low-dimensional classifications, representations, and semidirect constructions that illustrate their computational tractability.
• Their rich cohomology and deformation theory provides explicit control over first-order deformations and obstructions, linking algebra and module dynamics.

Weighted Rota–Baxter Jacobi–Jordan algebras are algebraic structures in which a Jacobi–Jordan algebra is equipped with a weighted Rota–Baxter operator, so that commutativity, the Jacobi identity, and a Rota–Baxter identity of weight $\lambda$ are imposed simultaneously. In the formulation studied by Anitchéou and Attan, the theory includes morphisms, low-dimensional classifications, representations carrying an additional compatible endomorphism, weighted Rota–Baxter paired operators, semidirect constructions, and a low-degree cohomology theory governing deformations and extensions (Anitchéou et al., 12 Aug 2025).

1. Algebraic definition

A Jacobi–Jordan algebra is a commutative, nonassociative algebra $(A,\ast)$ over a field $\K$ of characteristic $0$ satisfying

$$x\ast y \;=\;y\ast x, \quad (x\ast y)\ast z+(y\ast z)\ast x+(z\ast x)\ast y \;=\;0, \qquad \forall x,y,z\in A.$$

The second condition is the Jacobi identity in the commutative setting (Anitchéou et al., 12 Aug 2025).

Given a scalar $\lambda\in\K$, a linear map $\mathcal I\colon A\to A$ on any algebra $(A,\ast)$ is a $\lambda$-weighted Rota–Baxter operator if

$$\mathcal I(x)\ast \mathcal I(y) \;=\; \mathcal I\!\bigl(\,\mathcal I(x)\ast y \;+\; x\ast \mathcal I(y)\;+\;\lambda\,(x\ast y)\bigr), \quad\forall x,y\in A.$$

A pair $(A,\ast)$0 is a $(A,\ast)$1-weighted Rota–Baxter Jacobi–Jordan algebra precisely when $(A,\ast)$2 is Jacobi–Jordan and $(A,\ast)$3 is a $(A,\ast)$4-weighted Rota–Baxter operator on it (Anitchéou et al., 12 Aug 2025).

A morphism $(A,\ast)$5 is an algebra map $(A,\ast)$6 satisfying

$(A,\ast)$7

This places the operator $(A,\ast)$8 on the same structural footing as the product, so that functoriality is defined by simultaneous compatibility with both operations.

2. Structural operations and induced products

Several closure properties are established for weighted Rota–Baxter Jacobi–Jordan algebras. If $(A,\ast)$9, then $\K$0 is again a Rota–Baxter operator of the same weight. If $\K$1, then $\K$2 is a $\K$3-weighted operator. In addition, the twist

$\K$4

is again $\K$5-weighted (Anitchéou et al., 12 Aug 2025).

A central construction is the twisted Jacobi–Jordan product

$\K$6

For any weighted Rota–Baxter Jacobi–Jordan algebra $\K$7, the algebra $\K$8 is again Jacobi–Jordan, and $\K$9 still satisfies the Rota–Baxter identity with respect to $0$0. Moreover,

$0$1

is a morphism of weighted Rota–Baxter Jacobi–Jordan algebras (Anitchéou et al., 12 Aug 2025).

This induced product is significant because it shows that the operator $0$2 is not merely an auxiliary endomorphism. It generates a second Jacobi–Jordan multiplication canonically associated to the original one. A plausible implication is that the operator can be used to pass between related commutative nonassociative geometries while preserving the Rota–Baxter structure.

3. Examples and low-dimensional classification

The basic examples illustrate both the generality of the definition and the tractability of explicit classification problems.

The classical integration example takes $0$3 with pointwise product and

$0$4

This is a $0$5-weighted weighted Rota–Baxter Jacobi–Jordan algebra; in this case the algebra is in fact associative (Anitchéou et al., 12 Aug 2025).

For any Jacobi–Jordan algebra $0$6, the identity map $0$7 is a $0$8-weighted Rota–Baxter operator. This gives a universal elementary example on every Jacobi–Jordan algebra (Anitchéou et al., 12 Aug 2025).

In dimension two, let

$0$9

All $$x\ast y \;=\;y\ast x, \quad (x\ast y)\ast z+(y\ast z)\ast x+(z\ast x)\ast y \;=\;0, \qquad \forall x,y,z\in A.$$0-weighted Rota–Baxter operators $$x\ast y \;=\;y\ast x, \quad (x\ast y)\ast z+(y\ast z)\ast x+(z\ast x)\ast y \;=\;0, \qquad \forall x,y,z\in A.$$1 are obtained by solving

$$x\ast y \;=\;y\ast x, \quad (x\ast y)\ast z+(y\ast z)\ast x+(z\ast x)\ast y \;=\;0, \qquad \forall x,y,z\in A.$$2

The resulting classification yields two families depending on $$x\ast y \;=\;y\ast x, \quad (x\ast y)\ast z+(y\ast z)\ast x+(z\ast x)\ast y \;=\;0, \qquad \forall x,y,z\in A.$$3. In particular, for $$x\ast y \;=\;y\ast x, \quad (x\ast y)\ast z+(y\ast z)\ast x+(z\ast x)\ast y \;=\;0, \qquad \forall x,y,z\in A.$$4 and $$x\ast y \;=\;y\ast x, \quad (x\ast y)\ast z+(y\ast z)\ast x+(z\ast x)\ast y \;=\;0, \qquad \forall x,y,z\in A.$$5, one family has matrix form

$$x\ast y \;=\;y\ast x, \quad (x\ast y)\ast z+(y\ast z)\ast x+(z\ast x)\ast y \;=\;0, \qquad \forall x,y,z\in A.$$6

Similar classifications exist for the two nonisomorphic $$x\ast y \;=\;y\ast x, \quad (x\ast y)\ast z+(y\ast z)\ast x+(z\ast x)\ast y \;=\;0, \qquad \forall x,y,z\in A.$$7-dimensional Jacobi–Jordan algebras listed in the literature (Anitchéou et al., 12 Aug 2025).

The low-dimensional classifications are obtained by solving a small system of quadratic equations in the entries of $$x\ast y \;=\;y\ast x, \quad (x\ast y)\ast z+(y\ast z)\ast x+(z\ast x)\ast y \;=\;0, \qquad \forall x,y,z\in A.$$8. This makes the theory amenable to explicit computation in finite dimension, while also indicating that classification rapidly becomes a nonlinear algebra problem.

4. Representations and semidirect products

A representation of a Jacobi–Jordan algebra $$x\ast y \;=\;y\ast x, \quad (x\ast y)\ast z+(y\ast z)\ast x+(z\ast x)\ast y \;=\;0, \qquad \forall x,y,z\in A.$$9 is a vector space $\lambda\in\K$0 equipped with a linear map $\lambda\in\K$1 satisfying

$\lambda\in\K$2

For a weighted Rota–Baxter Jacobi–Jordan algebra $\lambda\in\K$3, a representation is a triple $\lambda\in\K$4 such that $\lambda\in\K$5 is a Jacobi–Jordan module and $\lambda\in\K$6 is linear with compatibility

$\lambda\in\K$7

Thus the representation theory involves not only the action $\lambda\in\K$8 but also a second operator $\lambda\in\K$9 linked to $\mathcal I\colon A\to A$0 by a twisted Rota–Baxter-type identity (Anitchéou et al., 12 Aug 2025).

Several module constructions are available. The adjoint module is $\mathcal I\colon A\to A$1 with $\mathcal I\colon A\to A$2. One may scale or twist the operator to obtain $\mathcal I\colon A\to A$3 for $\mathcal I\colon A\to A$4. If $\mathcal I\colon A\to A$5 is a representation of weight $\mathcal I\colon A\to A$6, then $\mathcal I\colon A\to A$7 is again a representation. The theory also includes direct sums, submodules, tensor products, and the endomorphism module

$\mathcal I\colon A\to A$8

with

$\mathcal I\colon A\to A$9

These constructions show that the category of representations has the expected closure properties found in operator-enriched algebraic settings (Anitchéou et al., 12 Aug 2025).

Given $(A,\ast)$0 and a representation $(A,\ast)$1, one forms the semidirect weighted Rota–Baxter Jacobi–Jordan algebra

$(A,\ast)$2

with

$(A,\ast)$3

and

$(A,\ast)$4

Conversely, any decomposition of a weighted Rota–Baxter Jacobi–Jordan algebra as $(A,\ast)$5 with that product and Rota–Baxter operator arises from a representation (Anitchéou et al., 12 Aug 2025). This equivalence identifies modules with split extensions in the standard semidirect sense.

5. Weighted Rota–Baxter paired operators

A $(A,\ast)$6-weighted Rota–Baxter paired operator on the pair $(A,\ast)$7 is a pair $(A,\ast)$8 satisfying simultaneously

$(A,\ast)$9

and

$\lambda$0

Equivalently, $\lambda$1 makes $\lambda$2 into a weighted Rota–Baxter Jacobi–Jordan algebra and $\lambda$3 is a representation of it (Anitchéou et al., 12 Aug 2025).

This equivalence is conceptually important because it packages the algebra and module compatibilities into a single paired notion. Rather than treating $\lambda$4 as an external datum attached to a pre-existing Rota–Baxter algebra, the formalism regards $\lambda$5 as one operator-theoretic object distributed across the algebra-module pair.

The paired-operator viewpoint also admits a graph characterization: the graph

$\lambda$6

is a Jacobi–Jordan subalgebra in a suitable semidirect sum (Anitchéou et al., 12 Aug 2025). This graph criterion provides a geometric reformulation of the compatibility equations and is typical of operator identities that can be encoded as subalgebra conditions in enlarged ambient objects.

6. Low-degree cohomology and deformation-theoretic meaning

For a Jacobi–Jordan algebra $\lambda$7 with module $\lambda$8, the theory begins with two parallel cochain complexes

$\lambda$9

where $$\mathcal I(x)\ast \mathcal I(y) \;=\; \mathcal I\!\bigl(\,\mathcal I(x)\ast y \;+\; x\ast \mathcal I(y)\;+\;\lambda\,(x\ast y)\bigr), \quad\forall x,y\in A.$$0 and $$\mathcal I(x)\ast \mathcal I(y) \;=\; \mathcal I\!\bigl(\,\mathcal I(x)\ast y \;+\; x\ast \mathcal I(y)\;+\;\lambda\,(x\ast y)\bigr), \quad\forall x,y\in A.$$1 is a second copy of it. For $$\mathcal I(x)\ast \mathcal I(y) \;=\; \mathcal I\!\bigl(\,\mathcal I(x)\ast y \;+\; x\ast \mathcal I(y)\;+\;\lambda\,(x\ast y)\bigr), \quad\forall x,y\in A.$$2,

$$\mathcal I(x)\ast \mathcal I(y) \;=\; \mathcal I\!\bigl(\,\mathcal I(x)\ast y \;+\; x\ast \mathcal I(y)\;+\;\lambda\,(x\ast y)\bigr), \quad\forall x,y\in A.$$3

while $$\mathcal I(x)\ast \mathcal I(y) \;=\; \mathcal I\!\bigl(\,\mathcal I(x)\ast y \;+\; x\ast \mathcal I(y)\;+\;\lambda\,(x\ast y)\bigr), \quad\forall x,y\in A.$$4 is the same but with a minus-sign in the second sum. One checks that

$$\mathcal I(x)\ast \mathcal I(y) \;=\; \mathcal I\!\bigl(\,\mathcal I(x)\ast y \;+\; x\ast \mathcal I(y)\;+\;\lambda\,(x\ast y)\bigr), \quad\forall x,y\in A.$$5

This is the zigzag cohomology framework used as input for the weighted Rota–Baxter theory (Anitchéou et al., 12 Aug 2025).

For a weighted Rota–Baxter Jacobi–Jordan algebra $$\mathcal I(x)\ast \mathcal I(y) \;=\; \mathcal I\!\bigl(\,\mathcal I(x)\ast y \;+\; x\ast \mathcal I(y)\;+\;\lambda\,(x\ast y)\bigr), \quad\forall x,y\in A.$$6 with representation $$\mathcal I(x)\ast \mathcal I(y) \;=\; \mathcal I\!\bigl(\,\mathcal I(x)\ast y \;+\; x\ast \mathcal I(y)\;+\;\lambda\,(x\ast y)\bigr), \quad\forall x,y\in A.$$7, let $$\mathcal I(x)\ast \mathcal I(y) \;=\; \mathcal I\!\bigl(\,\mathcal I(x)\ast y \;+\; x\ast \mathcal I(y)\;+\;\lambda\,(x\ast y)\bigr), \quad\forall x,y\in A.$$8 be the induced Jacobi–Jordan module over $$\mathcal I(x)\ast \mathcal I(y) \;=\; \mathcal I\!\bigl(\,\mathcal I(x)\ast y \;+\; x\ast \mathcal I(y)\;+\;\lambda\,(x\ast y)\bigr), \quad\forall x,y\in A.$$9 defined by

$(A,\ast)$00

There are then two zigzag complexes,

$(A,\ast)$01

In low degrees, the weighted Rota–Baxter Jacobi–Jordan cochains are

$(A,\ast)$02

and similarly for $(A,\ast)$03. The combined differentials are

$(A,\ast)$04

One checks

$(A,\ast)$05

so that

$(A,\ast)$06

is a cochain complex (Anitchéou et al., 12 Aug 2025).

Its low-degree cohomology has explicit descriptions. First,

$(A,\ast)$07

Second, $(A,\ast)$08 is the quotient of all antiderivations by the inner antiderivations. An antiderivation is a pair $(A,\ast)$09 such that

$(A,\ast)$10

and

$(A,\ast)$11

Inner antiderivations come from $(A,\ast)$12 via $(A,\ast)$13 (Anitchéou et al., 12 Aug 2025).

The deformation-theoretic interpretation is explicit: $(A,\ast)$14 classifies first-order deformations and abelian extensions of $(A,\ast)$15 by $(A,\ast)$16, while $(A,\ast)$17 parametrizes obstruction classes to extending a first-order deformation to higher order (Anitchéou et al., 12 Aug 2025). This places the cohomology in the standard role of controlling infinitesimal and obstruction data, specialized to the weighted Rota–Baxter Jacobi–Jordan context.

7. Conceptual position within the theory

The theory combines three ingredients: commutativity, the Jacobi identity, and the weighted Rota–Baxter identity. The Jacobi–Jordan component supplies the ambient nonassociative structure; the operator $(A,\ast)$18 generates the twisted product $(A,\ast)$19; and the representation endomorphism $(A,\ast)$20 makes module theory sensitive to the same weight parameter $(A,\ast)$21 (Anitchéou et al., 12 Aug 2025).

Several standard-looking constructions are therefore altered by the presence of $(A,\ast)$22 and $(A,\ast)$23. Duals require weight $(A,\ast)$24; semidirect products carry the operator $(A,\ast)$25; and the cohomology is not a single complex but a zigzag arrangement tying the original algebra to the induced algebra $(A,\ast)$26. This suggests that weighted Rota–Baxter Jacobi–Jordan algebras should be viewed not merely as Jacobi–Jordan algebras with an endomorphism, but as systems with two coupled algebraic layers.

A common misconception would be to identify the Rota–Baxter operator with a derivation-like map. The defining identity does not have derivation form; instead, it reconstructs the product of $(A,\ast)$27 and $(A,\ast)$28 from mixed terms involving $(A,\ast)$29, $(A,\ast)$30, and the weight term $(A,\ast)$31. Likewise, the module compatibility for $(A,\ast)$32 is not an independent axiom unrelated to $(A,\ast)$33; it is exactly the second component of the paired-operator formalism (Anitchéou et al., 12 Aug 2025).

Within the scope developed by Anitchéou and Attan, the subject is organized around a coherent progression: definitions, explicit low-dimensional examples, module theory, semidirect reconstruction, paired operators, and low-degree cohomology (Anitchéou et al., 12 Aug 2025). The resulting framework gives a structured representation theory and a deformation theory for weighted Rota–Baxter Jacobi–Jordan algebras without leaving the Jacobi–Jordan setting.