---
title: Weighted Rota-Baxter Jacobi-Jordan Algebras
url: https://www.emergentmind.com/topics/weighted-rota-baxter-jacobi-jordan-algebras
type: topic
---

# Weighted Rota-Baxter Jacobi-Jordan Algebras

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 [2508.09394].

## 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 [2508.09394].

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 $\bigl(A,\mathcal I\bigr)$ is a $\lambda$-weighted Rota–Baxter Jacobi–Jordan algebra precisely when $(A,\ast)$ is Jacobi–Jordan and $\mathcal I$ is a $\lambda$-weighted Rota–Baxter operator on it [2508.09394].

A morphism $\phi\colon (A,\mathcal I)\to (A',\mathcal I')$ is an algebra map $\phi\colon A\to A'$ satisfying
\[
\phi\circ\mathcal I=\mathcal I'\circ\phi.
\]
This places the operator $\mathcal I$ 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 $\psi\in\Aut(A)$, then $\psi^{-1}\circ\mathcal I\circ\psi$ is again a Rota–Baxter operator of the same weight. If $\mu\in\K$, then $\mu\mathcal I$ is a $(\mu\lambda)$-weighted operator. In addition, the twist
\[
-\lambda\,\id_A-\mathcal I
\]
is again $\lambda$-weighted [2508.09394].

A central construction is the twisted Jacobi–Jordan product
\[
x\ast_{\mathcal I}y
\;:=\;
\mathcal I(x)\ast y \;+\; x\ast\mathcal I(y)\;+\;\lambda\,x\ast y.
\]
For any weighted Rota–Baxter Jacobi–Jordan algebra $(A,\mathcal I)$, the algebra $\bigl(A,\ast_{\mathcal I}\bigr)$ is again Jacobi–Jordan, and $\mathcal I$ still satisfies the Rota–Baxter identity with respect to $\ast_{\mathcal I}$. Moreover,
\[
\mathcal I\colon (A,\ast_{\mathcal I})\to(A,\ast)
\]
is a morphism of weighted Rota–Baxter Jacobi–Jordan algebras [2508.09394].

This induced product is significant because it shows that the operator $\mathcal I$ 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 $A=C(\R)$ with pointwise product and
\[
\mathcal I(f)(x)=\int_0^x f(t)\,dt.
\]
This is a $0$-weighted weighted Rota–Baxter Jacobi–Jordan algebra; in this case the algebra is in fact associative [2508.09394].

For any Jacobi–Jordan algebra $A$, the identity map $\id_A$ is a $(-1)$-weighted Rota–Baxter operator. This gives a universal elementary example on every Jacobi–Jordan algebra [2508.09394].

In dimension two, let
\[
A=\operatorname{span}\{e_1,e_2\},
\qquad
e_1\ast e_1=e_2.
\]
All $\lambda$-weighted Rota–Baxter operators $\mathcal I$ are obtained by solving
\[
\mathcal I(e_i)\ast\mathcal I(e_j)
=\mathcal I\bigl(\mathcal I(e_i)\ast e_j+e_i\ast\mathcal I(e_j)+\lambda\,e_i\ast e_j\bigr).
\]
The resulting classification yields two families depending on $\lambda$. In particular, for $\lambda\neq0$ and $a_1\neq\lambda/2$, one family has matrix form
\[
\begin{pmatrix}
a_1&0\\
a_2 &a_1^2/(2a_1+\lambda)
\end{pmatrix}.
\]
Similar classifications exist for the two nonisomorphic $4$-dimensional Jacobi–Jordan algebras listed in the literature [2508.09394].

The low-dimensional classifications are obtained by solving a small system of quadratic equations in the entries of $\mathcal I$. 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 $(A,\ast)$ is a vector space $V$ equipped with a linear map $\rho\colon A\to\gl(V)$ satisfying
\[
\rho(x\ast y)
\;=\;
-\,\rho(x)\,\rho(y)\,-\,\rho(y)\,\rho(x),
\qquad \forall\,x,y\in A.
\]
For a weighted Rota–Baxter Jacobi–Jordan algebra $\bigl(A,\mathcal I\bigr)$, a representation is a triple $(V,\rho,\mathcal T)$ such that $(V,\rho)$ is a Jacobi–Jordan module and $\mathcal T\colon V\to V$ is linear with compatibility
\[
\rho\bigl(\mathcal I(x)\bigr)\bigl(\mathcal T(u)\bigr)
\;=\;
\mathcal T\!\bigl(
\rho(\mathcal I(x))u
\;+\;\rho(x)\bigl(\mathcal T(u)\bigr)
\;+\;\lambda\,\rho(x)u
\bigr),
\qquad
\forall x\in A,\;u\in V.
\]
Thus the representation theory involves not only the action $\rho$ but also a second operator $\mathcal T$ linked to $\mathcal I$ by a twisted Rota–Baxter-type identity [2508.09394].

Several module constructions are available. The adjoint module is $(A,\rho_\ad,\mathcal I)$ with $\rho_\ad(x)y=x\ast y$. One may scale or twist the operator to obtain $(V,\mu\mathcal T)$ for $\mu\in\K$. If $(V,\rho,\mathcal T)$ is a representation of weight $0$, then $(V^*,\rho^*,\mathcal T^*)$ is again a representation. The theory also includes direct sums, submodules, tensor products, and the endomorphism module
\[
\bigl(\gl(V),\hat\rho,\hat{\mathcal T}\bigr)
\]
with
\[
\hat\rho(x)f(u)=-f(\rho(x)u),
\qquad
\hat{\mathcal T}(f)(u)=-\lambda\,f(u)-f(\mathcal T(u)).
\]
These constructions show that the category of representations has the expected closure properties found in operator-enriched algebraic settings [2508.09394].

Given $(A,\mathcal I)$ and a representation $(V,\rho,\mathcal T)$, one forms the semidirect weighted Rota–Baxter Jacobi–Jordan algebra
\[
\bigl(A\ltimes V\,,\,\mathcal I\oplus\mathcal T\bigr)
\]
with
\[
(x,u)\ast (y,v)=(x\ast y,\;\rho(x)v+\rho(y)u)
\]
and
\[
\mathcal I\oplus\mathcal T(x,u)=(\mathcal I(x),\mathcal T(u)).
\]
Conversely, any decomposition of a weighted Rota–Baxter Jacobi–Jordan algebra as $A\oplus V$ with that product and Rota–Baxter operator arises from a representation [2508.09394]. This equivalence identifies modules with split extensions in the standard semidirect sense.

## 5. Weighted Rota–Baxter paired operators

A $\lambda$-weighted Rota–Baxter paired operator on the pair $\bigl((A,\ast),(V,\rho)\bigr)$ is a pair $(\mathcal I,\mathcal T)$ satisfying simultaneously
\[
\mathcal I(x)\ast\mathcal I(y)
=\mathcal I\!\bigl(\mathcal I(x)\ast y + x\ast \mathcal I(y)+\lambda x\ast y\bigr),
\]
and
\[
\rho\bigl(\mathcal I(x)\bigr)\mathcal T(u)
=\mathcal T\!\bigl(\rho(\mathcal I(x))u+\rho(x)\mathcal T(u)+\lambda\rho(x)u\bigr).
\]
Equivalently, $\mathcal I$ makes $(A,\mathcal I)$ into a weighted Rota–Baxter Jacobi–Jordan algebra and $\mathcal T$ is a representation of it [2508.09394].

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

The paired-operator viewpoint also admits a graph characterization: the graph
\[
\{(\mathcal I(x),x,\mathcal T(u),u)\} \subset A\oplus A\oplus V\oplus V
\]
is a Jacobi–Jordan subalgebra in a suitable semidirect sum [2508.09394]. 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 $(A,\ast)$ with module $(V,\rho)$, the theory begins with two parallel cochain complexes
\[
d^n\colon \C^n(A,V)\to\C^{n+1}(A,V),
\qquad
\delta^n\colon \A^n(A,V)\to\C^{n+1}(A,V),
\]
where $\C^n(A,V)=\Hom(A^{\otimes n},V)$ and $\A^n(A,V)$ is a second copy of it. For $f\in \C^n$,
\[
d^n f(x_1,\dots,x_{n+1})
=
\sum_{i=1}^{n+1}\rho(x_i)\,f(x_1,\dots,\widehat x_i,\dots,x_{n+1})
\;+\;
\sum_{i<j} f(x_i\ast x_j,\dots),
\]
while $\delta^n$ is the same but with a minus-sign in the second sum. One checks that
\[
d^{n+1}\circ d^n=0,
\qquad
\delta^{n+1}\circ \delta^n=0.
\]
This is the zigzag cohomology framework used as input for the weighted Rota–Baxter theory [2508.09394].

For a weighted Rota–Baxter Jacobi–Jordan algebra $\bigl(A,\mathcal I\bigr)$ with representation $(V,\rho,\mathcal T)$, let $\tilde V=(V,\tilde\rho)$ be the induced Jacobi–Jordan module over $\bigl(A,\ast_{\mathcal I}\bigr)$ defined by
\[
\tilde\rho(x)u
=\rho(\mathcal I(x))u \;-\;\mathcal T(\rho(x)u).
\]
There are then two zigzag complexes,
\[
\bigl(\C^\bullet(A,V),d^\bullet,\delta^\bullet\bigr)
\quad\text{and}\quad
\bigl(\C^\bullet(A_{\mathcal I},\tilde V),\tilde d^\bullet,\tilde\delta^\bullet\bigr).
\]
In low degrees, the weighted Rota–Baxter Jacobi–Jordan cochains are
\[
\C^0_{RB}(A,V)=V,
\qquad
\C^1_{RB}(A,V)=\C^1(A,V)\oplus\C^0(A_{\mathcal I},\tilde V),
\]
and similarly for $\A^n_{RB}$. The combined differentials are
\[
d^0_{RB}(v)=(d^0(v),-v),
\qquad
d^1_{RB}(f,g)=\bigl(d^1f,\;-\tilde d^0 g-\;(f\circ\mathcal I - \mathcal T\circ f)\bigr).
\]
One checks
\[
d^1_{RB}\circ\delta^0_{RB}=0,
\]
so that
\[
\bigl(\C^\bullet_{RB},\A^\bullet_{RB},d^\bullet_{RB},\delta^\bullet_{RB}\bigr)
\]
is a cochain complex [2508.09394].

Its low-degree cohomology has explicit descriptions. First,
\[
H^0_{RB}(A,V)=0.
\]
Second, $H^1_{RB}(A,V)$ is the quotient of all antiderivations by the inner antiderivations. An antiderivation is a pair $(\eta\colon A\to V,\;v\in V)$ such that
\[
\eta(x\ast y)
=-\,\bigl[\rho(x)\,\eta(y)+\rho(y)\,\eta(x)\bigr],
\]
and
\[
\eta\bigl(\mathcal I(x)\bigr)
-\mathcal T\bigl(\eta(x)\bigr)
=\mathcal T\bigl(\rho(x)v\bigr)
\;-\;\rho\bigl(\mathcal I(x)\bigr)v.
\]
Inner antiderivations come from $v\in V$ via $\eta(x)=\rho(x)(v)$ [2508.09394].

The deformation-theoretic interpretation is explicit: $H^1_{RB}(A,V)$ classifies first-order deformations and abelian extensions of $\bigl(A,\mathcal I\bigr)$ by $(V,\mathcal T)$, while $H^2_{RB}(A,V)$ parametrizes obstruction classes to extending a first-order deformation to higher order [2508.09394]. 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 $\mathcal I$ generates the twisted product $\ast_{\mathcal I}$; and the representation endomorphism $\mathcal T$ makes module theory sensitive to the same weight parameter $\lambda$ [2508.09394].

Several standard-looking constructions are therefore altered by the presence of $\mathcal I$ and $\mathcal T$. Duals require weight $0$; semidirect products carry the operator $\mathcal I\oplus\mathcal T$; and the cohomology is not a single complex but a zigzag arrangement tying the original algebra to the induced algebra $(A,\ast_{\mathcal I})$. 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 $\mathcal I(x)$ and $\mathcal I(y)$ from mixed terms involving $\mathcal I(x)$, $\mathcal I(y)$, and the weight term $\lambda x\ast y$. Likewise, the module compatibility for $\mathcal T$ is not an independent axiom unrelated to $\mathcal I$; it is exactly the second component of the paired-operator formalism [2508.09394].

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 [2508.09394]. 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.

Source: https://www.emergentmind.com/topics/weighted-rota-baxter-jacobi-jordan-algebras