Weighted Rota-Baxter Paired Operators

• Weighted Rota-Baxter paired operators are algebraic structures combining a weighted operator on an algebra with a compatible operator on its representation.
• They unify formulations in Jacobi-Jordan, Lie, and associative cases through equivalence theorems, subalgebra criteria, and graph characterizations.
• The framework underpins cohomology and deformation theory, offering insights into dendriform splitting, module compatibility, and group extensions.

Weighted Rota-Baxter paired operators are compatibility structures built from a weight parameter $\lambda$ and two linear operators whose roles depend on the ambient algebraic context. In the Jacobi-Jordan setting, a $\lambda$-weighted Rota-Baxter paired operator on a pair $((A,\ast),(V,\rho))$ consists of a $\lambda$-weighted Rota-Baxter operator $\mathcal I:A\to A$ on the algebra together with a linear map $\mathcal T:V\to V$ satisfying a representation-level compatibility identity; in this form, the notion is studied alongside representations and low-degree cohomology for weighted Rota-Baxter Jacobi-Jordan algebras (Anitchéou et al., 12 Aug 2025). In related literature, analogous paired constructions occur for associative algebras, Lie algebras, and group-theoretic extensions of Rota-Baxter theory, where the term “paired” may denote either two operators on one algebra or an operator on an algebra together with a compatible operator on a module or representation (Brzeziński, 2015, Das, 2021, Gao et al., 2024).

1. Definition in the Jacobi-Jordan setting

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

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

A representation of $\lambda$0 is a vector space $\lambda$1 together with a linear map

$\lambda$2

such that

$\lambda$3

Fix $\lambda$4. A $\lambda$5-weighted Rota-Baxter paired operator on the pair $\lambda$6 is a pair of linear maps

$\lambda$7

satisfying two conditions (Anitchéou et al., 12 Aug 2025). The first is the $\lambda$8-weighted Rota-Baxter identity on the Jacobi-Jordan algebra: $\lambda$9 The second is the representation-level compatibility condition: $((A,\ast),(V,\rho))$0

The formal content of the definition is that the algebra carries a weighted Rota-Baxter operator and the module carries a companion operator compatible with it. In the wording of the source, identity (An) says that $((A,\ast),(V,\rho))$1 is itself a $((A,\ast),(V,\rho))$2-weighted Rota-Baxter Jacobi-Jordan algebra, while identity (Am) says that $((A,\ast),(V,\rho))$3 is a representation of the Rota-Baxter algebra $((A,\ast),(V,\rho))$4 (Anitchéou et al., 12 Aug 2025). Hence no additional axioms are required beyond these two identities.

2. Paired compatibility and equivalent formulations

The basic equivalence theorem states that a pair $((A,\ast),(V,\rho))$5 is a $((A,\ast),(V,\rho))$6-weighted Rota-Baxter paired operator if and only if two conditions hold simultaneously: $((A,\ast),(V,\rho))$7 is a $((A,\ast),(V,\rho))$8-weighted Rota-Baxter Jacobi-Jordan algebra, and $((A,\ast),(V,\rho))$9 is a representation of $\lambda$0 (Anitchéou et al., 12 Aug 2025). This identifies the paired operator concept with the representation theory of weighted Rota-Baxter Jacobi-Jordan algebras rather than with an independent third layer of structure.

A further structural reformulation uses a direct-sum Jacobi-Jordan algebra. On $\lambda$1, define

$\lambda$2

Then $\lambda$3 becomes a representation of $\lambda$4 via

$\lambda$5

This construction packages the weight parameter into a semidirect-type enlargement of the original data (Anitchéou et al., 12 Aug 2025).

The graph characterization gives a geometric criterion. Let

$\lambda$6

Then $\lambda$7 is a $\lambda$8-weighted Rota-Baxter paired operator if and only if this graph is a Jacobi-Jordan subalgebra of the semidirect product

$\lambda$9

endowed with the product induced by $\mathcal I:A\to A$0 and $\mathcal I:A\to A$1 (Anitchéou et al., 12 Aug 2025). The proof is by direct verification that closure of the graph is equivalent to the two defining identities.

This subalgebra criterion is significant because it recasts operator identities as an internal closure property. A plausible implication is that it places weighted paired operators within a broader “graph subobject” paradigm familiar from operator-theoretic formulations in other algebraic categories.

3. Examples and basic constructions

The Jacobi-Jordan paper gives two explicit examples (Anitchéou et al., 12 Aug 2025).

The first uses a two-dimensional Jacobi-Jordan algebra $\mathcal I:A\to A$2 with basis $\mathcal I:A\to A$3 and product

$\mathcal I:A\to A$4

For a weight $\mathcal I:A\to A$5, choose

$\mathcal I:A\to A$6

Then $\mathcal I:A\to A$7 is a $\mathcal I:A\to A$8-weighted Rota-Baxter algebra. Taking the adjoint representation

$\mathcal I:A\to A$9

and setting $\mathcal T:V\to V$0, one obtains a $\mathcal T:V\to V$1-weighted Rota-Baxter paired operator.

The second begins with a $\mathcal T:V\to V$2-weighted Rota-Baxter associative algebra $\mathcal T:V\to V$3. Define a Jacobi-Jordan product by symmetrization: $\mathcal T:V\to V$4 Let $\mathcal T:V\to V$5 with regular action

$\mathcal T:V\to V$6

Then $\mathcal T:V\to V$7 is also a $\mathcal T:V\to V$8-weighted Rota-Baxter Jacobi-Jordan algebra, and with $\mathcal T:V\to V$9 the pair $(A,\ast)$0 satisfies the paired-operator condition (Anitchéou et al., 12 Aug 2025).

These examples show two recurrent mechanisms. One is internal, using the adjoint representation and the same operator on algebra and module. The other is transport of structure from associative Rota-Baxter theory by passing to the symmetrized product. This suggests that weighted paired operators are stable under standard passage from associative to Jordan-type or Jacobi-Jordan-type structures when the representation is chosen canonically.

4. Relation to representations

In the Jacobi-Jordan context, every $(A,\ast)$1-weighted Rota-Baxter paired operator $(A,\ast)$2 on $(A,\ast)$3 amounts exactly to the statement that $(A,\ast)$4 is a representation of the Rota-Baxter Jacobi-Jordan algebra $(A,\ast)$5 (Anitchéou et al., 12 Aug 2025). Equivalently,

$(A,\ast)$6

The source describes this as precisely the usual module-compatibility condition for $(A,\ast)$7 over the algebra with operator $(A,\ast)$8.

An analogous interpretation appears in the Lie setting. For a Lie algebra $(A,\ast)$9, a $\mathbb K$0-weighted Rota-Baxter operator $\mathbb K$1 satisfies

$\mathbb K$2

A representation of $\mathbb K$3 is a pair $\mathbb K$4 where $\mathbb K$5 is a $\mathbb K$6-representation and $\mathbb K$7 satisfies

$\mathbb K$8

The Lie-theoretic paper then treats a weighted Rota-Baxter paired operator as the simultaneous choice of such an operator $\mathbb K$9 and such a compatible map $0$0 (Das, 2021).

The recurrence of the same module-compatibility pattern in Jacobi-Jordan and Lie settings indicates that “paired operator” often functions as a representation-theoretic enhancement of a weighted Rota-Baxter operator. At the same time, the associative literature uses the same expression differently, as discussed below; the terminology is therefore context-dependent.

5. Cohomology and deformation theory

The Jacobi-Jordan paper develops a low-degree cohomology theory governing simultaneous deformations of the product $0$1, the operator $0$2, and the module map $0$3 (Anitchéou et al., 12 Aug 2025). It starts from two zigzag complexes: one for $0$4 with coefficients in $0$5, and one for the descendant algebra $0$6 with induced representation $0$7. In degrees $0$8 and $0$9, the Rota-Baxter cochain spaces are

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

with analogous spaces for $$(x\ast y)\ast z + (y\ast z)\ast x + (z\ast x)\ast y =0, \quad\forall x,y,z\in A.$$1, and differentials given in matrix form by

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

One checks $$(x\ast y)\ast z + (y\ast z)\ast x + (z\ast x)\ast y =0, \quad\forall x,y,z\in A.$$3, yielding well-defined cohomology groups $$(x\ast y)\ast z + (y\ast z)\ast x + (z\ast x)\ast y =0, \quad\forall x,y,z\in A.$$4 and $$(x\ast y)\ast z + (y\ast z)\ast x + (z\ast x)\ast y =0, \quad\forall x,y,z\in A.$$5.

In low degree, $$(x\ast y)\ast z + (y\ast z)\ast x + (z\ast x)\ast y =0, \quad\forall x,y,z\in A.$$6, while a $$(x\ast y)\ast z + (y\ast z)\ast x + (z\ast x)\ast y =0, \quad\forall x,y,z\in A.$$7-cocycle $$(x\ast y)\ast z + (y\ast z)\ast x + (z\ast x)\ast y =0, \quad\forall x,y,z\in A.$$8 is precisely an antiderivation

$$(x\ast y)\ast z + (y\ast z)\ast x + (z\ast x)\ast y =0, \quad\forall x,y,z\in A.$$9

satisfying

$\lambda$00

and

$\lambda$01

Boundaries are exactly the inner antiderivations $\lambda$02, where $\lambda$03, so

$\lambda$04

For the adjoint representation $\lambda$05, one recovers

$\lambda$06

The Lie-theoretic study supplies a fuller deformation-theoretic framework via a differential graded Lie algebra (DGLA) whose Maurer-Cartan elements are exactly the weighted Rota-Baxter paired operators (Das, 2021). For $\lambda$07,

$\lambda$08

and for a fixed pair $\lambda$09 the differential is

$\lambda$10

The paper states that

$\lambda$11

is a DGLA, and that a pair $\lambda$12 is a $\lambda$13-weighted Rota-Baxter paired operator if and only if it is a Maurer-Cartan element (Das, 2021). It also identifies $\lambda$14 with abelian extensions and with obstruction spaces for higher-order deformations.

Taken together, these constructions show that weighted paired operators are not only operator identities but deformation-governed objects. In the Jacobi-Jordan setting this is developed in low degree, whereas in the Lie setting it is organized by a full DGLA.

6. Associative paired operators, dendriform structures, and related variants

In associative algebra, Brzeziński introduced a weight-$\lambda$15 Rota-Baxter system on an associative algebra $\lambda$16 as a pair of endomorphisms $\lambda$17 satisfying

$\lambda$18

$\lambda$19

and then described a weighted generalization by

$\lambda$20

$\lambda$21

for arbitrary weight $\lambda$22 (Brzeziński, 2015). When $\lambda$23, this recovers the original system. If one imposes $\lambda$24, the first identity becomes the ordinary weight-$\lambda$25 Rota-Baxter identity and the second is automatic (Brzeziński, 2015).

This associative paired construction is equivalent, under non-degeneracy, to the existence of a $\lambda$26-dendriform algebra structure defined by

$\lambda$27

and it yields the associative product

$\lambda$28

(Brzeziński, 2015). At weight $\lambda$29, Brzeziński also obtained a left pre-Lie product

$\lambda$30

The same source places ordinary weighted Rota-Baxter operators, twisted Rota-Baxter operators, and quasitriangular examples from covariant bialgebras into the same framework. In particular, if $\lambda$31 is a $\lambda$32-twisted Rota-Baxter operator, one takes

$\lambda$33

to build a weighted paired operator (Brzeziński, 2015). Quasitriangular examples arise from associative Yang-Baxter pairs $\lambda$34, which define $\lambda$35 and $\lambda$36 by convolution-type formulas and automatically satisfy the system at weight $\lambda$37.

A comparison with the Jacobi-Jordan and Lie cases shows a shift in meaning. In the associative setting, “paired” refers to two operators on the same algebra. In the Jacobi-Jordan and Lie settings, it refers to an operator on the algebra together with a compatible operator on a representation (Anitchéou et al., 12 Aug 2025, Das, 2021). This suggests that the unifying idea is simultaneous Rota-Baxter data with a shared weight and a common compatibility law, even though the exact categorical placement of the second operator varies.

7. Broader extensions and mathematical context

Weighted Rota-Baxter structures admit further extensions beyond associative, Lie, and Jacobi-Jordan algebras. One such extension is the group-theoretic notion of a Rota-Baxter operator with pair-weight $\lambda$38 on a group $\lambda$39, where maps $\lambda$40 satisfy $\lambda$41 and an operator $\lambda$42 obeys

$\lambda$43

A sequence-limit version then produces a limit-weight zero theory, and for Lie groups its tangent map yields a weight-zero Rota-Baxter operator on the Lie algebra (Gao et al., 2024). The paper also develops multiplicative integration-by-parts and a group version of the First Fundamental Theorem of Calculus for suitable topological groups.

Although this group-theoretic construction is not called a paired operator in exactly the same way as the algebraic theories, it is built from a pair of maps encoding weight data. A plausible implication is that the passage from single operators to paired or pair-weighted data is a recurring device for accommodating nonzero weights, twisted variants, or nonlinear structures.

The cohomological viewpoint likewise extends across settings. For associative algebras, weighted relative Rota-Baxter operators $\lambda$44 are governed by a DGLA whose Maurer-Cartan elements are precisely the $\lambda$45-weighted relative Rota-Baxter operators (Das, 2021). Their cohomology admits a Hochschild interpretation, and linear, formal, and finite-order deformations are controlled by cocycles, Nijenhuis elements, and obstruction classes in $\lambda$46 (Das, 2021). In the Lie case, the analogous cohomology coincides with Chevalley-Eilenberg cohomology for a derived bracket and module structure (Das, 2021).

Within this broader landscape, weighted Rota-Baxter paired operators occupy the intersection of operator identities, representation theory, and deformation theory. The Jacobi-Jordan work (Anitchéou et al., 12 Aug 2025) places them explicitly within low-degree cohomology and representation theory; the Lie work (Das, 2021) places them inside a Maurer-Cartan formalism; and the associative work (Brzeziński, 2015) relates paired data to dendriform splitting, pre-Lie products, weak pseudotwistors, and covariant bialgebra constructions.