Rota-Baxter Operad

• Rota-Baxter operad is a universal algebraic structure that encodes associative algebras with a binary product and a unary Rota-Baxter operator defined by specific operadic relations.
• Its minimal model, RBₗ^∞, provides a resolution framework for homotopy Rota-Baxter algebras by generalizing A∞-relations and incorporating higher operators.
• The operadic perspective enables systematic deformation theory, splittings into dendriform-type algebras, and connections to Koszul duality and extended structures.

A Rota-Baxter operad is a universal algebraic structure that encodes Rota-Baxter associative algebras and their generalizations at the level of operads. The operad’s relations formalize both the associative product and the defining Rota-Baxter equation of a fixed weight λ, allowing for a systematic study of classical, homotopy, and extended Rota-Baxter structures, their deformations, and their Koszul duals. The operadic viewpoint unifies classical Rota-Baxter algebras, their homotopy upgrades (RB∞-algebras), deformation theory, and the theory of splittings and derived algebraic structures such as dendriform and tridendriform algebras.

1. Presentation of the Rota-Baxter Operad

Let $k$ be a field of characteristic zero and fix a weight parameter $\lambda \in k$. The Rota-Baxter operad RB$_\lambda$ is the quotient of the free (non-symmetric) operad $F(\mu, R)$ generated by:

• a binary operation $\mu$ (the associative product), $\mu(x, y) = x \cdot y$
• a unary operation $R$ (the Rota-Baxter operator)

subject to:

1. Associativity

$\mu \circ_1 \mu - \mu \circ_2 \mu = 0$

which expands to $\mu(\mu(x, y), z) - \mu(x, \mu(y, z)) = 0$ for all $x, y, z$.

2. Rota-Baxter relation (weight $\lambda$)

$$\mu(R(x), R(y)) - R(\mu(R(x), y) + \mu(x, R(y)) + \lambda \mu(x, y)) = 0$$

Thus,

$\mathrm{RB}_\lambda = F(\mu, R) / \langle \text{associativity}, \text{Rota–Baxter $\lambda$-relation} \rangle$

The case $\lambda=0$ yields the classical Rota-Baxter relation of weight zero; for arbitrary $\lambda$ the operad is inhomogeneous and not quadratic, hence not Koszul in the classical sense (Wang et al., 2021, Wang et al., 2022).

2. Minimal Model and Homotopy Rota-Baxter Operad

The minimal model RB$_\lambda^\infty$ ("homotopy Rota–Baxter operad") is the quasi-free dg operad $(F(M), \partial)$ constructed as follows (Wang et al., 2021, Wang et al., 2022):

• Generators:
  • For $n=1$: $T_1$ (homotopy $R$-operator), degree 0.
  • For $n\ge2$: $m_n$ (homotopy multiplications), degree $n-2$; $T_n$ (homotopy Rota–Baxter operators), degree $n-1$.
• Differential:
  • For $m_n$:

  $$\partial(m_n) = \sum_{i+j+k=n, j\ge 1} (-1)^{i + j k} m_{i+1+k} \circ (\mathrm{id}^{\otimes i} \otimes m_j \otimes \mathrm{id}^{\otimes k})$$

  (the Stasheff $A_\infty$-formula) - For $T_n$, an explicit formula involving sums over trees with $m$- and $T$-vertices, and terms encoding the weight $\lambda$.

There is a canonical projection $m_2 \mapsto \mu$, $T_1 \mapsto R$, all higher generators to $0$, inducing a quasi-isomorphism RB$_\lambda^\infty \twoheadrightarrow$ RB$_\lambda$. Thus, RB$_\lambda^\infty$ is a minimal resolution. This provides a foundation for the notion of homotopy Rota-Baxter algebras of any weight as algebras over RB$_\lambda^\infty$ (Wang et al., 2021, Wang et al., 2022).

3. Deformation Theory and L$_\infty$-structure

For an RB$_\lambda$-algebra $A$, the deformation complex is constructed as follows:

• Total cochain complex:

$$C^*(A) = C^*(A, A) \oplus C^*_{\mathrm{RBO}_\lambda}(A, A)$$

combining the Hochschild complex (for $\mu$) and Rota-Baxter operator cochains (for $R$), with a differential defined as the mapping cone of a natural chain map.

• L$_\infty$-algebra:

On $C^*(A)$, a family of multibrackets $l_n : C^*(A)^{\otimes n} \to C^*(A)[2-n]$ is constructed, extending the Gerstenhaber bracket and involving mixed terms in the R-coefficients. The Maurer–Cartan elements correspond precisely to pairs $(\mu, R)$ solving the operad relations.

• Cohomological interpretation:

$H^2(C^*(A))$ classifies $1$-parameter deformations and abelian extensions; higher cohomology groups encode obstructions.

The convolution $L_\infty$-algebra arising from $$\operatorname{Hom}_\mathbb{S}(\mathrm{RB}_\lambda^\infty, \operatorname{End}_A)$$ is canonically isomorphic (up to suspension) to the explicit $L_\infty$ above (Wang et al., 2021, Wang et al., 2022).

4. Splitting of Operads and Rota-Baxter Operators

The operadic machinery generalizes the construction of dendriform and tridendriform structures via Rota-Baxter operators to arbitrary operads:

• Operadic Rota-Baxter operators on a (possibly symmetric) operad $\mathcal{P}$ are defined via an adjunction of a unary operator $P$ to the operations of $\mathcal{P}$, together with generalized Rota-Baxter relations, parameterized by a configuration $C$ indexing the splittings (Pei et al., 2013).
• This gives rise to a split operad $\mathrm{CSp}(\mathcal{P})$, with new operations $\omega_I$ for each $I$ in the configuration set, and relations induced by "splitting trees."

Any $\mathcal{P}$-algebra with a Rota-Baxter operator becomes a $\mathrm{CSp}(\mathcal{P})$-algebra, and conversely, every splitting can be reconstructed from a (possibly relative) Rota-Baxter operator.

This framework naturally generalizes Loday's dendriform, tridendriform, pre-Lie, and post-Lie algebras as companions of the associative and Lie Rota-Baxter operads (Pei et al., 2013, Zheng et al., 11 Dec 2024).

5. Extended Rota-Baxter and Derived Operads

Extended Rota-Baxter algebras incorporate additional parameters, e.g., in the identity,

$$P(x) P(y) = P(x P(y)) + P(P(x) y) + \lambda P(x y) + \kappa x y$$

with the associated extended Rota-Baxter operad $\mathrm{ERB}_{(\lambda, \kappa)}$ being strictly binary-quadratic and Koszul in the operadic sense (Zheng et al., 11 Dec 2024).

Taking algebraic companions, one systematically obtains extended (tri)dendriform, pre-Lie, and post-Lie operads as quotients and images of the extension process. Free extended Rota-Baxter algebras are described via bracketed words with a recursive concatenation product, reflecting the operadic structure (Zheng et al., 11 Dec 2024).

6. Koszul Duality, Homotopy Cooperads, and Infinity Structures

For any Rota-Baxter operad $\mathrm{RB}_\lambda$, the Koszul dual structure is described via a homotopy cooperad. The cobar construction yields the minimal model, facilitating the study of RB$_\lambda^\infty$-algebras (homotopy Rota-Baxter algebras) (Wang et al., 2022, Qin et al., 3 Mar 2025).

The minimal model is constructed with generators and differentials determined by the combinatorics of planar trees:

• Higher multiplications (Stasheff $A_\infty$-type relations)
• Higher Rota-Baxter operators (encoding higher corrections to the classical relation)

The theory extends to Rota-Baxter systems (with multiple unary operators) and the study of associative $\infty$–Yang–Baxter pairs, linking homotopy Rota-Baxter systems and generalized Yang–Baxter equations (Qin et al., 3 Mar 2025).

7. Applications and Derived Structures

The operadic perspective enables:

• Construction and identification of companion algebraic structures (dendriform, tridendriform, pre-Lie, post-Lie).
• Functorial "splitting" of any algebraic operad along the lines induced by Rota-Baxter operators (Pei et al., 2013).
• Explicit minimal resolutions, enabling systematic deformation and obstruction computations.
• Construction of free objects (free Rota-Baxter algebras, free extended Rota-Baxter algebras) via combinatorial structure (bracketed words, planar trees).

The Rota-Baxter operad, its minimal resolution, and extensions underpin both classical and modern approaches to homotopy algebra, deformation theory, and the systematic study of splitting operations in universal algebra (Wang et al., 2021, Wang et al., 2022, Pei et al., 2013, Zheng et al., 11 Dec 2024, Qin et al., 3 Mar 2025).