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Deformations and homotopy theory of Rota-Baxter Lie algebras

This paper establishes a homotopy cooperad whose cobar construction yields the minimal model of the operad of Rota-Baxter Lie algebras via algebraic Morse theory, thereby deriving their deformation complex, $L_\infty$ -algebra structure, and the concept of homotopy Rota-Baxter Lie algebras.

Original authors: Jun Chen, Kai Wang, Guodong Zhou

Published 2026-08-19

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CC BY 4.0

Original authors: Jun Chen, Kai Wang, Guodong Zhou

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Mathematics often advances by studying how structures change. Just as a physicist might ask how a crystal deforms under pressure, algebraists ask how mathematical systems shift when their defining rules are tweaked slightly. This field, known as deformation theory, seeks to understand the flexibility of mathematical objects. For decades, a guiding principle has held that the behavior of any such object is governed by a specific type of algebraic structure that tracks these changes. When the object is a Lie algebra—a system used to describe continuous symmetries and transformations—mathematicians have long known how to describe its deformations using a structure called an $L_\infty$ -algebra. This structure acts as a control center, encoding all possible ways the system can bend or twist while remaining consistent. However, when the system includes an extra layer of complexity, such as a special operator that rearranges elements in a specific way, the rules become much harder to write down.

The paper at hand tackles one of these complex systems: the Rota-Baxter Lie algebra. These structures originated in probability theory but have since become vital in areas ranging from quantum physics to combinatorics. A Rota-Baxter Lie algebra consists of a standard Lie algebra paired with a linear operator that satisfies a particular identity, effectively acting as a bridge between different parts of the system. While mathematicians had previously understood how to deform the operator alone or how to handle cases where the operator's weight was zero, a complete picture for the general case—where the operator and the underlying algebra change simultaneously—remained elusive. The challenge was that the standard tools used to analyze these systems failed because the algebraic rules governing Rota-Baxter Lie algebras are too intricate to fit into the usual categories. The authors of this study, Jun Chen, Kai Wang, and Guodong Zhou, set out to build a new framework from the ground up to solve this problem.

The researchers began by constructing a "minimal model" for these algebras. In the language of algebra, a minimal model is a simplified, yet complete, blueprint that captures the essential behavior of a complex system. For many standard algebraic structures, this blueprint can be found using a technique called Koszul duality. However, the authors demonstrated that the system they were studying does not fit the criteria for this standard technique. Instead of giving up, they employed a sophisticated method known as algebraic Morse theory. This approach allowed them to analyze the vast space of possible algebraic configurations, identifying a specific set of "critical" points that represent the true, stable forms of the system. By mapping out these critical points, they successfully constructed the minimal model for Rota-Baxter Lie algebras of any weight, a feat that had previously been out of reach.

With this new blueprint in hand, the team was able to extract the precise rules that govern how these algebras deform. They showed that the space of all possible deformations is controlled by a specific $L_\infty$ -algebra, a structure that generalizes the familiar concept of a Lie algebra to include higher-order interactions. This discovery allowed them to define a new concept: the "homotopy Rota-Baxter Lie algebra." This is a flexible version of the original system where the strict rules are relaxed to hold only "up to homotopy," meaning they are satisfied in a way that allows for small, continuous errors that can be corrected. This new definition provides a rigorous way to study systems that are slightly broken or fluctuating, which is often the reality in physical applications.

The study also clarified the relationship between these Lie algebras and their associative counterparts. In mathematics, there is a standard way to turn an associative algebra (where order of multiplication matters) into a Lie algebra (where it does not) by using a commutator. The authors proved that this classical transformation extends perfectly to their new homotopy versions. They constructed a direct map between the minimal model of the associative system and the minimal model of the Lie system, showing that the complex homotopy structures on the Lie side are naturally induced by the simpler associative side. This connection confirms that the new theory is consistent with established mathematical principles while extending them into uncharted territory.

Ultimately, the paper provides a complete toolkit for understanding the deformations of Rota-Baxter Lie algebras. By proving that the minimal model exists and explicitly describing the controlling algebraic structure, the authors have removed the ambiguity that previously surrounded these objects. They have shown that the deformation theory is not just a collection of ad-hoc calculations but is governed by a single, coherent algebraic structure. This work lays the foundation for future research, allowing mathematicians to apply these robust deformation techniques to problems in quantum field theory and other fields where these algebras play a central role. The results are not merely suggestions or simulations; they are rigorous proofs that establish a new, solid framework for a class of mathematical objects that had resisted a unified description.

Technical Summary: Deformations and Homotopy Theory of Rota-Baxter Lie Algebras

Problem Statement
The paper addresses two interconnected problems in the deformation theory and homotopy theory of algebraic structures, specifically focusing on Rota-Baxter Lie algebras of arbitrary weight $\lambda$ .

Explicit Construction of the Controlling $L_\infty$ -algebra: While the general philosophy of deformation theory posits that deformations are controlled by an $L_\infty$ -algebra, constructing this algebra explicitly for specific structures is often difficult. The authors aim to construct the $L_\infty$ -algebra governing the simultaneous deformations of both the Lie bracket and the Rota-Baxter operator for Rota-Baxter Lie algebras of arbitrary weight.
Homotopy Version of the Structure: The paper seeks to define the "homotopy version" of Rota-Baxter Lie algebras (analogous to $L_\infty$ -algebras for Lie algebras or $A_\infty$ -algebras for associative algebras). This requires finding a minimal model for the operad $\lambda\text{RBL}$ governing these structures.

A significant challenge is that the operad $\lambda\text{RBL}$ is not Koszul (and not even quadratic), meaning classical Koszul duality theory cannot be directly applied to generate a minimal model. Previous works had largely focused on deforming the Rota-Baxter operator with a fixed bracket or restricted to weight zero. This paper aims to treat the simultaneous deformation of the bracket and operator for arbitrary weights.

Methodology
The authors employ a combination of operadic theory, algebraic Morse theory, and homotopy algebra techniques:

Minimal Models via Algebraic Morse Theory: Since $\lambda\text{RBL}$ is not Koszul, the authors construct a coaugmented symmetric homotopy cooperad, denoted $S(\lambda\text{RBL}^!)$ , which serves as the Koszul dual homotopy cooperad. They then utilize algebraic Morse theory to prove that the cobar construction $\Omega(\lambda\text{RBL}^!)$ is indeed the minimal model of $\lambda\text{RBL}$ . This involves defining a specific "right path-permutation extension" order on shuffle tree monomials to construct an acyclic Morse matching on the free operad generated by the desuspension of the dual cooperad.
Convolution Formalism: Once the minimal model (and its dual homotopy cooperad) is established, the authors apply the general convolution formalism. They define an $L_\infty$ -algebra on the space of cochains via the convolution of the dual cooperad with the endomorphism operad.
Twisting Procedure: To recover the deformation complex of a specific Rota-Baxter Lie algebra $(g, \ell, T)$ , the authors apply the twisting procedure to the constructed $L_\infty$ -algebra using the Maurer-Cartan element corresponding to the given structure.
Comparison with Associative Theory: The paper utilizes the commutator construction to relate Rota-Baxter associative algebras to Rota-Baxter Lie algebras, constructing a morphism between their respective minimal models.

Key Contributions and Results

Minimal Model of $\lambda\text{RBL}$ (Theorem A):
The authors explicitly construct a coaugmented symmetric homotopy cooperad $S(\lambda\text{RBL}^!)$ and prove that its cobar construction, $\lambda\text{RBL}_\infty := \Omega(\lambda\text{RBL}^!)$ , is the minimal model of the operad of Rota-Baxter Lie algebras of weight $\lambda$ . This is achieved by exhibiting an explicit acyclic Morse matching on the free operad, where the critical tree monomials correspond to the normal forms of Rota-Baxter Lie algebras.
Deformation Complex and $L_\infty$ -Structure (Theorem B):
- An $L_\infty$ -algebra $C_{\text{RBL}}^\lambda(V)$ is defined on the graded space of cochains of a graded space $V$ . The Maurer-Cartan elements of this $L_\infty$ -algebra are shown to be exactly the homotopy Rota-Baxter Lie algebra structures on $V$ .
- By twisting this $L_\infty$ -algebra with the Maurer-Cartan element corresponding to a specific Rota-Baxter Lie algebra $(g, \ell, T)$ , the authors recover the deformation complex $C^\bullet_{\text{RBL}}(g)$ .
- This complex is explicitly described as the negative shift of the mapping cone of a chain map $\Phi^\bullet$ from the Chevalley-Eilenberg complex of the Lie algebra $g$ to the cochain complex of the Rota-Baxter operator $T$ (with coefficients in a representation).
- The paper also establishes that the space governing the deformation of the operator alone (with fixed bracket) carries a dg Lie algebra structure.
Definition of Homotopy Rota-Baxter Lie Algebras (Theorem C context):
Based on the minimal model, the authors provide a rigorous definition of homotopy Rota-Baxter Lie algebras of weight $\lambda$ . These are defined as morphisms from the minimal model $\lambda\text{RBL}_\infty$ to the endomorphism operad. Explicitly, this structure consists of a family of anti-symmetric maps $\ell_n$ (defining an $L_\infty$ -algebra) and $T_n$ satisfying generalized compatibility conditions (Equations 19 and 20) that extend the Rota-Baxter identity to the homotopy level.
Relation to Associative Theory:
The paper constructs an explicit morphism of dg operads $\Phi: \lambda\text{RBL}_\infty \to \lambda\text{RBAS}_\infty$ from the minimal model of Rota-Baxter Lie algebras to the minimal model of Rota-Baxter associative algebras (constructed in prior work [40]). This morphism is given on generators by anti-symmetrization, demonstrating that the classical functor sending associative algebras to Lie algebras via the commutator deforms to the homotopy level.

Significance and Claims
The paper claims to provide the first comprehensive operadic-level treatment of Rota-Baxter Lie algebras of arbitrary weight, addressing the simultaneous deformation of the bracket and the operator.

It resolves the difficulty of non-Koszul operads by successfully applying algebraic Morse theory to construct a minimal model.
It unifies the deformation theory, cohomology theory, and homotopy theory of these structures under a single framework derived from the minimal model.
The results generalize previous work on relative Rota-Baxter Lie algebras (specifically those of weight zero) and extend the theory to arbitrary weights, showing that the cohomology theories coincide with existing constructions at the level of cochain complexes while providing the underlying $L_\infty$ -structure which was previously missing.
The work establishes a precise homotopy-theoretic link between Rota-Baxter associative algebras and Rota-Baxter Lie algebras, confirming that the passage from associative to Lie structures via commutators is compatible with the homotopy theory.

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