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Restricted (Relative) Rota-Baxter operators on restricted Lie algebras and restricted Lie triple systems and related structures

This paper defines and investigates restricted Rota-Baxter operators on restricted Lie algebras and restricted Lie triple systems, establishing their connections to pre-Lie structures, proving that the pre-Lie triple system operad splits the Lie triple system operad, and demonstrating that Jacobson identities hold in pre-Lie triple systems in positive characteristic to introduce the concept of restricted pre-Lie triple systems.

Original authors: Jon Beristain, Abdenacer Makhlouf

Published 2026-09-21

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Jon Beristain, Abdenacer Makhlouf

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

In the vast landscape of modern mathematics, there exists a branch dedicated to understanding the hidden symmetries and structural rules that govern how things combine. Think of this as the study of the grammar of algebra, where mathematicians look for the fundamental laws that dictate how objects interact, whether they are numbers, shapes, or abstract functions. For decades, researchers have been particularly interested in a specific type of algebraic structure known as a Lie algebra, which serves as a mathematical model for continuous symmetry, much like the way a spinning top maintains its balance. Within this field, a special kind of operator, named after mathematicians Baxter and Rota, has long been known to act as a powerful tool. This operator allows mathematicians to split complex systems into simpler, more manageable parts, revealing a deeper layer of structure often called a "pre-Lie" system. These pre-Lie systems are fascinating because they are almost associative but possess a subtle asymmetry that makes them incredibly useful in fields ranging from physics to computer science.

However, a significant gap remained in our understanding when these systems are studied under specific conditions involving prime numbers, a scenario known in mathematics as "positive characteristic." In this mathematical universe, the usual rules of arithmetic behave differently, and a special operation, akin to raising a number to a specific power, becomes a defining feature of the structure. While mathematicians had successfully defined how these special operators work in standard Lie algebras under these conditions, the same clarity had not been achieved for more complex, three-part systems known as Lie triple systems. These triple systems are the mathematical cousins of Lie algebras, appearing naturally in the study of curved spaces and symmetric geometries, but their behavior in this prime-number setting had remained elusive.

The researchers in this paper set out to fill that gap by defining and studying a new concept: the restricted Rota-Baxter operator specifically for these triple systems. Their work begins by establishing a rigorous definition for how this operator functions when the system is constrained by the special rules of positive characteristic. They demonstrated that when such an operator is applied to a restricted Lie triple system, it does not merely split the system; it transforms it into a new, related structure called a restricted pre-Lie triple system. This is a significant discovery because it proves that the deep connection between these splitting operators and pre-Lie structures, which was well-known in simpler cases, holds true even in this more complex and restrictive environment.

To ensure their findings were not just isolated examples but part of a broader mathematical truth, the authors employed a sophisticated framework known as operad theory. You can think of an operad as a blueprint or a master plan that describes all the possible ways operations can be combined within a system. Using this blueprint, the researchers showed that the mathematical structure of pre-Lie triple systems is essentially a "splitting" of the structure of Lie triple systems. This means that the more complex pre-Lie system contains all the information of the original system, but organized in a way that reveals the hidden asymmetry introduced by the operator. This theoretical proof provides a solid foundation, confirming that the definition they proposed is the natural and correct one.

A crucial part of their investigation involved verifying a set of identities, named after the mathematician Jacobson, which describe how the special power operation behaves when applied to the sum of two elements. In many algebraic systems, these identities are essential for the structure to be considered "restricted." The authors proved that these identities hold true within the new pre-Lie triple systems they defined. This was a necessary step to validate their entire framework. By confirming that the power operation behaves consistently, they were able to introduce the formal concept of a "restricted pre-Lie triple system," a structure that had not been clearly defined before.

The paper concludes by extending these results to an even more general setting involving what are called relative operators, which act between a system and a separate space of values. They showed that the same principles apply here as well, providing a unified theory that connects restricted Lie algebras, restricted Lie triple systems, and their pre-Lie counterparts. The work does not claim to solve every problem in the field, nor does it propose immediate applications in engineering or physics. Instead, it offers a precise, proven map of a previously uncharted territory in abstract algebra. By defining these new operators and proving their properties, the researchers have provided the mathematical community with the necessary tools to explore these complex systems with greater confidence and clarity, ensuring that the intricate dance of algebraic structures in prime-number worlds is now better understood.

Technical Summary: Restricted (Relative) Rota-Baxter Operators on Restricted Lie Algebras and Restricted Lie Triple Systems

Problem Statement
The paper addresses the gap in the theory of algebraic structures in positive characteristic $p > 0$ . While Rota-Baxter operators and their connections to pre-Lie structures are well-established for Lie algebras and Lie triple systems in characteristic zero, the extension of these concepts to the "restricted" setting (where a $p$ -map exists) requires new definitions and verifications. Specifically, the authors aim to:

Define restricted Rota-Baxter operators on restricted Lie algebras and restricted Lie triple systems.
Establish the connection between these operators and restricted pre-Lie structures.
Verify whether Jacobson's identities (which govern the $p$ -map in restricted algebras) hold within pre-Lie triple systems.
Interpret the relationship between Lie triple systems and pre-Lie triple systems through the lens of operad theory.

Methodology
The authors employ a combination of algebraic definitions, structural proofs, and operadic techniques:

Definitions in Positive Characteristic: The paper introduces restricted Rota-Baxter operators by imposing a condition on the $p$ -map of the underlying algebra. For a restricted Lie algebra $(L, [\cdot, \cdot], (\cdot)^{[p]})$ and a Rota-Baxter operator $P$ , the operator is restricted if $P(x)^{[p]} = P(\text{ad}_{Px}^{p-1}(x))$ . This definition is motivated by the requirement that the graph of $P$ forms a restricted Lie subsystem of the semi-direct product $L \oplus L$ .
Operadic Splitting: To justify the definition of pre-Lie triple systems, the authors utilize the method of "splitting" operads developed in prior literature. They define the operad of Lie triple systems ( $LTS$ ) and apply a configuration-based splitting procedure (specifically using the configuration $A$ ) to derive the operad of pre-Lie triple systems ($pre-LTS$).
Verification of Identities: The core technical work involves proving that Jacobson's identities, which describe the expansion of $(x+y)^{[p]}$ , hold in pre-Lie triple systems. This is achieved by analyzing the representation theory of Lie triple systems and computing the coefficients of the polynomial expansion of the $p$ -th power of a sum in the induced Lie triple system structure.
Construction of Restricted Structures: The authors define "trivially restricted" pre-Lie triple systems (where the $p$ -map is defined via iterated operations) and a more general notion of "restricted" pre-Lie triple systems (involving an abstract $p$ -map satisfying specific axioms).

Key Contributions and Results

Restricted Rota-Baxter Operators on Lie Algebras:
- The authors define a restricted Rota-Baxter operator $P$ on a restricted Lie algebra $L$ by the condition $P(x)^{[p]} = P(\text{ad}_{Px}^{p-1}(x))$ .
- They prove that the graph of such an operator is a restricted Lie subsystem of the semi-direct product $L \oplus L$ .
- It is shown that a restricted Lie algebra with a restricted Rota-Baxter operator induces a trivially restricted pre-Lie algebra structure, and $P$ acts as a morphism of restricted Lie algebras between the induced structure and the original one.
Operadic Justification of Pre-Lie Triple Systems:
- The paper demonstrates that the operad of pre-Lie triple systems is the "splitting" of the operad of Lie triple systems. By applying the splitting method to the relations defining Lie triple systems, the authors recover the defining identities of pre-Lie triple systems (Equations 7 and 8 in the text), providing a rigorous operadic foundation for their existence.
Jacobson's Identities in Pre-Lie Triple Systems:
- A central result is the proof that Jacobson's identities hold in pre-Lie triple systems in positive characteristic. Specifically, for a pre-Lie triple system $(T, \{\cdot, \cdot, \cdot\})$ , the induced Lie triple system bracket $[\cdot, \cdot, \cdot]_C$ satisfies the expansion $(x+y)^{\{p\}} = x^{\{p\}} + y^{\{p\}} + \sum s_i(x,y)$ , where $s_i$ are the standard coefficients.
- This result allows for the definition of restricted pre-Lie triple systems, equipped with a $p$ -map satisfying specific compatibility conditions with the ternary product.
Restricted Rota-Baxter Operators on Restricted Lie Triple Systems:
- The concept is extended to restricted Lie triple systems. A Rota-Baxter operator $P$ on a restricted Lie triple system $(T, [\cdot, \cdot, \cdot], (\cdot)^{[p]})$ is restricted if $P(x)^{[p]} = P((x, Px, \dots, Px))$ , where the right-hand side denotes the iterated bracket with $p-1$ copies of $Px$ .
- The authors prove that such an operator induces a restricted pre-Lie triple system structure on $T$ via the product $\{x, y, z\} = [x, Py, Pz]$ .
- The operator $P$ is shown to be a morphism of restricted Lie triple systems from the induced structure to the original one.
Relative (O-operators) Extension:
- The results are generalized to relative Rota-Baxter operators (O-operators) acting between a representation space $V$ and the algebra $T$ . The definition of a restricted relative Rota-Baxter operator is given, and it is shown that its graph forms a restricted subsystem of the semi-direct product $T \oplus V$ .

Significance and Claims
The paper claims to provide a unified framework for studying Rota-Baxter operators in the context of restricted algebras and triple systems. By establishing that Jacobson's identities hold in pre-Lie triple systems, the authors successfully introduce the notion of restricted pre-Lie triple systems, filling a theoretical gap in positive characteristic algebra.

The significance lies in:

Generalization: Extending the well-known correspondence between Rota-Baxter operators and pre-Lie structures from the binary (Lie algebra) case to the ternary (Lie triple system) case, and further to the restricted setting.
Structural Insight: Using operad theory to formally derive pre-Lie triple systems as a splitting of Lie triple systems, thereby validating the definition of pre-Lie triple systems beyond ad-hoc algebraic manipulation.
Consistency: Demonstrating that the "restricted" condition (compatibility with the $p$ -map) is preserved under the transition from Lie triple systems to pre-Lie triple systems via Rota-Baxter operators, mirroring the behavior observed in the binary case.

The authors maintain a modest tone, focusing on the definition, verification of identities, and structural relationships without proposing new applications or experimental proposals. The work serves as a foundational step for further research into cohomology, deformations, and representations of these restricted structures in positive characteristic.

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