Reynolds LieDer Pair Overview

• Reynolds LieDer pair is a structure that combines a Lie algebra with a Reynolds operator and a commuting derivation, ensuring compatibility in averaging and differentiation.
• Its cohomology framework supports the classification of abelian extensions, analysis of formal deformations, and determination of rigidity through innovative operator methods.
• The concept bridges classical invariant theory with modern quantum algebra, providing new insights into deformation theory and operator-algebraic extensions.

A Reynolds LieDer pair consists of a Reynolds Lie algebra equipped with a derivation that commutes with the Reynolds operator. This concept blends the algebraic structures of Reynolds operators, which encode invariant-theoretic and averaging properties, with derivations characteristic of Lie and differential algebra. The Reynolds LieDer pair framework is central for cohomological analysis, deformation theory, and classification of algebraic extensions, and serves as a bridge between the theory of Reynolds Lie algebras and operator-based structures in classical and quantum algebraic settings.

1. Definition and Structure of Reynolds LieDer Pairs

Let $(L, [\cdot,\cdot])$ be a Lie algebra over a field, and $R: L \to L$ a Reynolds operator, i.e.,

$$[R x, R y] = R( [x, R y] + [R x, y] - [R x, R y] ), \qquad \forall x, y \in L.$$

A (linear) derivation $d: L \to L$ is a map satisfying

$d([x, y]) = [d(x), y] + [x, d(y)].$

A Reynolds LieDer pair is a triple $(L, R, d)$ such that $R$ is a Reynolds operator, $d$ is a derivation, and $R$ and $d$ commute,

$R \circ d = d \circ R.$

This compatibility ensures that $d$ restricts to the subalgebras of $L$ preserved by $R$ and that the action of $d$ respects the averaging process encoded by $R$ (Imed et al., 23 Apr 2025). If $R$ is invertible, $(L, R, R^{-1} - \mathsf{id})$ gives a canonical example of a Reynolds LieDer pair (Hou et al., 2021).

2. Cohomology of Reynolds LieDer Pairs

To paper deformations and extensions in this operator-enhanced context, a dedicated cohomology theory is constructed. The key is the combination of the Chevalley–Eilenberg cochain complex for Lie algebras and additional structures to incorporate both $R$ and $d$.

The cochain groups are given by

$$C^n_{\operatorname{RLieDer}}(L; V) = C^n(L; V) \oplus C^{n-1}(L; V),$$

where $C^n(L; V)$ is the skew-symmetric $n$-linear maps from $L$ to $V$ for a representation $V$ (with its own Reynolds operator $R_V$ and derivation $d_V$). The differential is

$DRLieDer(f) = ( DR(f), -A(f) ),$

where $DR$ encodes the combined coboundary for the Reynolds and Lie structures, and $A$ acts by taking sums over derivations applied to each argument minus the action of $d_V$ (Imed et al., 23 Apr 2025).

The cohomology $H^n_{\operatorname{RLieDer}}(L; V)$ captures cocycles which are both Reynolds-cocycles and invariant under $d$: $f \circ d_L = d_V \circ f.$ This construction generalizes the standard Lie algebra cohomology and extends deformation theory and extension classification to Reynolds LieDer pairs.

3. Formal Deformations and Rigidity

The deformation theory of Reynolds LieDer pairs uses formal power series expansions: $$\mu_t = \mu + \mu_1 t + \mu_2 t^2 + \dots, \quad R_t = R + R_1 t + R_2 t^2 + \dots, \quad d_t = d + d_1 t + d_2 t^2 + \dots$$ First-order deformations are governed by cohomology: the triple $(\mu_1, R_1, d_1)$ forms a 2-cocycle in $C^2_{\operatorname{RLieDer}}(L; L)$. The deformation constraints (to order $t$) are: $$\delta_{\operatorname{CE}}(\mu_1) = 0 \qquad \delta_R(R_1) + \phi(\mu_1) = 0 \qquad \delta_{\operatorname{CE}}(d_1) + A(\mu_1) = 0 \qquad R_1 d + R d_1 - d_1 R - d R_1 = 0$$ (Imed et al., 23 Apr 2025). Rigidity of a Reynolds LieDer pair is equivalent to the vanishing of the second cohomology group: $H^2_{\operatorname{RLieDer}}(L; L) = 0$ implies every deformation is trivial up to equivalence.

4. Abelian Extensions and Classification via Cohomology

An abelian extension of Reynolds LieDer pairs is an exact sequence

$$0 \rightarrow (V, R_V, d_V) \rightarrow (\hat{L}, \hat{R}, \hat{d}) \rightarrow (L, R, d) \rightarrow 0$$

with $(V, R_V, d_V)$ abelian and the maps compatible. Upon choosing a section $s: L \to \hat{L}$, one defines cocycle representatives for the extension via

$$\Theta(x, y) = [s(x), s(y)]_{\hat{L}} - s([x, y]), \quad x(a) = \hat{d}(s(a)) - s(d(a)), \quad \xi(a) = \hat{R}(s(a)) - s(R(a))$$

[(Imed et al., 23 Apr 2025), §5]. The extension class is then characterized by a 2-cocycle in the cohomology $H^2_{\operatorname{RLieDer}}(L; V)$. The cohomology thus classifies equivalence classes of abelian extensions, generalizing the classical theory to the Reynolds LieDer setting.

5. Obstructions to Lifting Derivations

Given a central extension of Reynolds Lie algebras, the theory provides criteria for lifting compatible derivations to the extension. For a central extension

$$0 \to (V, R_V) \to (\hat{L}, \hat{R}) \to (L, R) \to 0$$

with derivations $d$ and $d_V$ on $L$ and $V$, the obstruction to finding $\hat{d}$ with $\hat{d}(V) \subset V$ and commuting with $\hat{R}$ arises as a 2-cocycle in ordinary Reynolds Lie algebra cohomology $H^2_R(L; V)$ [(Imed et al., 23 Apr 2025), §6]. This criterion mirrors the obstruction-theoretic role of cohomology in the classical cases and provides an explicit computational framework for extension problems.

6. Connections with Other Operator-Algebraic Structures

The Reynolds LieDer pair concept is related to several other algebraic structures:

• LieDer pairs: Lie algebras with derivations, classified and studied by their own deformation and extension cohomology (Tang et al., 2019).
• Reynolds Lie algebras: Lie algebras with Reynolds operators, with induced algebra structures $[x, y]_R = [x, Ry] + [Rx, y] - [Rx, Ry]$; Reynolds operators arise both in invariant theory and in the averaging of algebraic or differential structures (Imed et al., 2023, Goncharov et al., 5 Aug 2025).
• Reynolds $n$-Lie algebras: Generalizations to higher arity, where Reynolds operators satisfy a modified identity and have an explicit cohomology and deformation theory (Hou et al., 2021).
• Rota–Baxter and endomorphism operators: Used to construct compatible LieDer and Reynolds structures and central in generating new indecomposable brackets and transfering averaging properties (Imed et al., 2023).
• Quadratic Reynolds Lie algebras and bialgebras: When the Lie algebra is quadratic and the Reynolds operator is compatible with the invariant form, a rich bialgebra and Manin triple theory emerges, paralleling the Drinfeld double in quantum algebra (Goncharov et al., 5 Aug 2025).

These relationships provide a unified operator-theoretic framework that subsumes averaging, compatibility, and differential structures.

7. Applications and Broader Significance

Reynolds LieDer pairs and their cohomology underpin formal deformation theory, extension and lifting problems, as well as rigidity and obstruction theory in operator-augmented Lie algebras. The framework captures the nuances of simultaneous averaging and differentiation in noncommutative settings and finds applications in:

• Classification and construction of algebra extensions and bialgebras (Goncharov et al., 5 Aug 2025)
• Cohomological control of deformations and rigidity phenomena (Imed et al., 23 Apr 2025)
• Systematic generalization to higher-arity algebras (e.g., $n$-Lie structures) (Hou et al., 2021)
• The paper of the classical Yang–Baxter equation and quantum group theory, with Reynolds operators enabling new classes of solutions (Goncharov et al., 5 Aug 2025)
• Bridging invariant theory, representation theory, and operator algebra

The Reynolds LieDer pair thus provides a natural setting for the paper and classification of algebraic structures subject to both averaging and differentiation, with a mature cohomological apparatus for extension and deformation problems, and tight interconnections with active research directions in modern algebra.