Quadratic Reynolds Lie Algebra

• Quadratic Reynolds Lie algebras are Lie algebras equipped with a Reynolds operator and a nondegenerate symmetric invariant bilinear form that satisfy strict compatibility conditions.
• They merge classical Lie theory with averaging operator techniques to yield novel insights in representation theory, graded structures, and deformation cohomology.
• Applications extend to invariant theory, Lie bialgebra constructions, and solutions to the classical Yang–Baxter equation, enriching both mathematics and physics.

A quadratic Reynolds Lie algebra is a Lie algebra equipped with both a Reynolds operator—a linear averaging operator satisfying a specific quadratic identity—and a nondegenerate symmetric invariant bilinear form (a quadratic form) linked by stringent compatibility conditions. The theory of these algebras encompasses representation-theoretic, structural, cohomological, and bialgebraic aspects, connecting classical Lie theory, invariant theory, and operator algebra. Quadratic Reynolds Lie algebras and their graded versions play a central role in bridging the theory of quadratic Lie algebras, Reynolds operators (from both fluid dynamics and algebra), and the structure of Lie bialgebras, with direct implications for the paper of symmetries, invariant forms, and classical Yang–Baxter equations.

1. Definition and Core Structure

A quadratic Reynolds Lie algebra is a Lie algebra $(\mathfrak{g}, [\cdot,\cdot])$ over a field (typically of characteristic zero), equipped with:

• A nondegenerate, symmetric, invariant bilinear form $S$:

$$S([x,y],z) + S(y, [x,z]) = 0 \quad \forall\, x, y, z \in \mathfrak{g}.$$

• A Reynolds operator $R\colon \mathfrak{g} \to \mathfrak{g}$ satisfying the quadratic Reynolds identity:

$$[R x, R y] = R\left( [R x, y] + [x, R y] - [R x, R y] \right), \qquad \forall x, y \in \mathfrak{g}.$$

• A compatibility condition:

$$S(R x, y) + S(x, R y) = 0 \qquad \forall\, x, y \in \mathfrak{g}.$$

Often, $S$ is called the invariant metric or quadratic form, and $R$ plays the role analogous to an averaging operator, ensuring a form of "invariance under projection" compatible with the Lie structure and the quadratic form (Goncharov et al., 5 Aug 2025).

The nondegeneracy and invariance of $S$ lead to a canonical identification $S^{\sharp}\colon \mathfrak{g} \to \mathfrak{g}^*$ via $x \mapsto S(x, -)$, and $R$ is required to satisfy

$S^{\sharp} \circ R = -R^* \circ S^{\sharp}$

where $R^*$ is the adjoint of $R$ with respect to $S$ (Goncharov et al., 5 Aug 2025).

2. Representation Theory and Local Graded Structures

Quadratic Reynolds Lie algebras generalize quadratic Lie algebras, whose representation theory is strongly governed by the existence of an invariant form. For a quadratic Lie algebra $(\mathfrak{g}_0, B_0)$ and a finite-dimensional representation $(\rho, V)$, local Lie algebras of the form

$$\Gamma(\mathfrak{g}_0, B_0, V) = V^* \oplus \mathfrak{g}_0 \oplus V$$

can be equipped with a bracket determined by $B_0$ and the representation $\rho$, leading to the construction of $\mathbb{Z}$-graded Lie algebras whose local part is $\Gamma(\mathfrak{g}_0, B_0, V)$. Minimal and maximal extensions, ${\mathfrak{g}}_{\min}$ and ${\mathfrak{g}}_{\max}$, arise such that any graded Lie algebra with the same local part is a suitable quotient of ${\mathfrak{g}}_{\max}$, and ${\mathfrak{g}}_{\min}$ is the unique quotient with trivial intersection with the local part (Rubenthaler, 2014).

For reductive quadratic Lie algebras and completely reducible faithful representations, the minimal graded algebra is finite-dimensional and semisimple when transitivity holds.

The presence of a Reynolds operator introduces an additional structure by splitting or projecting representations, often yielding further decompositions of modules and leading to the isomorphism of the adjoint and coadjoint representations:

$$(\mathfrak{g}; R, \mathrm{ad}) \cong (\mathfrak{g}^*; -R^*, \mathrm{ad}^*)$$

by $S^\sharp$ (Goncharov et al., 5 Aug 2025).

3. Structure Theory, Cohomology, and Deformation

Classification and rigidity of quadratic Reynolds Lie algebras are controlled by an associated cohomology theory. Consider a Reynolds operator $R$ and, possibly, a derivation $d$ commuting with $R$ ($R \circ d = d \circ R$), forming a Reynolds LieDer pair $(\mathfrak{g}, R, d)$. The induced bracket

$[x, y]_R = [x, R y] + [R x, y] - [R x, R y]$

defines a new Lie algebra $\mathfrak{g}_R$, and cochains $$(f,g) \in \mathrm{Hom}(\wedge^n \mathfrak{g}, V) \oplus \mathrm{Hom}(\wedge^n \mathfrak{g}_R, V)$$ fit into an explicit bicomplex:

$$D_R(f, g) = \big( \delta_{CE} f,\, -\partial_R g - \phi(f) \big)$$

with $$H^{n}_{RLieDer}(\mathfrak{g}; V) = \ker D_R / \operatorname{im} D_R$$ (Imed et al., 23 Apr 2025).

Vanishing of $H^2_{RLieDer}(\mathfrak{g}; V)$ signals rigidity of the Reynolds LieDer pair, and nontrivial classes detect obstructions to formal deformations, abelian extensions, and extensions of derivations. This approach refines the classical Chevalley–Eilenberg cohomology to accommodate the extra operator structure of $R$ and is sensitive to the quadratic structure via the interplay with the invariant metric.

In the case of quadratic Reynolds Lie algebras, these cohomology groups become even more restrictive since quadraticity imposes symmetry relations on the cochains, further constraining possible deformations and extensions (Imed et al., 23 Apr 2025).

4. Bialgebraic Structures, Matched Pairs, and Manin Triples

Quadratic Reynolds Lie algebras serve as a natural domain for Reynolds Lie bialgebra structures. A Reynolds Lie bialgebra consists of a quadratic Reynolds Lie algebra $(\mathfrak{g}, [\cdot,\cdot], R, S)$ and a cobracket $$\delta\colon \mathfrak{g} \to \wedge^2 \mathfrak{g}$$ compatible with both the Reynolds and Lie algebraic structures. The quadratic form $S$ produces, via $S^\sharp$, an isomorphism from the adjoint–Reynolds structure to the coadjoint–Reynolds structure (Goncharov et al., 5 Aug 2025).

Key concepts include:

• Matched pairs: Two Reynolds Lie algebras $(\mathfrak{g}, R)$ and $(\mathfrak{g}', R')$ acting on one another via suitably compatible representations, so that the direct sum $\mathfrak{g} \oplus \mathfrak{g}'$ admits both a Lie bracket and a direct sum Reynolds operator.
• Manin triples: A quadratic Reynolds Lie algebra $(\mathfrak{d}, R, S)$ together with isotropic subalgebras $\mathfrak{g}$ and $\mathfrak{g}^*$ such that the restriction of $R$ and $S$ matches with the respective structures. The equivalence of Manin triples and matched pairs in this context yields a direct route to Reynolds Lie bialgebras.
• Drinfeld doubles and CYBE: Solutions to the classical Yang–Baxter equation compatible with the Reynolds operator structure yield Reynolds Lie bialgebras, with the r-matrix required to satisfy both the standard CYBE and an averaging property: $(R \otimes I + I \otimes R)(r) = 0$.

A significant structural result is that a Reynolds operator $R$ on a quadratic Rota–Baxter Lie algebra (with $S$ also compatible with the Rota–Baxter structure) naturally produces a Reynolds Lie bialgebra structure (Goncharov et al., 5 Aug 2025).

5. Graded Variants and Polynomial/Prehomogeneous Constructions

Graded quadratic Reynolds Lie algebras arise in several geometric and invariant-theoretic constructions. For instance, given a reductive quadratic Lie algebra $\mathfrak{g}_0$ and a finite-dimensional representation $V$, one defines a local Lie algebra $$\Gamma(\mathfrak{g}_0, B_0, V) = V^* \oplus \mathfrak{g}_0 \oplus V$$ and builds from it:

• Maximal and minimal $\mathbb{Z}$-graded Lie algebras $g_{\max}$ and $g_{\min}$, capturing the entire family of graded structures whose local part is given by $\Gamma$ (Rubenthaler, 2014).
• Geometric scenarios such as polynomial type (symplectic) graded Lie algebras, where $V$ is taken as a space of homogeneous polynomials on a vector space $W$, and $\mathfrak{g}_0 = \mathfrak{gl}(W)$, admit identifications with classical Lie algebras (such as $\mathfrak{sl}_{n+1}$, $\mathfrak{sp}(n)$, etc.), where the presence of relative invariants and dual pairs is determined through nontrivial $sl_2$-triples.

These graded structures are frequently studied in the context of representation theory, invariant theory, and dual pairs. The existence of an associated $sl_2$-triple is equivalent to the presence of nontrivial relative invariants on some representation orbit, tightly linking graded algebraic properties to geometric invariants.

6. Operator Theory, Pre-Lie Structures, and Yang–Baxter Theory

Quadratic Reynolds Lie algebras are situated at the confluence of several operator identities. For Reynolds operators arising as special cases of twisted Rota–Baxter operators or O-operators (with a suitable 2-cocycle), the associated quadratic identities manifest as:

$$[R(x), R(y)] = R([R(x), y] + [x, R(y)] - [R(x), R(y)]),$$

so that the image of the operator features quadratically both in the bracket and the right-hand terms (Das, 2020). In the module-theoretic setting, the concept of relative Rota–Baxter operators transfers the Reynolds condition to modules. Similarly, Reynolds pre-Lie algebras, with a product $\{\cdot, \cdot\}$ satisfying a Reynolds-type averaging property, provide structures whose commutator bracket yields a quadratic Reynolds Lie algebra. The left multiplication representation $L(x) = \{x, -\}$ induces corresponding representation-theoretic features.

From a bialgebraic viewpoint, these structures underpin constructions of solutions to the Classical Yang–Baxter Equation (CYBE) compatible with the Reynolds operator:

• The $r$-matrix must obey both the CYBE and the Reynolds averaging condition $(R \otimes I + I \otimes R)(r) = 0$.
• Relative Rota–Baxter operators and compatible pre-Lie structures systematically yield such $r$-matrices (Goncharov et al., 5 Aug 2025).

7. Cohomological Rigidity, Extensions, and Applications

The explicit cohomology theory for Reynolds LieDer pairs is essential for the deformation, extension, and rigidity theory of quadratic Reynolds Lie algebras:

• First- and second-cohomology groups govern infinitesimal and formal deformations, abelian extensions, and obstructions to lifting derivations or extensions (Imed et al., 23 Apr 2025).
• The presence of the quadratic form $S$ infuses extra symmetry into these cohomology groups, yielding particularly rigid or constrained algebraic objects.
• In applications arising from left Leibniz algebras with associative, $L$-invariant, or $R$-invariant metrics, the core structure always descends to a quadratic Lie algebra—often a quadratic Reynolds Lie algebra (Abid et al., 2023).
• In nilpotent and double extension settings, the existence and classification of quadratic Reynolds Lie algebras rely on explicit matrix-based algorithms and the analysis of quotient and derivation structures (Benito et al., 25 Jan 2024, Benito et al., 2018).

In summary, quadratic Reynolds Lie algebras represent the intersection of averaging operator theory, quadratic and graded Lie algebra theory, invariant and cohomological algebra, and Lie bialgebra theory. The integration of these perspectives yields a flexible yet tightly constrained setting ideal for the paper of symmetry, representations, invariants, and quantization in both mathematics and mathematical physics.