Quadratic Reynolds Lie Algebras

• Quadratic Reynolds Lie algebras are defined as Lie algebras equipped with a Reynolds operator and a nondegenerate symmetric invariant bilinear form, establishing a robust algebraic structure.
• They feature key invariants, such as the dup-number, and employ classification methods via double extensions and central extensions linked with Manin triples and matched pairs.
• These algebras connect to the classical Yang–Baxter equation and cohomological deformations, offering insights into integrable systems, quantum groups, and representation theory.

A quadratic Reynolds Lie algebra is a Lie algebra (𝔤, [·,·]) equipped with both a Reynolds operator—a linear map R : 𝔤 → 𝔤 satisfying the Reynolds identity—and a nondegenerate symmetric invariant bilinear form S, making (𝔤, S) a quadratic Lie algebra. The synthesis of these structures imposes rich algebraic, cohomological, and representation-theoretic properties, with notable connections to the theory of double extensions, Manin triples, bialgebra theory, and the classical Yang–Baxter equation.

1. Definition and Foundational Structure

A quadratic Reynolds Lie algebra is defined as a triple (𝔤, R, S), where:

• 𝔤 is a Lie algebra with bracket [·,·],
• R : 𝔤 → 𝔤 is a Reynolds operator satisfying the Reynolds relation

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

• S is a nondegenerate symmetric invariant bilinear form, i.e.,

$$S([x, y], z) = S(x, [y, z]) \quad \text{for all } x, y, z \in \mathfrak{g}.$$

The compatibility condition between S and R is described by the skew-adjointness property

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

(Goncharov et al., 5 Aug 2025).

This compatibility enables the construction of an isomorphism $S^\sharp : \mathfrak{g} \to \mathfrak{g}^*$ given by $\langle S^\sharp(x), y \rangle = S(x, y)$ and ensures intertwining between the adjoint representation with R and the coadjoint representation with –R*:

$S^\sharp \circ R = - R^* \circ S^\sharp.$

2. Invariants and Classification Principles

Quadratic Reynolds Lie algebras can be studied using invariants originating in quadratic Lie algebra theory. A key invariant is the so-called dup-number (decomposability invariant), defined for any quadratic Lie algebra (𝔤, B) as

$$\operatorname{dup}(\mathfrak{g}) = \dim \{ \alpha \in \mathfrak{g}^* \mid \alpha \wedge I = 0 \},$$

where $I(X,Y,Z) = B([X,Y], Z)$ is the canonical 3-form. For non-Abelian cases, $\operatorname{dup}(\mathfrak{g}) \in \{0,1,3\}$, partitioning algebras into ordinary (dup = 0) and singular (dup = 1 or 3) types (Thanh et al., 2010).

The presence of R introduces the potential for refined invariants. For example, extension to a "Reynolds dup-number" is plausible by measuring how the extra structure modulates the decomposability of I, specifically with respect to 1-forms compatible with R.

Classification results for singular cases typically reduce to analyzing double extensions by skew-symmetric derivations and O(n)-adjoint orbits in 𝔬(n). The isomorphism class of a singular quadratic Lie algebra constructed as a double extension is determined by the O(n)-adjoint orbit of the extension map (Thanh et al., 2010, Pham et al., 2012).

3. Double Extensions, Central Extensions, and Canonical Constructions

The double extension technique underpins both the construction and classification of quadratic and quadratic Reynolds Lie algebras. Let (𝔮, B_𝔮) be a quadratic vector space and $C \in \mathfrak{o}(\mathfrak{q})$ a skew-symmetric derivation. The double extension is realized on $\mathfrak{g} = \mathfrak{q} \oplus t$, with $t$ a 2-dimensional space, and the Lie bracket incorporates the data of C:

• $[e, X] = C(X)$,
• $$[X, Y] = [X, Y]_{\mathfrak{q}} + B_{\mathfrak{q}}(C(X), Y) f$$.

This structure, when compatible with a Reynolds operator R (e.g., R extended from C via appropriate averaging properties), provides a systematic approach to realizing quadratic Reynolds Lie algebras (Pham et al., 2012, Benito et al., 6 Jan 2024, Benito et al., 25 Jan 2024).

Central extensions with invariant metrics also play a central role. A quadratic Lie algebra admits a central extension with an invariant metric if and only if the cocycle defining the extension is derived from skew-symmetric derivations. When the extension kernel is isotropic, the extension can be expressed as a double extension, preserving the quadratic structure (García-Delgado et al., 2018).

4. Reynolds Operators, Rota–Baxter Operators, and Pre-Lie Structures

The Reynolds operator is an averaging operator that satisfies a quadratic identity, providing a direct generalization of Rota–Baxter operators in the Lie algebraic setting:

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

(Goncharov et al., 5 Aug 2025). When combined with a quadratic form, the self-adjointness property ensures intertwining of the adjoint and coadjoint actions.

Connections to Rota–Baxter Lie algebras are particularly noteworthy. A quadratic Rota–Baxter Lie algebra of weight λ includes an additional operator B for which $S(x, B y) + S(B x, y) + \lambda S(x, y) = 0$, and commutation or compatibility between R and B enables the construction of Reynolds Lie bialgebras (Goncharov et al., 5 Aug 2025, Goncharov, 2019).

Quadratic Reynolds Lie algebras also interface with pre-Lie and NS-Lie algebra structures. A Reynolds pre-Lie algebra is endowed with a Reynolds operator R satisfying an analogue of the Reynolds identity for pre-Lie products. The sub-adjacent Lie algebra then inherits a Reynolds structure (Goncharov et al., 5 Aug 2025).

5. Matched Pairs, Manin Triples, and Bialgebra Structures

Quadratic Reynolds Lie algebras naturally participate in broader algebraic machines:

• Matched pairs of Reynolds Lie algebras: two Reynolds Lie algebras with compatible mutual actions, such that their direct sum (as vector spaces) is a Lie algebra respecting both Reynolds structures.
• Manin triples in the Reynolds context: a quadratic Reynolds Lie algebra (𝔡, R, S) with two maximal isotropic Reynolds subalgebras 𝔤 and 𝔤′, forming a direct sum decomposition 𝔤 ⊕ 𝔤′ = 𝔡. The equivalence between matched pairs and Manin triples holds under appropriate compatibility conditions (Goncharov et al., 5 Aug 2025).

Such structures are foundational in bialgebra theory, where a Reynolds Lie bialgebra consists of a Lie algebra with both a Reynolds operator R and a compatible cobracket (or dual operator –R*) (Goncharov et al., 5 Aug 2025).

6. The Classical Yang–Baxter Equation and Reynolds Lie Bialgebras

The classical Yang–Baxter equation (CYBE) receives a Reynolds-theoretic enhancement. For r ∈ 𝔤 ⊗ 𝔤 in a quadratic Reynolds Lie algebra (𝔤, R, S), the CYBE is considered together with an invariance (or compatibility) with R:

• The invariance condition: $(ad_x \otimes + \otimes ad_x)(r + \sigma(r)) = 0$.
• The CYBE: $[[r, r]] = 0$.
• The Reynolds condition: $(R \otimes + \otimes R)(r) = 0$.

Solutions r then yield Reynolds Lie bialgebra structures (Goncharov et al., 5 Aug 2025). Further, relative Rota–Baxter operators produce such r-matrices in the context of matched-pair and semidirect product constructions.

7. Cohomology, Deformations, and Classification Problems

Cohomology theories specialized to Reynolds Lie algebras, and by extension to quadratic Reynolds Lie algebras, provide a formalism for studying deformations, extensions, and structural rigidity. The cohomology of Reynolds LieDer pairs (triples (L, R, d) with a derivation d commuting with R) encodes formal deformations, abelian extensions, and derivation extensions via explicit cocycle and coboundary conditions (Imed et al., 23 Apr 2025). The preservation of the quadratic form in such deformations imposes additional algebraic constraints, informing both the deformation theory and classification of these objects.

A key aspect is that the presence of an invariant quadratic form, compatible with both the Lie bracket and the Reynolds operator, provides a powerful restriction, often yielding isomorphisms between the adjoint and coadjoint representations, and, in the context of extensions, frequently necessitating triviality of certain cocycles for invariance to be preserved (Thanh et al., 2010, Imed et al., 23 Apr 2025).

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Quadratic Reynolds Lie algebras thus synthesize invariant-theoretic, representation-theoretic, and cohomological structures. Their analysis leverages deep connections to double extensions, matched pairs, Manin triples, and the operator theory of Rota–Baxter and Reynolds type, supported by a robust cohomological framework for questions of rigidity, deformation, and bialgebraic extensions. These structures provide a natural algebraic setting for generalizations in invariant theory, quantum groups, and the theory of integrable systems.