Reynolds Lie Bialgebra Theory

• Reynolds Lie bialgebras are Lie bialgebras equipped with Reynolds operators that average and deform both the bracket and cobracket under strict compatibility conditions.
• They unify various algebraic constructs such as quadratic structures, matched pairs, and Manin triples, offering a versatile framework for novel operator-theoretic generalizations.
• These structures facilitate constructing solutions to the classical Yang–Baxter equation and advancing cohomological and deformation theories in integrable systems and quantization.

A Reynolds Lie bialgebra is a Lie bialgebra equipped with a Reynolds operator subject to compatibility constraints between the algebraic and coalgebraic structures. This framework extends the classical theory of Lie bialgebras by integrating Reynolds operators, which act as averaging operators and deform both the bracket and cobracket in a controlled manner. Reynolds Lie bialgebras unify diverse algebraic constructs—quadratic structures, matched pairs, Manin triples, and various operator-theoretic generalizations—and provide avenues for constructing solutions to the classical Yang–Baxter equation (CYBE) in new algebraic contexts (Goncharov et al., 5 Aug 2025).

1. Quadratic Reynolds Lie Algebras

A quadratic Reynolds Lie algebra consists of a Lie algebra $(\mathfrak{g}, [\cdot,\cdot])$ endowed with a Reynolds operator $R:\mathfrak{g}\rightarrow \mathfrak{g}$ and a nondegenerate symmetric invariant bilinear form $S$. The Reynolds operator satisfies

$[Rx,Ry] = R\Big([Rx,y]+[x,Ry]-[Rx,Ry]\Big)$

for all $x,y$. The bilinear form $S$ further obeys

$S(Rx,y)+S(x,Ry)=0$

ensuring compatibility between $R$ and $S$. This structure enables a canonical identification of the adjoint and coadjoint representations. Specifically, the map $S^\sharp: \mathfrak{g}\to \mathfrak{g}^*$, defined by $S^\sharp(x)(y)=S(x,y)$, intertwines $R$ and its dual $R^*$ via

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

and similarly for the adjoint and coadjoint actions: $$S^\sharp \circ \operatorname{ad}_x = \operatorname{ad}_x^* \circ S^\sharp$$ for all $x$ (Goncharov et al., 5 Aug 2025). This establishes an isomorphism between the representations, which is central in constructing Manin triples and doubles for Reynolds Lie bialgebras.

2. Matched Pairs and Manin Triples

Matched pairs of Reynolds Lie algebras generalize the notion of matched pairs for classical Lie algebras by including compatible Reynolds operators. Given two Reynolds Lie algebras $(\mathfrak{g},R)$ and $(\mathfrak{g}',R')$ together with representation data $(\rho,\mu)$, compatibility is encoded in equations such as

$$\rho(Rx)(R\xi) = R\left(\rho(x)(R\xi) + \rho(Rx)(\xi) - \rho(Rx)(R\xi)\right)$$

for all $x\in\mathfrak{g}$, $\xi\in\mathfrak{g}'$. Manin triples for Reynolds Lie algebras are triples $(\mathfrak{d},R,S)$, with $\mathfrak{d} = \mathfrak{g}\oplus \mathfrak{g}’$—both Reynolds Lie subalgebras—each being isotropic with respect to $S$ (Goncharov et al., 5 Aug 2025).

There exists an equivalence between Manin triples and matched pairs in the Reynolds framework, mirroring the classical correspondence. Specifically, Reynolds Lie bialgebras are precisely those that arise from matched pairs $(\mathfrak{g},R)$ and $(\mathfrak{g}^*,-R^*)$ with compatibility in both Lie and Reynolds structures.

3. Reynolds Operators on Quadratic Rota–Baxter Lie Algebras

Rota–Baxter Lie algebras $(\mathfrak{g}, B)$ generalize Lie algebras by allowing operators $B$ satisfying

$[Bx,By]=B([Bx,y]+[x,By]+\lambda[x,y])$

for $\lambda$ fixed. In the quadratic setting, one also imposes

$S(x,By) + S(Bx,y) + \lambda S(x,y) = 0$

for a symmetric invariant form $S$. If a Reynolds operator $R$ commutes with $B$ and satisfies $B\circ R^* = -R\circ B$, then $R$ naturally induces a Reynolds Lie bialgebra (Goncharov et al., 5 Aug 2025). The integrability and deformation properties of such algebras yield explicit constructions of r-matrices solving the CYBE (see Section 4).

4. Classical Yang–Baxter Equation in Reynolds Context

The classical Yang–Baxter equation (CYBE) in a Reynolds Lie algebra requires that a skew-symmetric element $r\in\mathfrak{g}\otimes\mathfrak{g}$ satisfy both

$$[[r,r]] = [r_{12}, r_{13}] + [r_{13}, r_{23}] + [r_{12}, r_{23}] = 0$$

and

$(R \otimes \text{Id} + \text{Id} \otimes R)(r) = 0$

This dual condition ensures compatibility with both the Lie algebraic and Reynolds structures, so $r$ yields a valid Reynolds Lie bialgebra structure. The quadratic form $S$ induces the proper identification between the coalgebra structure and the adjoint/coadjoint actions (Goncharov et al., 5 Aug 2025).

5. Relative Rota–Baxter Operators and Reynolds Pre-Lie Algebras

Relative Rota–Baxter operators generalize Reynolds operators in representations $(W,T,\rho)$ of a Reynolds Lie algebra. A map $K:W\to \mathfrak{g}$ is said to be a relative Rota–Baxter operator if

$[Ku, Kv] = K(\rho(Ku)v - \rho(Kv)u)$

and $R\circ K = K\circ T$. This construction produces induced Reynolds Lie algebra structures on $W$, and solutions to the CYBE via appropriate skew-symmetrizations of $K$ (Goncharov et al., 5 Aug 2025).

Reynolds pre–Lie algebras $(\mathfrak{g},\{\cdot,\cdot\},R)$ possess a binary product $\{\cdot,\cdot\}$ satisfying a pre–Lie identity and an associated Reynolds operator. Their sub-adjacent Lie algebras inherit compatible Reynolds structures, further enriching the field of operadic Lie bialgebra generalizations.

6. Cohomology Theory for Reynolds Lie Bialgebras

Recent advances establish a cohomological framework for Reynolds Lie algebras and bialgebras, including derivations and extensions. Cohomology theories classify deformations, abelian extensions, and rigidity properties, with second cohomology groups $H^2_{RLY}$ and $H^2_{RLieDer}$ parameterizing infinitesimal deformations and extension classes (Teng et al., 6 Mar 2024Imed et al., 23 Apr 2025).

Cohomology is computed using complexes that merge classical Chevalley–Eilenberg cohomology with additional differentials dictated by the Reynolds operator. Extension theory for Reynolds LieDer pairs is also governed by explicit obstruction cocycles, whose vanishing characterizes extensibility of derivations and operator structures.

7. Connections to General Bialgebra Theory and Applications

Reynolds Lie bialgebras provide an operator-theoretic generalization applicable across integrable systems, quantization, and higher gauge theories. They admit broad categorical generalizations, including Reynolds n–Lie algebras, NS–Lie algebras, and triple systems, all benefiting from the operator splitting approach.

Solutions to the CYBE constructed from Reynolds or relative Rota–Baxter operators yield new bialgebra structures, integrable models, and quantum group deformations. Moreover, the interplay of quadratic forms, matched pair constructions, and operator identities establishes a flexible paradigm for further algebraic and geometric applications.

Table: Reynolds Lie Bialgebra Structures (from (Goncharov et al., 5 Aug 2025))

Structure Type	Defining Feature	Compatibility Condition
Quadratic Reynolds Lie Algebra	$R:\mathfrak{g}\to\mathfrak{g}$, $S$ invariant	$S(Rx,y)+S(x,Ry)=0$
Matched Pair of Reynolds LAs	$(\mathfrak{g},R),(\mathfrak{g}',R')$, representations $\rho$	$\rho(Rx)(R\xi)=R(\cdots)$
Manin Triple of Reynolds LAs	$\mathfrak{d}=\mathfrak{g}\oplus\mathfrak{g}'$, $S$ isotropic	$R$ on each subalgebra, $S$-paired
Reynolds Operator on Rota–Baxter	$B,R$ operators, quadratic $S$	$R\circ B = B\circ R$
CYBE Solution	$r\in\mathfrak{g}\otimes\mathfrak{g}$, $[[r,r]]=0$	$(R\otimes +\otimes R)(r)=0$

The Reynolds Lie bialgebra theory systematically generalizes Lie bialgebra theory, providing operator-driven splittings and a powerful formalism for compatibility, quantization, and integrable systems.