Manin Triples in Nijenhuis BiHom–Lie Algebras

• Manin triples of Nijenhuis BiHom–Lie algebras are advanced structures that unify Lie bialgebras, matched pairs, and Nijenhuis operators within a generalized algebraic paradigm.
• Key methodologies include the use of Yau twists and canonical double constructions that establish equivalences among BiHom bialgebras, matched pairs, and Manin triples.
• This framework facilitates practical applications in integrable systems and Poisson geometry, providing new avenues for exploring symmetry and deformation theories.

A Manin triple of Nijenhuis BiHom–Lie algebras is a structure that unifies and generalizes the concepts of Lie bialgebras, Manin triples, matched pairs, and Nijenhuis operators in the context of BiHom–Lie algebras. This advanced framework, developed in "Nijenhuis BiHom-Lie bialgebras and differential Lie bialgebras," establishes precise equivalences between these objects and provides a foundation for further study of integrable systems, Poisson geometry, and generalized symmetries within the BiHom-algebraic paradigm (Liu et al., 20 Jan 2026).

1. BiHom–Lie Algebras and Nijenhuis Operators

A BiHom–Lie algebra is a quadruple $(L,\,[\,,\,],\alpha,\beta)$ where $L$ is a $\mathbf k$–vector space, $[\,,\,]\colon L\otimes L\to L$ is a bilinear bracket, and $\alpha,\beta\colon L\to L$ are commuting linear maps. These satisfy:

• Multiplicativity: $\alpha([x,y])\;=\;[\alpha(x),\alpha(y)]$, $\beta([x,y])\;=\;[\beta(x),\beta(y)]$.
• BiHom–antisymmetry: $[\beta(x),\alpha(y)] = -[\beta(y),\alpha(x)]$.
• BiHom–Jacobi identity: $$[\beta^2(x),[\beta(y),\alpha(z)]] + [\beta^2(y),[\beta(z),\alpha(x)]] + [\beta^2(z),[\beta(x),\alpha(y)]] = 0$$ for all $x,y,z\in L$.

If $\alpha^2 = \beta^2 = \text{Id}_L$, the algebra is called involutive.

A Nijenhuis operator $N:L\to L$ commutes with $\alpha,\beta$ and satisfies:

$$[N(x),N(y)] = N\big([N(x),y] + [x,N(y)] - N([x,y])\big).$$

Equivalently:

$$[N(x),N(y)] + N^2([x,y]) - N([N(x),y] + [x,N(y)]) = 0.$$

2. Definition of Manin Triples for Nijenhuis BiHom–Lie Algebras

Given involutive Nijenhuis BiHom–Lie algebras $(L,[\,,],N,\alpha,\beta)$, $(L^*,[\,,]_*,S^*,\alpha^*,\beta^*)$, dual via $\langle\cdot,\cdot\rangle:L^*\times L\to \mathbf k$, define $D = L \oplus L^*$ with structure maps $\alpha_D = \alpha \oplus \alpha^*$, $\beta_D = \beta \oplus \beta^*$, and total Nijenhuis map $N_D = N \oplus S^*$. There is a nondegenerate symmetric bilinear form:

$$\mathcal B_D(x+a^*,y+b^*) = \langle a^*,y\rangle + \langle b^*,x\rangle.$$

The bracket is given by the standard double formula: $\begin{split} [x+a^*,y+b^*]_D =\; &[x,y] + \ad^*(x)(b^*) - \ad^*(y)(a^*) \ &+ [a^*,b^*]_* + \mathfrak{ad}^*(a^*)(y) - \mathfrak{ad}^*(b^*)(x) \end{split}$ where $\ad^*$ and $\mathfrak{ad}^*$ are the coadjoint actions. The data

$$\bigl(D=L\oplus L^*,\,[\,,]_D,\,N_D,\,\alpha_D,\,\beta_D\bigr),\quad L,L^*\subset D$$

form a Manin triple of Nijenhuis BiHom–Lie algebras if:

• $D$ is a Nijenhuis BiHom–Lie algebra,
• $\mathcal B_D$ is invariant and $L,L^*$ are isotropic,
• $L, L^*$ are closed under $N_D$.

3. Equivalence Theorem and Canonical Constructions

A foundational result is the three-way equivalence for finite-dimensional $L$: (i) Manin triple of Nijenhuis BiHom–Lie algebras, (ii) Nijenhuis BiHom–Lie bialgebra, (iii) matched pair of Nijenhuis BiHom–Lie algebras.

Explicitly,

• A Nijenhuis BiHom–Lie bialgebra consists of $(L,[\,,],\Delta,N,S,\alpha,\beta)$ where $S$ is adjoint–admissible to $S^*$ and $N,S$ satisfy certain compatibility (co-Nijenhuis, cocycle, etc.).
• The cobracket $\Delta = ([\,,]_*)^*: L\to L\otimes L$ is the dual of $[\,,]_*$.
• A matched pair involves $(L,L^*,N,S^*,\mathfrak{ad}^*,\ad^*)$ such that $L$ and $L^*$ act on each other compatibly with BiHom–representations and Nijenhuis structures.

Standard constructions effect these correspondences:

• (ii)$\Rightarrow$(i): The double $D=L\oplus L^*$, bracket and form as specified, $N$ extended to $N_D=N\oplus S^*$.
• (ii)$\Leftrightarrow$(iii): $\Delta$ is the coadjoint of $[\,,]_*$, with actions defined by the coadjoint representations.

4. Structural Properties and Proof Summary

The Yau–twist technique demonstrates any BiHom–Lie (bi)algebra arises as a twist of an ordinary Lie (bi)algebra by commuting automorphisms. Classical results (Drinfeld–Semenov-Tian-Shansky) yield

$$(\text{Manin triple}) \Longleftrightarrow (\text{Lie bialgebra}) \Longleftrightarrow (\text{matched pair})$$

which, upon twisting, extend exactly to the BiHom and Nijenhuis setting.

Structural properties:

• Invariant form: $\mathcal B_D$ is preserved by $N_D$ because $S$ is the adjoint of $N$ relative to a nondegenerate invariant form.
• Isotropy: $L \perp L^*$ with respect to $\mathcal B_D$.
• Closure: $L$ and $L^*$ are stable under $N_D$.
• The verification involves detailed component-wise bracket computations.

5. Examples, Reductions, and Explicit Cases

Several corollaries and explicit constructions illustrate the theory:

• If $\alpha=\beta=\text{Id}$, the structure reduces to the classical Manin triples of Nijenhuis Lie algebras (Corollary 2.11).
• In Section 2.4, a two-dimensional involutive BiHom–Lie algebra $(A,[\,,],\alpha,\beta)$, with an explicit Nijenhuis operator $N:A\to A$, is worked out fully; its dual $A^*$ and resulting 4-dimensional Manin triple $A\oplus A^*$ exemplify the theory.
• Proposition 2.7: If $L$ admits a nondegenerate invariant form and $N$ preserves it up to an adjoint $S$, then $S$ satisfies the co-Nijenhuis condition.
• These results verify that every Nijenhuis BiHom–Lie bialgebra (or matched pair) yields a canonical Manin triple double, and conversely, every Manin triple splits into a pair of dual Nijenhuis BiHom–Lie bialgebras.

6. Impact and Further Directions

The Manin triple framework for Nijenhuis BiHom–Lie algebras organizes bialgebraic, matched-pair, and double constructions within the generalized BiHom–Lie paradigm. The results generalize classical Lie and Nijenhuis geometric structures, enabling systematic study of symmetry, reductions, and bialgebraic deformations in new algebraic settings. This approach connects with integrability, noncommutative geometry, and the theory of differential Lie bialgebras, suggesting multiple avenues for further exploration in both mathematical physics and pure algebra (Liu et al., 20 Jan 2026).