Frölicher–Nijenhuis Bracket Overview

• The Frölicher–Nijenhuis bracket is a graded Lie bracket on vector-valued forms, characterized by graded skew-symmetry and a Jacobi identity.
• It generalizes classical Lie and Schouten–Nijenhuis brackets, unifying derivations of the de Rham algebra and encoding integrability conditions in complex and special holonomy geometries.
• Widely applicable in smooth manifolds, Lie algebroids, groupoids, and operadic settings, it offers essential tools for deformation theory and the analysis of integrable PDEs.

The Frölicher–Nijenhuis bracket is a bilinear graded Lie bracket on the space of vector-valued differential forms on manifolds and related geometric or algebraic objects. It generalizes the Lie bracket of vector fields and the Schouten–Nijenhuis bracket of multivectors, simultaneously encoding infinitesimal symmetries, the theory of derivations of the de Rham algebra, and compatibility among geometric structures such as almost complex or special holonomy tensors. The bracket admits rigorous formulations in smooth, algebraic, Lie algebroid, groupoid, and operadic contexts, and has become fundamental in secondary calculus, integrability theory, and the cohomology of geometric structures.

1. Formal Definition and Algebraic Structure

Let $M$ be a smooth manifold. The space of $TM$-valued k-forms is $$\Omega^k(M, TM) = \Gamma(\Lambda^k T^*M \otimes TM)$$. For $K \in \Omega^p(M, TM)$, define the contraction $i_K : \Omega^q(M) \to \Omega^{q+p-1}(M)$ and the Nijenhuis–Lie derivative $\mathcal{L}_K := [i_K, d]$, a graded derivation of degree $p$.

The Frölicher–Nijenhuis bracket is the unique bilinear map

$$[\cdot, \cdot]_{FN} : \Omega^p(M, TM) \times \Omega^q(M, TM) \to \Omega^{p+q}(M, TM)$$

satisfying: $$[\mathcal{L}_K, \mathcal{L}_L] = \mathcal{L}_{[K, L]_{FN}}.$$ Expanding in local frames yields (for $K = \alpha \otimes X, L = \beta \otimes Y$): $$\begin{aligned} [K, L]_{FN} =\; &\alpha \wedge \beta \otimes [X, Y] + \alpha \wedge \mathcal{L}_X \beta \otimes Y - (-1)^{pq} \beta \wedge \mathcal{L}_Y \alpha \otimes X \ &+ (-1)^p d\alpha \wedge (i_X \beta) \otimes Y + (-1)^p (i_Y \alpha) \wedge d\beta \otimes X. \end{aligned}$$ This bracket is graded skew-symmetric: $[K, L]_{FN} = -(-1)^{pq} [L, K]_{FN},$ and satisfies the graded Jacobi identity

$$\sum_{\mathrm{cycl}} (-1)^{p_1 p_3} [K_1, [K_2, K_3]_{FN}]_{FN} = 0$$

for $K_i \in \Omega^{p_i}(M, TM)$ (Kosmann-Schwarzbach, 2021).

The same construction generalizes to microlinear Frölicher spaces (Nishimura, 2011), Lie algebroids (Nicola et al., 2014), algebraic modules (Krasil'shchik, 2008), Hom-Lie algebras (Baishya et al., 2024), nonsymmetric operads (Baishya et al., 4 May 2025), and groupoids (Bursztyn et al., 2017).

2. Geometric and Algebraic Interpretation

In the algebraic language, for a commutative algebra $A$, the bracket is defined on the module of vector-valued forms $D_1(\Lambda^p(A))$ by

$$[\Omega, \Omega']_{FN} = \text{unique element such that } [L_\Omega, L_{\Omega'}] = L_{[\Omega, \Omega']_{FN}}$$

where $L_\Omega = [d, i_\Omega]$ and $i_\Omega$ is the contraction operator (Krasil'shchik, 2008).

The FN-bracket unifies the classical Lie bracket (on vector fields), the Nijenhuis torsion (for endomorphism-valued 1-forms), and the Schouten–Nijenhuis bracket (on multivectors) (Kosmann-Schwarzbach, 2021, Bursztyn et al., 2017). On vector-valued forms of degree 1, it recovers the Nijenhuis torsion $N_J = [J, J]_{FN}/2$. For multivectors, it agrees (up to degree shift) with the Schouten–Nijenhuis bracket.

On groupoids, multiplicative vector-valued forms are closed under the FN-bracket, yielding a graded Lie subalgebra (Bursztyn et al., 2017). On Lie algebroids, the FN-bracket is realized via covariant Lie derivatives associated to flat, torsion-free connections: for $\phi \in \Omega^k(A, A), \psi \in \Omega^l(A, A)$,

$$[\phi, \psi]_{FN} = L_\phi \psi - (-1)^{kl} L_\psi \phi,$$

with $L_\phi$ the covariant Lie derivative (Nicola et al., 2014).

3. Cohomological and Deformation-Theoretical Applications

The integrability of geometric structures is encoded by Maurer–Cartan elements for the FN-bracket. For an almost complex structure $J$, integrability is $[J, J]_{FN} = 0$. More generally, for any $N\in D_1(\Lambda^1(A))$ with $[N, N]_{FN} = 0$, the operator $d_{FN} := [N, \cdot]_{FN}$ squares to zero, yielding the Frölicher–Nijenhuis cohomology (Krasil'shchik, 2008).

This construction applies to complex, Poisson, $G$-structure, and special holonomy geometries (Kawai et al., 2018, Kawai et al., 2016) and to bi-differential graded algebras (Müller-Hoissen, 2024). Deformation theory of Hom-Lie morphisms and operadic Nijenhuis or Rota–Baxter operators is controlled by the FN-bracket in Hom-Lie and operadic settings (Baishya et al., 2024, Baishya et al., 4 May 2025).

Key examples:

• In Kähler geometry, $d_\Psi := \mathcal{L}_{\partial_g \Psi}$ and $\mathrm{ad}_{\hat\Psi} := [\hat\Psi, -]^{FN}$ yield Dolbeault and $d^c$ cohomology (Kawai et al., 2018).
• On a $G_2$ or $\mathrm{Spin}(7)$ manifold, torsion-freeness is equivalent to vanishing of explicit FN-brackets of defining forms (Kawai et al., 2016):
  • $G_2$: $[\operatorname{Cr}, X]_{FN} = 0$ or $[X, X]_{FN} = 0$
  • $\mathrm{Spin}(7)$: $[P, P]_{FN} = 0$

4. Connections to Integrability and PDEs

The FN-bracket provides a natural language for integrability in differential systems. If two (1,1)-tensor fields $N_1, N_2$ satisfy pairwise vanishing FN-brackets, the algebra of differential forms becomes a bi-differential graded algebra $(\Omega^*(M), d_{N_1}, d_{N_2})$, with $d_{N_i}^2 = 0$, $d_{N_1}d_{N_2} + d_{N_2}d_{N_1} = 0$ (Müller-Hoissen, 2024). This underlies the geometry of integrable PDEs, including chiral models and self-dual Yang–Mills hierarchies, as well as their Darboux–dressing transformations.

In matrix settings, this leads to zero-curvature conditions and new classes of non-autonomous integrable matrix PDEs, in all of which the FN-bracket structures the commutation relations of the generalized differentials and their curvature conditions (Müller-Hoissen, 2024).

5. Lie Algebroid, Groupoid, and Operator-Theoretic Extensions

For a Lie algebroid $A\to M$, vector-valued forms $\Omega^k(A, A)$ admit a FN-bracket using flat, torsion-free connections. The bracket yields compatibility conditions for Nijenhuis operators, deformations, and generalizations to structures such as Poisson–Nijenhuis and Dirac–Nijenhuis (Nicola et al., 2014).

On Lie groupoids, multiplicative $TG$-valued forms close under the FN-bracket, with immediate applications to complex and holomorphic groupoids, principal connections, and their curvature. In principal bundle geometry, the FN-bracket provides an extension of the gauge algebra, unifying the Cartan calculus, extended “field-dependent” Lie brackets on gauge parameters, and the Lie–algebroid structure (François, 2023).

Operator-theoretic generalizations (e.g., in nonsymmetric operads (Baishya et al., 4 May 2025) or Hom-Lie algebras (Baishya et al., 2024)) formulate the FN-bracket through cup products, contractions, and semidirect products, identifying Maurer–Cartan elements with Nijenhuis or Rota–Baxter operators and constructing new graded Lie algebraic structures relevant for deformation and cohomology theories.

6. Special Holonomy, Derived Brackets, and Further Developments

In manifolds with special holonomy, the FN-bracket encodes not only integrability conditions, but also yields secondary cohomologies, $L_\infty$–algebras, and a derived-bracket formalism for associative or coassociative submanifolds (Kawai et al., 2018, Kawai et al., 2016). These constructions support the analysis of infinitesimal and higher obstructions to geometric deformations and bridge FN-bracket theory with homotopy algebras and derived geometry.

Fernandez–Gray and Fernandez classifications of $G_2$ and $\text{Spin}(7)$-structures are realized in terms of vanishing or non-vanishing projections of the FN-bracket, revealing new perspectives and computational tools for classifying special geometric structures (Kawai et al., 2016).

7. Summary Table: Essential Formulations

Context	FN-Bracket Definition	Graded Lie Algebra Property
Smooth manifolds	$$[\mathcal{L}_K,\mathcal{L}_L]=\mathcal{L}_{[K,L]_{FN}}$$	$[K,L]_{FN} = -(-1)^{pq}[L,K]_{FN}$
Algebraic (modules over $A$)	$$[L_\Omega, L_{\Omega'}] = L_{[\Omega, \Omega']_{FN}}$$	$$[\Omega,\Omega']_{FN} = -(-1)^{pq}[\Omega',\Omega]_{FN}$$
Lie algebroids	$$[\phi, \psi]_{FN} = L_\phi \psi - (-1)^{kl} L_\psi \phi$$	As above, using covariant Lie derivatives
Hom-Lie/Operad	$[P, Q]_{FN} = [P, Q]_c + \cdots$	As above, with cup product and contractions

This structure endows the collection of vector-valued forms (in each category) with a robust and flexible graded Lie algebra framework, fundamental for secondary calculus, deformation theory, integrability, and modern differential geometry (Krasil'shchik, 2008, Kosmann-Schwarzbach, 2021, Nishimura, 2011, Nicola et al., 2014, Bursztyn et al., 2017, Kawai et al., 2016, Kawai et al., 2018, François, 2023, Baishya et al., 2024, Baishya et al., 4 May 2025, Müller-Hoissen, 2024).