gl-Regular Nijenhuis Operator

• gl-Regular Nijenhuis Operator is a tensor with vanishing Nijenhuis torsion and a unique eigenvalue structure, ensuring each eigenvalue has geometric multiplicity one.
• Its algebraic regularity supports complete local normal forms and effective coordinate classification, enabling decomposition into invariant subbundles.
• This operator underpins integrable systems, geodesic equivalence, and Lie algebra structures by providing explicit symmetries, conservation laws, and classification results.

A gl-Regular Nijenhuis Operator is a linear (or, more generally, tensorial) object of central importance in Nijenhuis geometry, with rich implications in the theory of integrable systems, differential geometry, and the structure theory of tensors and algebras. Its defining properties are governed by two joint conditions: (1) vanishing Nijenhuis (Frolicher–Nijenhuis) torsion, and (2) an algebraic regularity condition where each eigenvalue has geometric multiplicity one (i.e., the minimal polynomial and characteristic polynomial coincide, yielding a maximal-dimensional $\GL(n)$-orbit). This gl-regularity ensures highly structured local normal forms, effective coordinate classifications, and an abundance of symmetries and conservation laws in analytic settings.

1. Definition and Basic Properties

Let $L$ be a $(1,1)$-tensor field (operator) on a manifold $M^n$ (real or complex). The Nijenhuis torsion $N_L$ is defined for vector fields $X,Y$ by

$$N_L(X,Y) = [LX, LY] - L([LX, Y] + [X, LY]) + L^2[X,Y].$$

A Nijenhuis operator is a $(1,1)$-tensor $L$ for which $N_L \equiv 0$ identically. The operator is called gl-regular if, at every point $p \in M$ (or, for algebras, at the level of the vector space), the linear map $L(p)$ satisfies any of the following equivalent conditions:

• Each eigenvalue of $L(p)$ has geometric multiplicity one (a single Jordan block per eigenvalue).
• The set $\{\operatorname{Id}, L, L^2, ..., L^{n-1}\}$ forms a basis for the commutant (centralizer) of $L$ in $\GL(n)$.
• The minimal and characteristic polynomials of $L$ coincide; the operator has cyclic vector(s).
• The $\GL(n)$-orbit of $L$ has maximal dimension $n^2-n$.

The property of gl-regularity is local: it applies at single points but can be extended over analytic open sets wherever these conditions persist. In Lie algebraic contexts, algebraic Nijenhuis operators $N$ are gl-regular if they are diagonalizable with simple (distinct) spectrum.

2. Local Classification and Normal Forms

At gl-regular (algebraically generic) points, a complete local normal form is available (Bolsinov et al., 2019, Akpan, 14 Mar 2025, Bolsinov et al., 2020, Antonov et al., 2023):

• Splitting Theorem: If the characteristic polynomial of $L$ splits into mutually coprime factors, then $TM$ splits as a direct sum of $L$-invariant, integrable subbundles, and $L$ is locally block-diagonal with each block corresponding to a single eigenvalue (real or complex pair).
• On each block, $L$ takes a block-Jordan normal form with a single Jordan block per eigenvalue. In local coordinates $(x^1, ..., x^n)$, $L$ is conjugated to a companion matrix:

$$L = \begin{pmatrix} f_1 & 1 & 0 & \dots & 0 \ f_2 & 0 & 1 & \dots & 0 \ \vdots & & \ddots & \ddots & \vdots \ f_{n-1} & 0 & \dots & 0 & 1 \ f_n & 0 & \dots & 0 & 0 \ \end{pmatrix}$$

with analytic conditions on the $f_i$ ensuring vanishing torsion.

• In the strictly gl-regular (differentially nondegenerate) case, the elementary symmetric polynomials $\sigma_i$ of the eigenvalues serve as coordinate charts, and the normal form is determined uniquely up to coordinate changes (Akpan, 14 Mar 2025).
• At singularities (e.g., Morse-type singularities of the determinant or eigenvalue collision), locally rigid models arise, fully classified in small dimensions (Akpan, 14 Mar 2025, Bolsinov et al., 2020, Antonov et al., 2023).

3. Symmetry Algebra and Conservation Laws

gl-Regular Nijenhuis operators possess rich symmetry and conservation law structures (Bolsinov et al., 2023):

• The commutant (symmetry algebra) $\mathcal{S}(L)$ of $L$ is finite-dimensional, commutative, associative under pointwise multiplication, and every element is Nijenhuis and a strong symmetry of every other symmetry.
• Every conservation law (closed $1$-form $\omega$ such that $d(L^*\omega)=0$) arises from the action of a symmetry on a fixed conservation law.
• At algebraically generic points, all symmetries and conservation laws are parameterized by exactly $n$ analytic functions of one variable, with explicit representatives in local normal forms.
• The splitting theorem applies: symmetries and conservation laws decompose according to invariant subbundles determined by the factorization of the characteristic polynomial.

4. Applications: Integrable Systems and Geodesic Equivalence

gl-Regular Nijenhuis operators play an essential role in the construction and analysis of integrable systems (Bolsinov et al., 2023):

• Given a gl-regular Nijenhuis operator $L$, there is a large space of (pseudo-)Riemannian metrics $g$ geodesically compatible with $L$, classified by the so-called Sinjukov–Topalov hierarchy or, more generally, by strong symmetries of $L$.
• The associated integrable quasilinear PDE system ("hydrodynamic type") has flows

$$\frac{\partial U}{\partial t_i} = A_i(U) \frac{\partial U}{\partial x},$$

where $A_i$ are recursively built from $L$ via the characteristic polynomial. These flows are integrable in quadratures even in non-diagonalizable cases (Bolsinov et al., 2023).

• The reduction of such systems onto spaces of compatible geodesics yields finite-dimensional completely integrable Hamiltonian systems, with explicit invariants and commuting flows generated by quadratic integrals constructed from $L$ and compatible $g$.

5. gl-Regular Nijenhuis Operators in Lie Algebras

For finite-dimensional Lie algebras, particularly in dimension three, a complete classification has been achieved (Sergeevna, 14 Oct 2024):

• A linear operator $N$ is an algebraic Nijenhuis operator (on a Lie algebra $\mathfrak{g}$) if $$T_N(X, Y) = [NX, NY] - N([NX, Y] + [X, NY]) + N^2[X,Y]$$ vanishes on $\mathfrak{g}$.
• gl-regularity here requires $N$ diagonalizable with distinct eigenvalues over the ground field.
• Only specific Bianchi types admit regular semisimple (gl-regular) algebraic Nijenhuis operators:
  • Real case: $\mathfrak{sl}(2,\mathbb{R})$, $\mathrm{aff}(\mathbb{R})$, and certain solvable types $r_{3,b}$;
  • Complex case: $\mathfrak{sl}(2,\mathbb{C})$, $\mathrm{aff}(\mathbb{C})$, and $r_{3,a}$ for $a\ne0$.
• In each admitted case, explicit Nijenhuis eigenbases and diagonal forms are constructed, with uniqueness and classification up to automorphisms detailed (Sergeevna, 14 Oct 2024).

Algebraic Setting	Admits gl-Regular $N$?	Eigenbasis and Diagonalization
$$\mathfrak{sl}(2,\mathbb{R}), \mathfrak{sl}(2,\mathbb{C})$$	Yes	Yes, all such $N$ equivalent under automorphism
$\mathrm{aff}(\mathbb{R}), \mathrm{aff}(\mathbb{C})$	Yes	Continuous family of inequivalent eigenbases
Solvable $r_{3,b}, r_{3,a}$	Yes ($b\neq0, a\neq0$)	Explicit eigenbases, parameterized spectrum
Others	No (real), sometimes yes (complexified)	N/A

6. gl-Regular Nijenhuis Operators with Unity and $F$-Manifolds

When additional structures are imposed, such as a cyclic vector field $e$ satisfying $\mathcal{L}_e L = \operatorname{Id}$, a richer geometric structure emerges (Antonov et al., 2023):

• The splitting theorem adapts to the presence of a unity, reducing the study to indecomposable cases (single real or complex-conjugate eigenvalue blocks).
• Canonical upper-triangular Toeplitz forms for $(L, e)$ exist in the gl-regular case, providing an explicit link to regular $F$-manifolds and commutative, associative multiplications.
• Complete semi-normal form classifications exist for dimensions two and three, specifying all singularity types and functional, integer moduli arising in the analytic category.

7. Dimensional Constraints, Singularity Theory, and Global Aspects

gl-Regularity imposes sharp constraints on possible global structures and singularities:

• On surfaces: only tori, Klein bottles, and certain complex surfaces admit nontrivial gl-regular Nijenhuis operators; e.g., on $S^2$ all such operators are constant-coefficient and trivial (Bolsinov et al., 2020).
• At singularities, companion-plus-twist and Morse-type normal forms fully classify local models, with rigidity results excluding nontrivial functional moduli except in special low-dimensional cases (Akpan, 14 Mar 2025).
• In the context of left-symmetric algebras (LSA), gl-regularity corresponds to the functional independence of characteristic polynomial coefficients, leading to explicit classification of all such LSA structures in dimension three (Scapucci, 17 Apr 2024).

• Zhikhareva, "Three-dimensional Lie algebras admitting regular semisimple algebraic Nijenhuis operators" (Sergeevna, 14 Oct 2024)
• Bolsinov, Konyaev, Matveev, various works, including (Bolsinov et al., 2019, Bolsinov et al., 2020, Bolsinov et al., 2023, Bolsinov et al., 2023, Akpan, 14 Mar 2025)
• Scapucci, "Classification of differentially non-degenerate left-symmetric algebras in dimension 3" (Scapucci, 17 Apr 2024)
• Antonov, Konyaevt, "Nijenhuis operators with a unity and F-manifolds" (Antonov et al., 2023)