# Generalized Nijenhuis Torsions and block-diagonalization of operator fields

Daniel Reyes Nozaleda  Departamento de Física Teórica, Facultad de Ciencias
Físicas, Universidad Complutense de Madrid, 28040 – Madrid, Spain
and Instituto de Ciencias Matemáticas, C/ Nicolás Cabrera, No 13–15, 28049
Madrid, Spain<EMAIL_ADDRESS><EMAIL_ADDRESS>,  Piergiulio Tempesta
Departamento de Física Teórica, Facultad de Ciencias Físicas, Universidad
Complutense de Madrid, 28040 – Madrid, Spain
and Instituto de Ciencias Matemáticas, C/ Nicolás Cabrera, No 13–15, 28049
Madrid, Spain<EMAIL_ADDRESS><EMAIL_ADDRESS>and  Giorgio
Tondo  Dipartimento di Matematica e Geoscienze, Università degli Studi di
Trieste, piaz.le Europa 1, I–34127 Trieste, Italy<EMAIL_ADDRESS>
(Date: May 15, 2022)

###### Abstract.

The theory of generalized Nijenhuis torsions, which extends the classical
notions due to Nijenhuis and Haantjes, offers new tools for the study of
normal forms of operator fields. We propose a general result ensuring that,
given a family of commuting operator fields whose generalized Nijenhuis
torsion of level $l$ vanishes, there exists a local chart where all operators
can be simultaneously block-diagonalized. We also introduce the notion of
generalized Haantjes algebra, consisting of operators with a vanishing higher-
level torsion, as a new algebraic structure naturally generalizing standard
Haantjes algebras.

###### 2010 Mathematics Subject Classification:

MSC: 53A45, 58C40, 58A30.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries on the Nijenhuis and Haantjes geometry
  3. 3 The generalized Nijenhuis tensors and block-diagonalization
    1. 3.1 Eigen-distributions and spectral properties of generalized Nijenhuis operators
    2. 3.2 Block-diagonalization
  4. 4 Generalized Haantjes algebras
    1. 4.1 Definitions
    2. 4.2 Cyclic generalized Haantjes algebras
  5. 5 Generalized Nijenhuis torsions and simultaneous block-diagonalization
  6. 6 Block-diagonalization of generalized Haantjes algebras
    1. 6.1 A level-three generalized Haantjes algebra
    2. 6.2 Block-diagonalization
  7. 7 A level-four generalized Haantjes algebra
    1. 7.1 Spectral analysis
    2. 7.2 Block-diagonalization

##  1\. Introduction

The purpose of this article is to establish a new geometric setting for the
block-diagonalization of operator fields on a differentiable manifold.

The problem of finding normal forms of (1,1)-tensor fields on a differentiable
manifold $M$ is relevant in many theoretical and applicative contexts. Indeed,
it is useful in the determination of a reduced form for a system of partial
differential equations [1, 3, 4] and in the theory of hydrodynamic-type
systems [8, 9]. Also, it finds applications in classical mechanics, since it
may allow to construct separation variables for completely integrable
Hamiltonian systems [6].

In this article, we address this problem with a tensorial approach, based on
the notion of Nijenhuis and Haantjes tensors and their generalizations,
recently introduced in [22].

The interest in the geometry of Nijenhuis and Haantjes tensors recently has
considerably increased. Indeed, new applications have been found in many
different contexts: for instance, the characterization of integrable chains of
partial differential equations of hydrodynamic type (see e.g. [8], [9]) and
their integrable reductions [10], as well as the study of topological field
theories [14]. Also, Nijenhuis geometry, which plays a crucial role in the
theory of almost complex structures [15], has been extended to contexts as the
study of analytic matrix functions, linearisation theory, operator algebras
[5], etc.

In [20], two of the present authors proposed the notion of Haantjes algebra.
It has been shown that for an Abelian algebra of semisimple operator fields
with vanishing Haantjes torsion there exist local coordinate charts where the
simultaneous diagonalization of all the operators of the algebra is allowed.
In [21], [19], the notion of $\omega\mathscr{H}$ manifolds (namely symplectic
manifolds endowed with an algebra of Haantjes operators), has been proposed as
a natural setting for the formulation of the theory of finite-dimensional
integrable Hamiltonian systems. The class of $P\mathscr{H}$ manifolds has been
introduced in [24].

A new, infinite family of higher-order torsions of level $m$, generalizing
both the Nijenhuis and the Haantjes torsions, have been defined and discussed
in [22]. These torsions can also be derived from a family of higher-order
Haantjes brackets, which generalize the Frölicher-Nijenhuis (FN) one. As the
FN bracket is directly related with the Nijenhuis torsion, so the higher
Haantjes brackets are related with our higher-order torsions. Operator fields
with a vanishing generalized torsion will be said to be generalized Nijenhuis
operators.

We remind that a different formulation of the theory of generalized Nijenhuis
torsions has been proposed in [13].

Besides, a generalization of the classical Haantjes theorem was proposed.
Precisely, it has been proved that, given an operator field
$\boldsymbol{A}:\mathfrak{X}(M)\rightarrow\mathfrak{X}(M)$, the vanishing of
any generalized Nijenhuis torsion is sufficient to guarantee the mutual
integrability of the eigen-distributions of $\boldsymbol{A}$, namely the
integrability of each of them and of all of their direct sums.

From a theoretical point of view, our main result is a theorem stating that
for a set of commuting operators on a differentiable manifold, if at least one
of them is a generalized Nijenhuis operator, then there exists a local
coordinate chart where they take simultaneously a block-diagonal form.

This theorem is further refined by requiring that all of the commuting
operators are generalized Nijenhuis operators of a (common) level $m$. In this
case, on a local coordinate chart, all of them take a block-diagonal form with
finer blocks.

These results naturally suggest the notion of generalized Haantjes algebra. It
consists of a set of generalized Nijenhuis operators forming a $C^{\infty}$
module; besides, this set is closed under the product of operators.

We will discuss in detail the properties of the class of cyclic generalized
Haantjes algebras. They are generated by the independent powers of a given
generalized Nijenhuis tensor. We shall also prove that any polynomial
$P(\boldsymbol{A})$ in an operator $\boldsymbol{A}$ such that
$\mathcal{\tau}^{(n)}_{\boldsymbol{A}}(X,Y)=0$, whose coefficients are
functions in $C^{\infty}(M)$, satisfies
$\mathcal{\tau}^{(n)}_{P(\boldsymbol{A})}(X,Y)=0$. This property allows us to
construct generalized Haantjes algebras in a very natural and direct way. We
shall present some nontrivial examples of cyclic generalized Haantjes algebras
of level three and four. In particular, we will show how to construct local
coordinate charts allowing the simultaneous block-diagonalization of all the
operators forming these algebras.

The approach proposed in this work offers a new perspective concerning the
classical problem of the normal form of operator fields. We remind that a
relevant contribution to this problem, in the spirit of tensor analysis, was
given by Bogoyavlenskij in [3, 4]. In particular, necessary and sufficient
conditions were proposed for the existence of local charts were a given
operator acquires a block-diagonal form. The main difference with respect to
those results is that we solve the problem for a full family of commuting
operators. At the same time, no knowledge a priori of the eigen-distributions
of the given operators is required. This aspect is crucial, since the spectral
analysis of operator fields becomes rapidly intractable by increasing the
dimension of the underlying differentiable manifold. Thus, we offer sufficient
tensor conditions for the simultaneous block-diagonalization of a family of
operators, which can be easily checked with the aid of computer algebra,
without entering the construction of their eigen-distributions. Of course, for
the explicit construction of the local chart where the simultaneous block-
diagonalization of the family of operators considered takes place, it is
necessary to enter into their spectral analysis. However, our result can be
interpreted as a tensor test, ensuring a priori the block-diagonalizability of
the full family. Once this property is ascertained for a concrete family, then
one can proceed in the construction of the local chart.

A thorough study of the rich algebraic properties of the generalized Haantjes
algebras represents an interesting research perspective, that will be
developed in a forthcoming work.

The paper is organized as follows. In Section 2, the theory of Nijenhuis and
Haantjes tensors is briefly reviewed. In Section 3, the concept of generalized
Nijenhuis operator is discussed. In Section 4, the new notion of generalized
Haantjes algebra is presented. In Section 5, the main theorems of the work,
concerning simultaneous block-diagonalization of non-semisimple operator
fields, are proved. In Sections 6 and 7, examples of generalized Haantjes
algebras are discussed and local charts ensuring simultaneous block-
diagonalization are explicitly constructed.

##  2\. Preliminaries on the Nijenhuis and Haantjes geometry

In this section, we shall review some basic notions concerning the geometry of
Nijenhuis or Haantjes torsions, following the original papers [12, 16] and the
related ones [17, 11]. Here we shall focus only on the aspects of the theory
which are relevant for the subsequent discussion.

Let $M$ be a differentiable manifold of dimension $n$, $\mathfrak{X}(M)$ the
Lie algebra of vector fields on $M$ and
$\boldsymbol{A}:\mathfrak{X}(M)\rightarrow\mathfrak{X}(M)$ be a smooth
$(1,1)$-tensor field. In the following, all tensors will be considered to be
smooth.

######  Definition 1.

The Nijenhuis torsion of $\boldsymbol{A}$ is the the vector-valued $2$-form
defined by

(2.1)
$\tau_{\boldsymbol{A}}(X,Y):=\boldsymbol{A}^{2}[X,Y]+[\boldsymbol{A}X,\boldsymbol{A}Y]-\boldsymbol{A}\Big{(}[X,\boldsymbol{A}Y]+[\boldsymbol{A}X,Y]\Big{)},$

where $X,Y\in\mathfrak{X}(M)$ and $[\ ,\ ]$ denotes the commutator of two
vector fields.

######  Definition 2.

The Haantjes torsion associated with $\boldsymbol{A}$ is the vector-valued
$2$-form defined by

(2.2)
$\mathcal{H}_{\boldsymbol{A}}(X,Y):=\boldsymbol{A}^{2}\tau_{\boldsymbol{A}}(X,Y)+\tau_{\boldsymbol{A}}(\boldsymbol{A}X,\boldsymbol{A}Y)-\boldsymbol{A}\Big{(}\tau_{\boldsymbol{A}}(X,\boldsymbol{A}Y)+\tau_{\boldsymbol{A}}(\boldsymbol{A}X,Y)\Big{)}.$

The Haantjes (Nijenhuis) vanishing condition inspires the following
definition.

######  Definition 3.

A Haantjes (Nijenhuis) operator is a (1,1)-tensor field whose Haantjes
(Nijenhuis) torsion identically vanishes.

A simple, relevant case of Haantjes operator is that of a tensor field
$\boldsymbol{A}$ which takes a diagonal form in a local chart
$\boldsymbol{x}=(x^{1},\ldots,x^{n})$:

(2.3)
$\boldsymbol{A}(\boldsymbol{x})=\sum_{i=1}^{n}\lambda_{i}(\boldsymbol{x})\frac{\partial}{\partial
x^{i}}\otimes\mathrm{d}x^{i}\ ,$

where $\lambda_{i}(\boldsymbol{x}):=\lambda^{i}_{i}(\boldsymbol{x})$ are the
eigenvalue fields of $\boldsymbol{A}$ and $\left(\frac{\partial}{\partial
x^{1}},\ldots,\frac{\partial}{\partial x^{n}}\right)$ are the fields forming
the so called natural frame associated with the local chart
$(x^{1},\ldots,x^{n})$. As is well known, the Haantjes torsion of the diagonal
operator (2.3) vanishes.

We also recall that two frames $\\{X_{1},\ldots,X_{n}\\}$ and
$\\{Y_{1},\ldots,Y_{n}\\}$ are said to be equivalent if $n$ nowhere vanishing
smooth functions $f_{i}$ exist, such that

$X_{i}=f_{i}(\boldsymbol{x})Y_{i}\ ,\qquad\qquad i=1,\ldots,n\ .$

######  Definition 4.

An _integrable_ frame is a reference frame equivalent to a natural frame.

It is interesting to observe that the algebraic properties of Haantjes
operators are richer that those of Nijenhuis operators. Hereafter,
$\boldsymbol{I}:\mathfrak{X}(M)\rightarrow\mathfrak{X}(M)$ will denote the
identity operator. A useful result is the following.

######  Proposition 1.

[1]. Let $\boldsymbol{A}$ be a (1,1)-tensor field. The following identity
holds:

(2.4)
$\mathcal{H}_{f\boldsymbol{I}+g\boldsymbol{A}}(X,Y)=g^{4}\,\mathcal{H}_{\boldsymbol{A}}(X,Y),$

where $f,g:M\rightarrow\mathbb{R}$ are $C^{\infty}(M)$ functions.

###### Proof.

See Proposition 1, p. 255 of [1]. ∎

Interestingly enough, such a property does not hold in the case of a Nijenhuis
operator.

Many more examples of Haantjes operators, relevant in classical mechanics and
in Riemannian geometry, can be found for instance in the works [19]–[23].

##  3\. The generalized Nijenhuis tensors and block-diagonalization

In this section, for the sake of clarity, we shall briefly review some of the
main algebro-geometric properties of the new class of generalized Nijenhuis
tensors introduced in [20] and further studied in [22].

######  Definition 5.

Let $\boldsymbol{A}:\mathfrak{X}(M)\rightarrow\mathfrak{X}(M)$ be a
(1,1)-tensor field. The generalized Nijenhuis torsion of $\boldsymbol{A}$ of
level $m$, for each integer $m\geq 1$, is the skew-symmetric (1,2)-tensor
field defined by

$\displaystyle\mathcal{\tau}^{(m)}_{\boldsymbol{A}}(X,Y)=\boldsymbol{A}^{2}\mathcal{\tau}^{(m-1)}_{\boldsymbol{A}}(X,Y)+\mathcal{\tau}^{(m-1)}_{\boldsymbol{A}}(\boldsymbol{A}X,\boldsymbol{A}Y)-\boldsymbol{A}\Big{(}\mathcal{\tau}^{(m-1)}_{\boldsymbol{A}}(X,\boldsymbol{A}Y)+\mathcal{\tau}^{(m-1)}_{\boldsymbol{A}}(\boldsymbol{A}X,Y)\Big{)},$
$\displaystyle\quad X,Y\in\mathfrak{X}(M)\ .$ (3.1)

Here the notation $\tau_{\boldsymbol{A}}^{(0)}(X,Y):=[X,Y]$,
$\tau_{\boldsymbol{A}}^{(1)}(X,Y):=\tau_{\boldsymbol{A}}(X,Y)$ and
$\tau_{\boldsymbol{A}}^{(2)}(X,Y):=\mathcal{H}_{\boldsymbol{A}}(X,Y)$ is
adopted.

######  Definition 6.

Let $\boldsymbol{A}:\mathfrak{X}(M)\rightarrow\mathfrak{X}(M)$ be a
(1,1)-tensor field. If $\mathcal{\tau}^{(m)}_{\boldsymbol{A}}(X,Y)=0$ for some
$m\in\mathbb{N}\backslash\\{0\\}$, we shall say that $\boldsymbol{A}$ is a
generalized Nijenhuis operator of level $m$.

We recall a result which is crucial in the following analysis.

######  Theorem 1.

[22] Let $\boldsymbol{A}:\mathfrak{X}(M)\to\mathfrak{X}(M)$ be an operator.
Assume that

(3.2) $\tau^{(m)}_{\boldsymbol{A}}(X,Y)=\boldsymbol{0},\qquad
X,Y\in\mathfrak{X}(M)$

for some $m\geq 1$. Then, each eigen-distribution of $\boldsymbol{A}$, as well
as each direct sum of its eigen-distributions, is integrable.

###  3.1. Eigen-distributions and spectral properties of generalized Nijenhuis
operators

We shall recall now some of the spectral properties of non-semisimple
operators on a manifold. We shall focus on the case of generalized Nijenhuis
operators. Let us denote by
$Spec(\boldsymbol{A}):=\\{\lambda_{1}(\boldsymbol{x}),\lambda_{2}(\boldsymbol{x}),\ldots,\lambda_{s}(\boldsymbol{x})\\}$,
$\boldsymbol{x}\in M$, the set of the eigenvalues of an operator
$\boldsymbol{A}:\mathfrak{X}(M)\to\mathfrak{X}(M)$, which will always be
assumed to be pointwise distinct. We denote by

(3.3)
$\mathcal{D}_{i}(\boldsymbol{x})=\ker\Big{(}\boldsymbol{A}(\boldsymbol{x})-\lambda_{i}(\boldsymbol{x})\boldsymbol{I}\Big{)}^{\rho_{i}},\qquad
i=1,\ldots,s$

the i-th eigen-distribution of index $\rho_{i}\geq 1$, which includes all the
(generalized) eigenvectors corresponding to the eigenvalue $\lambda_{i}$. In
Eq. (3.3), $\rho_{i}$ stands for the Riesz index of $\lambda_{i}$, which is
the minimum integer such that

(3.4)
$\ker\Big{(}\boldsymbol{A}(\boldsymbol{x})-\lambda_{i}(\boldsymbol{x})\boldsymbol{I}\Big{)}^{\rho_{i}}\equiv\ker\Big{(}\boldsymbol{A}(\boldsymbol{x})-\lambda_{i}(\boldsymbol{x})\boldsymbol{I}\Big{)}^{\rho_{i}+1}\
.$

In the forthcoming considerations, we shall always suppose a regularity
condition, namely that the rank of the distributions and $\rho_{i}$ are
(locally) independent of $\boldsymbol{x}$. When $\rho_{i}=1$,
$\mathcal{D}_{i}$ is a proper eigen-distribution.

It is also useful, from an applicative point of view, to consider the explicit
action of a generalized Nijenhuis torsion on a pair of generalized
eigenvectors of $\boldsymbol{A}$. In [22] the following formula has been
proved by induction over the integers $m\geq 2$:

######  Proposition 2.

Let $\boldsymbol{A}$ be a (1,1)-tensor and $X_{\alpha}\in\mathcal{D}_{\mu}$,
$Y_{\beta}\in\mathcal{D}_{\nu}$ be two of its generalized eigenvectors
corresponding to the eigenvalues $\mu$, $\nu$ respectively. Then, for any
integer $m\geq 2$ the following formula holds:

(3.5)
$\mathcal{\tau}^{(m)}_{\boldsymbol{A}}(X_{\alpha},Y_{\beta})=\sum_{i,j=0}^{m}(-1)^{i+j}\binom{m}{i}\binom{m}{j}\Big{(}\boldsymbol{A}-\mu\mathbf{I}\Big{)}^{m-i}\Big{(}\boldsymbol{A}-\nu\mathbf{I}\Big{)}^{m-j}[X_{\alpha-i},Y_{\beta-j}]\
,$

where

(3.6) $\boldsymbol{A}X_{\alpha}=\mu
X_{\alpha}+X_{\alpha-1},\qquad\boldsymbol{A}Y_{\beta}=\nu
Y_{\beta}+Y_{\beta-1}\ ,\qquad 1\leq\alpha\leq\rho_{\mu},\quad
1\leq\beta\leq\rho_{\nu}\ ,$

and $X_{0}$ and $Y_{0}$ are, by definition, null vector fields.

###  3.2. Block-diagonalization

As a nontrivial application of Theorem 1, one can also prove that, given an
operator $\boldsymbol{A}$, condition (3.2) is also sufficient to ensure the
existence of a local chart where the operator $\boldsymbol{A}$ can be block-
diagonalized. We envisage relevant applications, for instance, in the theory
of hydrodynamic-type systems [3], in the study of partial separability of
Hamiltonian systems [6] and, more generally, in the context of Courant’s
problems for first-order hyperbolic systems of partial differential equations
[7].

Let $\boldsymbol{A}$ be an operator satisfying condition (3.2); we denote by
$r_{i}$ the rank of the distribution $\mathcal{D}_{i}$ of $\boldsymbol{A}$. We
also introduce the distribution (of corank $r_{i}$)

(3.7)
$\mathcal{E}_{i}:=Im\Bigl{(}\boldsymbol{A}-\lambda_{i}\mathbf{I}\Bigr{)}^{\rho_{i}}=\bigoplus_{{j=1,\,j\neq
i}}^{s}\mathcal{D}_{j},\qquad\qquad i=1,\ldots,s$

which is spanned by all the generalized eigenvectors of $\boldsymbol{A}$,
except those associated with the eigenvalue $\lambda_{i}$ (we remind that by
hypothesis $\boldsymbol{A}$ has real eigenvalues). We shall say that
$\mathcal{E}_{i}$ is a _characteristic distribution_ of $\boldsymbol{A}$. Let
$\mathcal{E}^{\circ}_{i}$ denote the annihilator of the distribution
$\mathcal{E}_{i}$. The cotangent spaces of $M$ can be decomposed as

(3.8)
$T_{\boldsymbol{x}}^{*}M=\bigoplus_{i=1}^{s}\mathcal{E}_{i}^{\circ}(\boldsymbol{x}).$

As a consequence of Theorem 1, each characteristic distribution
$\mathcal{E}_{i}$ is integrable. By $\mathrm{E}_{i}$ we denote the foliation
associated with $\mathcal{E}_{i}$ and by $E_{i}(\boldsymbol{x})$ the connected
leave through $\boldsymbol{x}$, belonging to $\mathrm{E}_{i}$. Given the set
of distributions
$\\{\mathcal{E}_{1},\mathcal{E}_{2},\ldots,\mathcal{E}_{s}\\}$, we have
associated an equal number of foliations
$\\{\mathrm{E}_{1},\mathrm{E}_{2},\ldots,\mathrm{E}_{s}\\}$. This set of
foliations is said to be the characteristic web of $\boldsymbol{A}$. The
leaves $E_{i}(\boldsymbol{x})$ of each foliation $\mathrm{E}_{i}$ are usually
referred to as the _characteristic fibers_ of the web.

######  Definition 7.

Let $\boldsymbol{A}:\mathfrak{X}(M)\to\mathfrak{X}(M)$ be an operator
satisfying Eq. (3.2). A collection of $r_{i}$ smooth functions will be said to
be adapted to the foliation $\mathrm{E}_{i}$ of the characteristic web of
$\boldsymbol{A}$ if the level sets of such functions coincide with the
characteristic fibers of $\mathrm{E}_{i}$.

######  Definition 8.

Let $\boldsymbol{A}:\mathfrak{X}(M)\to\mathfrak{X}(M)$ be an operator
satisfying Eq. (3.2). A parametrization of the characteristic web of
$\boldsymbol{A}$ is an ordered set of $n$ independent smooth functions listed
as $(\boldsymbol{f}^{1},\ldots,\boldsymbol{f}^{i},\ldots,\boldsymbol{f}^{s})$,
such that for any $i=1,\ldots,s$, the ordered subset
$\boldsymbol{f}^{i}=(f^{i,1},\ldots,f^{i,r_{i}})$ is adapted to the $i$-th
characteristic foliation of the web:

(3.9) $f^{i,k}_{|E_{i}(\mathbf{x})}=c^{i,k}\qquad\forall
E_{i}(\mathbf{x})\in\mathrm{E}_{i}\ ,\quad k=1,\ldots,r_{i}\ .$

Here $c^{i,k}$ are real constants depending on the indices $i$ and $k$ only.
In this case, we shall say that the collection of these functions is adapted
to the web and that each of them is a _characteristic function_.

In the case of a single operator with a vanishing higher-order torsion, the
following result gives a simple and very general tensorial criterion for the
existence of local coordinates ensuring block-diagonalizability.

######  Theorem 2.

[22] Let $\boldsymbol{A}:\mathfrak{X}(M)\to\mathfrak{X}(M)$ be an operator. If

(3.10) $\tau^{(m)}_{\boldsymbol{A}}(X,Y)=\boldsymbol{0}\ ,\qquad
X,Y\in\mathfrak{X}(M)$

for some $m\geq 1$, then $\boldsymbol{A}$ admits local charts where it takes a
block-diagonal form.

One of the main achievement of this work is to generalize this result to the
case of families of commuting operators. To this aim, we shall introduce a new
class of operator algebras.

##  4\. Generalized Haantjes algebras

###  4.1. Definitions

The notion of Haantjes algebra has been introduced and discussed in [20]. In
this section, we define a class of new, generalized Haantjes algebras.

######  Definition 9.

A generalized Haantjes algebra of level $l$ is a pair $(M,\mathscr{H}^{(l)})$
with the following properties:

  * •

$M$ is a differentiable manifold of dimension $\mathrm{n}$;

  * •

$\mathscr{H}^{(l)}$ is a set of Haantjes operators
$\boldsymbol{K}:\mathfrak{X}(M)\to\mathfrak{X}(M)$ whose Haantjes torsion of
level $l$ vanishes: $\tau^{(l)}_{\boldsymbol{K}}=\boldsymbol{0}$. Also, they
generate:

    * –

a free module over the ring of smooth functions on $M$:

(4.1)
$\mathcal{\tau}^{(l)}_{\bigl{(}f\boldsymbol{K}_{1}+g\boldsymbol{K}_{2}\bigr{)}}(X,Y)=\mathbf{0}\
,\qquad\forall\,X,Y\in\mathfrak{X}(M)\ ,\quad\,f,g\in C^{\infty}(M)\
,\quad\forall\leavevmode\nobreak\
\boldsymbol{K}_{1},\boldsymbol{K}_{2}\in\mathscr{H}^{(l)};$

    * –

a ring w.r.t. the composition operation

(4.2)
$\mathcal{\tau}^{(l)}_{\bigl{(}\boldsymbol{K}_{1}\,\boldsymbol{K}_{2}\bigr{)}}(X,Y)=\mathbf{0}\
,\qquad\forall\,\boldsymbol{K}_{1},\boldsymbol{K}_{2}\in\mathscr{H}^{(l)},\quad\forall\,X,Y\in\mathfrak{X}(M)\
.$

If

(4.3)
$\boldsymbol{K}_{1}\,\boldsymbol{K}_{2}=\boldsymbol{K}_{2}\,\boldsymbol{K}_{1}\
,\quad\qquad\ \boldsymbol{K}_{1},\boldsymbol{K}_{2}\in\mathscr{H}^{(l)}\ ,$

the algebra $(M,\mathscr{H}^{(l)})$ will be said to be an Abelian generalized
Haantjes algebra. Moreover, if the identity operator
$\boldsymbol{I}\in\mathscr{H}^{(l)}$, then $(M,\mathscr{H}^{(l)})$ will be
said to be a generalized Haantjes algebra with identity.

In other words, the set $\mathscr{H}^{(l)}$ can be regarded as an associative
algebra of generalized Haantjes operators.

The case $l=2$, namely that of Haantjes algebras [20], possesses several
important properties. Among them, we recall that for a given Abelian Haantjes
algebra $\mathscr{H}^{(2)}\equiv\mathscr{H}$ there exists an associated set of
coordinates, called Haantjes coordinates, by means of which all
$\boldsymbol{K}\in\mathscr{H}$ can be written simultaneously in a block-
diagonal form. In particular, if $\mathscr{H}$ is also semisimple, on each set
of Haantjes coordinates all $\boldsymbol{K}\in\mathscr{H}$ can be written
simultaneously in a purely diagonal form [20]. We mention that Haantjes
algebras play a relevant role in the theory of classical separable and
multiseparable Hamiltonian systems [19].

The following result simplifies the study of the integrability of the eigen-
distributions of a family of operators forming an Abelian, generalized
Haantjes algebra.

######  Lemma 1.

Let $(M,\mathscr{H}^{(l)})$ be an Abelian generalized Haantjes algebra of
level $l$. We shall assume that the rank of the eigen-distributions of the
operators belonging to $\mathscr{H}^{(l)}$ is independent of
$\boldsymbol{x}\in M$. Then, each nontrivial intersection of these eigen-
distributions is integrable.

###### Proof.

Let $\\{\boldsymbol{K}_{1},\ldots,\boldsymbol{K}_{w}\\}$ be a basis of
$(M,\mathscr{H}^{(l)})$, and
$(\mathcal{D}_{i_{1}}^{(1)},\ldots,\mathcal{D}_{i_{w}}^{(w)})$,
$i_{1}=1,\ldots,s_{1}$, $i_{w}=1,\ldots,s_{w}$, be the set of their (proper)
eigen-distributions. Let

(4.4)
$\mathcal{V}_{a}=\mathcal{D}_{i_{1}}^{(1)}(\boldsymbol{x})\bigcap\ldots\bigcap\mathcal{D}_{i_{w}}^{(w)}(\boldsymbol{x})\qquad
a=1,\ldots,v,\qquad v\leq n$

denote a nontrivial intersection of eigen-distributions of the operators
$\\{\boldsymbol{K}_{1},\ldots,\boldsymbol{K}_{w}\\}$. Consequently, this
distribution, being the intersection of distributions which are involutive due
to Theorem 1 [22], is also involutive.

∎

###  4.2. Cyclic generalized Haantjes algebras

A special class of commutative generalized Haantjes algebras are those
generated by a single operator and its independent powers.

######  Definition 10.

A generalized Haantjes algebra $(M,\mathscr{H}^{(l)})$ will be said to be
cyclic if there exists an operator
$\boldsymbol{L}:\mathfrak{X}(M)\rightarrow\mathfrak{X}(M)$, with
$\tau_{\boldsymbol{L}}^{(l)}(X,Y)=0$, such that
$\mathscr{H}^{(l)}\subseteq\mathcal{L}(\boldsymbol{L})$, where
$\mathcal{L}(\boldsymbol{L})=\langle\boldsymbol{I},\boldsymbol{L},\ldots,\boldsymbol{L}^{k},\ldots\rangle$
is the algebra generated by all the independent powers of $\boldsymbol{L}$.

Given a cyclic algebra $(M,\mathscr{H}^{(l)})$ generated by an operator
$\boldsymbol{L}$, its rank is less than or equal to the degree of the minimal
polynomial of $\boldsymbol{L}$.

######  Remark 1.

Cyclic generalized Haantjes algebras as algebraic structures are not rare.
Indeed, as we will prove, given an operator $\boldsymbol{L}$ with
$\tau_{\boldsymbol{L}}^{(l)}(X,Y)=0$, any polynomial in $\boldsymbol{L}$
having coefficients in $C^{\infty}(M)$ is still an operator having the same
level-$l$ vanishing torsion. Thus, the independent powers of any operator with
a given vanishing generalized torsion provide naturally a basis of a cyclic
generalized algebra of the same level.

In the following, we shall prove this fundamental fact. To this aim, we shall
use and adapt to our general case several algebraic techniques and results
introduced and proved by Bogoyavlenskij in [2] for the case of Nijenhuis and
Haantjes operators. In particular, we shall use a useful representation of
(1,2)-tensors in the ring $S_{3}$ of polynomials of three independent
variables $z,\lambda,\mu$ with coefficients depending on $\boldsymbol{x}\in
M$. A generic polynomial of this ring has the form

$S(z,\lambda,\mu):=\sum_{i,j,k}^{N}s_{ijk}(\boldsymbol{x})z^{i}\lambda^{j}\mu^{k},\qquad\boldsymbol{x}\in
M$

with $N\in\mathbb{N}\backslash\\{0\\}$, $s_{ijk}(\boldsymbol{x})\in
C^{\infty}(M)$. Given an operator $\boldsymbol{A}$ and a (1,2)-tensor
$T(X,Y)$, we introduce the representation defined by [2]

(4.5)
$R_{S}(T)(X,Y)=\sum_{i,j,k}^{N}s_{ijk}(\boldsymbol{x})\boldsymbol{A}^{i}T(\boldsymbol{A}^{j}X,\boldsymbol{A}^{k}Y)\
.$

Thus, the action of $\lambda$ and $\mu$ is associated with the first and
second arguments of $T(X,Y)$, whereas the action of $z$ is associated with the
valued of $T(X,Y)$. Representation (4.5) satisfies the basic properties

(4.6) $R_{S_{1}+S_{2}}=R_{S_{1}}+R_{S_{2}},$ (4.7) $\qquad R_{S_{1}\cdot
S_{2}}=R_{S_{2}\cdot S_{1}}=R_{S_{1}}\cdot R_{S_{2}}\ .$

This framework allows us to represent the “tower” of generalized Nijenhuis
torsions in a direct way.

######  Lemma 2.

Let $\boldsymbol{A}:\mathfrak{X}(M)\rightarrow\mathfrak{X}(M)$ be an operator.
We have:

(4.8)
$\mathcal{\tau}^{(m+1)}_{\boldsymbol{A}}(X,Y)=R_{\sigma}\tau^{(m)}_{\boldsymbol{A}}(X,Y)$

where $\sigma$ is the polynomial $\sigma(z,\lambda,\mu)=(z-\lambda)(z-\mu)$.

###### Proof.

It is an immediate consequence of the formula (5), which indeed can be written
as

(4.9) $\displaystyle
R_{(z^{2}-z\lambda-z\mu+\lambda\mu)}\mathcal{\tau}^{(m)}_{\boldsymbol{A}}(X,Y)$
$\displaystyle=$
$\displaystyle\boldsymbol{A}^{2}\mathcal{\tau}^{(m)}_{\boldsymbol{A}}(X,Y)+\mathcal{\tau}^{(m)}_{\boldsymbol{A}}(\boldsymbol{A}X,\boldsymbol{A}Y)$
$\displaystyle-$
$\displaystyle\boldsymbol{A}\Big{(}\mathcal{\tau}^{(m)}_{\boldsymbol{A}}(X,\boldsymbol{A}Y)+\mathcal{\tau}^{(m)}_{\boldsymbol{A}}(\boldsymbol{A}X,Y)\Big{)}$
$\displaystyle=$ $\displaystyle\mathcal{\tau}^{(m+1)}_{\boldsymbol{A}}(X,Y)\
.$

∎

When $m=2$, we recover the formula

$\mathcal{H}_{\boldsymbol{A}}(X,Y)=R_{\sigma}\tau_{\boldsymbol{A}}(X,Y)$

first stated in [2]. We also remind that, given a polynomial
$P(z)=\sum_{k=0}^{N}c_{k}(\boldsymbol{x})z^{k}$, the Bézout identity holds:

(4.10) $P(z)-P(\lambda)=(z-\lambda)Q_{P}(z,\lambda)$

where

$Q_{P}(z,\lambda)=\sum_{k=1}^{N}c_{k}(\boldsymbol{x})\sum_{p+q=k-1}z^{p}\lambda^{q}\
.$

We prove now a useful technical result.

######  Lemma 3.

Let $\boldsymbol{A}:\mathfrak{X}(M)\rightarrow\mathfrak{X}(M)$ be an operator.
Let
$\boldsymbol{P}:=P(\boldsymbol{A})=\sum_{k=0}^{N}c_{k}(\boldsymbol{x})\boldsymbol{A}^{k}$
be a polynomial in $\boldsymbol{A}$ with variable coefficients. We have

(4.11)
$\mathcal{\tau}_{\boldsymbol{P}}^{(m)}(X,Y)=R_{(Q_{P}(z,\lambda))^{m}(Q_{P}(z,\mu))^{m}}\mathcal{\tau}_{\boldsymbol{A}}^{(m)}(X,Y),\qquad
m\geq 2\ .$

###### Proof.

We shall proceed by induction over $m\geq 2$. The case $m=2$, corresponding to
the Haantjes torsion, has been proved in [2]. Thus, we assume that the
property is true for the case of a torsion of level $m-1$ and we prove that it
holds true for the case of a torsion of level $m$. According to Lemma 2, we
have

(4.12)
$\mathcal{\tau}^{(m)}_{\boldsymbol{A}}(X,Y)=R_{\sigma}\mathcal{\tau}^{(m-1)}_{\boldsymbol{A}}(X,Y)$

where $\sigma=(z-\lambda)(z-\mu)$. Also, in terms of the operator
$\boldsymbol{P}$ expression (4.12) can be written as

(4.13)
$\mathcal{\tau}^{(m)}_{\boldsymbol{P}}(X,Y)=R_{(P(z)-P(\lambda))(P(z)-P(\mu))}\mathcal{\tau}^{(m-1)}_{\boldsymbol{P}}(X,Y)\
.$

Thus, applying twice the Bezout identity (4.10), we get

$(P(z)-P(\lambda))(P(z)-P(\mu))=(z-\lambda)(z-\mu)Q_{P}(z,\lambda)Q_{P}(z,\mu)\
.$

Exploiting property (4.7), we obtain

(4.14)
$\mathcal{\tau}^{(m)}_{\boldsymbol{P}}(X,Y)=R_{Q_{P}(z,\lambda)Q_{P}(z,\mu)}R_{\sigma}\mathcal{\tau}^{(m-1)}_{\boldsymbol{P}}(X,Y)\
.$

By the induction hypothesis, we have

$\mathcal{\tau}^{(m-1)}_{\boldsymbol{P}}(X,Y)=R_{(Q_{P}(z,\lambda))^{m-1}(Q_{P}(z,\mu))^{m-1}}\mathcal{\tau}_{\boldsymbol{A}}^{(m-1)}(X,Y)\
.$

Consequently, due to eq. (4.14) and Lemma 2, we deduce that

(4.15)
$\mathcal{\tau}^{(m)}_{\boldsymbol{P}}(X,Y)=R_{(Q_{P}(z,\lambda))^{m}(Q_{P}(z,\mu))^{m}}R_{\sigma}\tau^{(m-1)}_{\boldsymbol{A}}(X,Y)=R_{(Q_{P}(z,\lambda))^{m}(Q_{P}(z,\mu))^{m}}\tau^{(m)}_{\boldsymbol{A}}(X,Y)\
,$

which completes the proof. ∎

We can now state the main result of this section.

######  Theorem 3.

Let $\boldsymbol{A}:\mathfrak{X}(M)\rightarrow\mathfrak{X}(M)$ be an operator
with $\tau_{\boldsymbol{A}}^{(m)}(X,Y)=0$, $m\geq 2$. Then

(4.16)
$\tau^{(m)}_{\big{(}\sum_{k=0}^{N}c_{k}(\boldsymbol{x})\boldsymbol{A}^{k}\big{)}}(X,Y)=0,\qquad
X,Y\in\mathfrak{X}(M)\ ,$

where $c_{k}(\boldsymbol{x})\in C^{\infty}(M)$.

###### Proof.

Let
$\boldsymbol{P}:=P(\boldsymbol{A})=\sum_{k=0}^{N}c_{k}(\boldsymbol{x})\boldsymbol{A}^{k}$.
It is sufficient to apply Lemma 3, taking into account that
$\tau_{\boldsymbol{A}}^{(m)}(X,Y)=0$. ∎

Interestingly enough, Theorem 3 does not hold for the case of the Nijenhuis
torsion (i.e., $m=1$). As is well-known, the relation (4.16) is valid for the
Nijenhuis torsion only if the coefficients $c_{k}(\boldsymbol{x})$ are all
constant.

##  5\. Generalized Nijenhuis torsions and simultaneous block-diagonalization

######  Theorem 4.

Let $\mathcal{S}=\\{\boldsymbol{K}_{1},\ldots,\boldsymbol{K}_{w}\\}$,
$\boldsymbol{K}_{\alpha}:\mathfrak{X}(M)\rightarrow\mathfrak{X}(M)$,
$\alpha=1,\ldots,w$ be a set of commuting operator fields; we assume that one
of them, say $\boldsymbol{K}_{1}$, satisfies the condition

(5.1) $\tau^{(l)}_{\boldsymbol{K}_{1}}(X,Y)=\boldsymbol{0},\qquad
X,Y\in\mathfrak{X}(M)\ ,$

for some $l\geq 1$. Then, there exist local charts in which all of the
operators $\boldsymbol{K}_{\alpha}$ can be written simultaneously in a block-
diagonal form.

###### Proof.

Assuming that condition (5.1) is satisfied, Proposition 2 ensures the
existence of an equivalence class of integrable frames and local charts where
the operator $\boldsymbol{K}_{1}$ takes a block-diagonal form. Such
coordinates are adapted to the characteristic web associated with the spectral
decomposition of $\boldsymbol{K}_{1}$:

(5.2)
$T_{\boldsymbol{x}}M=\mathcal{D}_{i}(\boldsymbol{x})\bigoplus\mathcal{E}_{i}(x)=\bigoplus_{i=1}^{s}\mathcal{D}_{i}(\boldsymbol{x})\
.$

with

(5.3)
$\boldsymbol{x}=(\boldsymbol{x}^{1},\ldots,\boldsymbol{x}^{i},\ldots,\boldsymbol{x}^{s})\
.$

The variables $\boldsymbol{x}^{i}=(x^{i,1},\ldots,x^{i,r_{i}})$ are defined
over the integral leaves of the eigen-distribution $\mathcal{D}_{i}$, whereas
the remaining ones, namely

$(\boldsymbol{x}^{1},\ldots,\boldsymbol{x}^{i-1},\boldsymbol{x}^{i+1},\ldots,\boldsymbol{x}^{s})$

are coordinates of the leaves, i.e., are constant
$(\boldsymbol{x}^{1}=\boldsymbol{c}^{1},\ldots,\boldsymbol{x}^{i-1}=\boldsymbol{c}^{i-1},\boldsymbol{x}^{i+1}=\boldsymbol{c}^{i+1},\ldots,\boldsymbol{x}^{s}=\boldsymbol{c}^{s})$
on each leaf $D_{i}(\boldsymbol{c})$ of the foliation. Here
$\boldsymbol{c}:=(\boldsymbol{c}^{1},\ldots,\boldsymbol{c}^{i-1},\boldsymbol{c}^{i+1},\ldots,\boldsymbol{c}^{s})$.
Since all operators of the set $\mathcal{F}$ commute, every distribution
$\mathcal{D}_{i}$ is invariant under the action of the operators
$\\{\boldsymbol{K}_{2},\ldots,\boldsymbol{K}_{w}\\}$. As a direct consequence
of this property, all the operators $\boldsymbol{K}_{\alpha}\in\mathcal{F}$ in
the local chart (5.3) take a block-diagonal form, where the $i$-th
($r_{i}\times r_{i}$) block matrix
$[\boldsymbol{K}_{\alpha}^{(i)}]_{|_{D_{i}(\boldsymbol{c})}}$ coincides with
the matrix $[\boldsymbol{K}_{\alpha|_{D_{i}(\boldsymbol{c})}}]$,
$i=1,\ldots,s$, which is the representation of the restricted operator
$\boldsymbol{K}_{\alpha}$ to the leaf $D_{i}(\boldsymbol{c})$. ∎

######  Theorem 5.

Let $\mathcal{S}=\\{\boldsymbol{K}_{1},\ldots,\boldsymbol{K}_{w}\\}$,
$\boldsymbol{K}_{\alpha}:\mathfrak{X}(M)\rightarrow\mathfrak{X}(M)$, be a set
of commuting operator fields. If

(i) all the operators of the family have vanishing generalized torsion of
level $l$:

(5.4) $\tau^{(l)}_{\boldsymbol{K}_{\alpha}}(X,Y)=\boldsymbol{0}\
,\qquad\alpha=1,\ldots,q,\qquad
X,Y\in\mathfrak{X}(M)\rightarrow\mathfrak{X}(M)$

(ii) all possible nontrivial intersections of their generalized eigen-
distributions

(5.5)
$\mathcal{V}_{a}(\boldsymbol{x}):=\bigoplus_{i_{1},\ldots,i_{w}}^{s_{1},\ldots,s_{w}}\mathcal{D}_{i_{1}}^{(1)}(\boldsymbol{x})\bigcap\ldots\bigcap\mathcal{D}_{i_{w}}^{(w)}(\boldsymbol{x})\
,\qquad a=1,\ldots,v\leq n$

are mutually integrable, then there exist sets of local coordinates, adapted
to the decomposition

(5.6)
$T_{\boldsymbol{x}}M=\bigoplus_{a=1}^{v}\mathcal{V}_{a}(\boldsymbol{x})\qquad\boldsymbol{x}\in
M,$

in which all operators $\boldsymbol{K}_{\alpha}$ admit simultaneously a block-
diagonal form with possibly finer blocks.

###### Proof.

According to Theorem 4, since by hypothesis
$\tau^{(l)}_{\boldsymbol{K}_{1}}(X,Y)=\boldsymbol{0}$,
$X,Y\in\mathfrak{X}(M)$, all the operators take simultaneously a block-
diagonal form in a suitable local chart. Assume also that
$\tau^{(l)}_{\boldsymbol{K}_{2}}(X,Y)=\boldsymbol{0}$. Then, the tangent space
at any point $\boldsymbol{x}$ admits the finer decomposition

(5.7)
$T_{\boldsymbol{x}}M=\bigoplus_{i_{1},i_{2}}^{s_{1},s_{2}}\mathcal{D}_{i_{1}}^{(1)}(\boldsymbol{x})\cap\mathcal{D}_{i_{2}}^{(2)}(\boldsymbol{x})\
,$

where $\mathcal{D}_{i_{2}}^{(2)}$ are the eigen-distributions of
$\boldsymbol{K}_{2}$. These eigen-distributions are integrable by virtue of
Theorem 1. Consequently, the Haantjes Theorem can also be applied to the
restriction of $\boldsymbol{K}_{2}$ to $D^{(1)}_{i_{1}}(\boldsymbol{c}_{1})$.
Therefore, there exists a transformation of coordinates, acting only on the
coordinates over the leaves of the foliation $D^{(1)}_{i_{1}}$

(5.8) $\Phi:M\rightarrow
M,\qquad(\boldsymbol{x}^{1},\ldots,\boldsymbol{x}^{i_{1}},\ldots,\boldsymbol{x}^{s_{1}})\mapsto(\boldsymbol{x}^{1},\ldots,\boldsymbol{y}^{i_{1}},\ldots,\boldsymbol{x}^{s_{1}})\
,$

such that the new coordinates
$\boldsymbol{y}^{i_{1}}=(y^{i_{1},1},\ldots,y^{i_{1},r_{i_{1}}})=\boldsymbol{f}^{i_{1}}(\boldsymbol{x}^{i_{1}})$
are adapted to the decomposition

(5.9)
$T_{\boldsymbol{x}}D^{(1)}_{i_{1}}(\boldsymbol{c}_{1})=\bigoplus_{i_{2}}^{s_{2}}\mathcal{D}_{i_{1}}^{(1)}(\boldsymbol{x})\cap\mathcal{D}_{i_{2}}^{(2)}(\boldsymbol{x}),\qquad\boldsymbol{x}\in
D^{(1)}_{i_{1}}(\boldsymbol{c}_{1})\ .$

Thus, we have

(5.10)
$[\boldsymbol{K}^{(i_{1})}_{\alpha}]=\left[\begin{array}[]{c|c|c}\boldsymbol{K}_{\alpha}^{(i_{1},1)}&0&0\\\
\hline\cr 0&\ddots&0\\\ \hline\cr
0&0&\boldsymbol{K}_{\alpha}^{(i_{1},s_{2})}\end{array}\right]\
,\qquad\alpha=1,\dots,w$

where
$[\boldsymbol{K}_{\alpha}^{(i_{1},j)}]_{|_{D_{i_{1}}^{(1)}(\boldsymbol{c}_{1})\cap
D_{j}^{(2)}(\boldsymbol{c}_{2})}}=\left[\boldsymbol{K}_{\alpha|_{D_{i_{1}}^{(1)}(\boldsymbol{c}_{1})\cap
D_{j}^{(2)}(\boldsymbol{c}_{2})}}\right]$, $j=1,\ldots,s_{2}$. Let us consider
the decomposition

(5.11)
$T_{\boldsymbol{x}}M=\bigoplus_{i_{1},i_{2}}^{s_{1},s_{2}}\mathcal{D}_{i_{1}}^{(1)}(\boldsymbol{x})\cap\mathcal{D}_{i_{2}}^{(2)}(\boldsymbol{x})=\bigoplus_{\gamma=1}^{u}\mathcal{U}_{\gamma}(\boldsymbol{x})\qquad\boldsymbol{x}\in
M\ ,$

where in the direct sum (5.11) $\mathcal{U}_{\gamma}\neq\\{\boldsymbol{0}\\}$,
$u\leq n$ and $r_{\gamma}$ denotes the rank of $\mathcal{U}_{\gamma}$
($\sum_{\gamma=1}^{u}r_{\gamma}=n$). Clearly, the distributions
$\mathcal{U}_{\gamma}$ are invariant under the action of each operator
$\boldsymbol{K}_{\alpha}\in\mathcal{S}$. Besides, these distributions are
involutive and are realized as the intersection of involutive distributions.
By assumption, they are also _mutually_ integrable. Therefore, there exist
local charts in $M$ of the form

(5.12)
$\\{U,(y^{1,1},\ldots,y^{1,r_{1}};\dots;y^{i_{1},1},\ldots,y^{i_{1},r_{i_{1}}};\ldots;y^{s_{1},1},\dots,y^{s_{1},r_{s_{1}}})\\}$

adapted to the decomposition (5.11), where all the operators
$\boldsymbol{K}_{\alpha}\in\mathcal{S}$ admit simultaneously a (possibly)
finer block-diagonal form. By extending the previous procedure to the Haantjes
operators $\boldsymbol{K}_{3},\ldots,\boldsymbol{K}_{w}$, we obtain the
decomposition

(5.13)
$T_{\boldsymbol{x}}M=\bigoplus_{i_{1},\ldots,i_{w}}^{s_{1},\ldots,s_{w}}\mathcal{D}_{i_{1}}^{(1)}(\boldsymbol{x})\bigcap\ldots\bigcap\mathcal{D}_{i_{w}}^{(w)}(\boldsymbol{x})=\bigoplus_{a=1}^{v}\mathcal{V}_{a}(\boldsymbol{x}),$

where in the direct sum (5.13) $\mathcal{V}_{a}\neq\\{\boldsymbol{0}\\}$,
$v\leq n$ and $r_{a}$ denotes the rank of $\mathcal{V}_{a}$
($\sum_{a=1}^{v}r_{a}=n$). Then, as the involutive distributions
$\mathcal{V}_{a}$ by assumption are also mutually integrable, there exist
local charts

(5.14)
$\\{U,(\boldsymbol{y}^{1},\ldots,\boldsymbol{y}^{a},\ldots,\boldsymbol{y}^{v})\\}$

adapted to the decomposition (5.13), such that

(5.15) $\mathcal{V}_{a}=\left\langle\frac{\partial}{\partial
y^{a,1}},\ldots,\frac{\partial}{\partial y^{a,r_{a}}}\right\rangle\qquad
a=1,\ldots,v,$

where the natural frame $\big{\\{}\frac{\partial}{\partial
y^{a,1}},\ldots,\frac{\partial}{\partial y^{a,r_{a}}}\big{\\}}$ over the
leaves of $\mathcal{V}_{a}$ is formed by joint _generalized_ eigenvector
fields of the operators $\\{\boldsymbol{K}_{1},\ldots,\boldsymbol{K}_{w}\\}$.
∎

######  Remark 2.

Given a cyclic generalized Haantjes algebra $(M,\mathscr{H}^{(l)})$ with
generator $\boldsymbol{L}$, the hypothesis $(ii)$ of Theorem 5 is
automatically satisfied. Indeed, the intersections $\mathcal{V}_{a}$ of eigen-
distributions of the operators forming the algebra $\mathscr{H}^{(l)}$ either
coincide with the eigen-distributions of the generator $\boldsymbol{L}$ or are
direct sums of them. In both cases, since (due to Theorem 1) the eigen-
distributions of $\boldsymbol{L}$ are mutually integrable, it turns out that
the direct sums of the intersections $\mathcal{V}_{a}$ are integrable as well.

In the following sections, we shall illustrate the results proposed. First, we
will show some examples of generalized Haantjes algebras of operator fields.
Then, we shall present the explicit construction of local charts of
coordinates allowing their simultaneous block-diagonalization.

##  6\. Block-diagonalization of generalized Haantjes algebras

As stated in Theorems 4 and 5, we can block-diagonalize families of commuting
generalized Nijenhuis operators by constructing a suitable coordinate chart.
To this aim, we propose the following procedure.

1) Determine the joint eigen-distributions of the given family of operator
fields
2) Determine a basis of one-forms for each of the the corresponding
annihilators
3) Integrate them to find the characteristic coordinates
4) Compute the expression of the operators of the given family in these
coordinates

To illustrate this procedure, we shall first construct families of operator
fields, depending on arbitrary functions, having a prescribed vanishing
higher-level torsion. A direct approach consists in considering an operator
field whose entries are all arbitrary functions, and imposing both that a
certain higher-level torsion is vanishing, and simultaneously that the lower-
level ones are not. In this way, one can obtain a set of differential
constraints that can be solved (possibly with suitable ansätze) to obtain
families of operator fields with a vanishing higher-level torsion, still
depending on arbitrary functions. Generally speaking, there is a lot of
freedom in this approach.

From this analysis it emerges that generalized Haantjes algebras are not rare.
Thus, by specializing the arbitrary functions contained in the families of
operators determined according to the procedure proposed, we can obtain
infinitely many examples of new, higher-level Haantjes algebras.

###  6.1. A level-three generalized Haantjes algebra

Let $M$ be a five-dimensional differentiable manifold. A first example is
given, in local coordinates $\boldsymbol{x}=(x^{1},x^{2},x^{3},x^{4},x^{5})$
by the family of operators

(6.1)
$\boldsymbol{L}^{(\boldsymbol{x})}=\left[\begin{array}[]{ccccc}f_{1}&1&0&1&0\\\
f_{1}-f_{2}+1&f_{1}+1&-f_{3}&f_{1}-f_{2}+1&-f_{3}\\\
1&0&f_{2}+f_{3}&1&f_{3}\\\ f_{2}-f_{1}&-1&f_{3}&f_{2}-1&f_{3}\\\
-1&0&-f_{3}-1&-1&f_{2}-f_{3}-1\\\ \end{array}\right],$

where $f_{1}=f_{1}(x^{1},x^{2})$, $f_{2}=f_{2}(x^{3},x^{4},x^{5})$ and
$f_{3}=f_{3}(x^{3})$ are arbitrary functions.

A direct computation allows us to prove the following result.

######  Proposition 3.

The family of operators $\boldsymbol{L}$ satisfies the following properties:

  * •

$\tau^{(3)}_{\boldsymbol{L}}(X,Y)=0$

  * •

$\mathcal{H}_{\boldsymbol{A}}(X,Y)\neq 0$ if and only if $f_{3}^{\prime}\neq
0$.

The minimal polynomial of $\boldsymbol{L}$ is of fifth degree and reads

(6.2) $m_{\boldsymbol{L}}(\boldsymbol{x},\lambda)=(\lambda-f_{1}-1)(\lambda-
f_{1}+1)\left(\lambda-f_{2}+1\right)(\lambda-f_{2})^{2}\ .$

Thus, it coincides with the characteristic polynomial; therefore
$\boldsymbol{L}$ is a cyclic operator. Also, notice that $\boldsymbol{L}$ is
generically non-semisimple. The eigenvalues and generalized eigenvectors of
$\boldsymbol{L}$ are

(6.3) $\displaystyle\lambda_{1}=f_{1}+1,\quad\rho_{1}=1$
$\displaystyle\Longrightarrow\mathcal{D}_{1}=\bigg{\langle}\dfrac{\partial}{\partial
x^{1}}+2\dfrac{\partial}{\partial x^{2}}-\dfrac{\partial}{\partial
x^{4}}\bigg{\rangle},$ $\displaystyle\lambda_{2}=f_{1}-1,\quad\rho_{2}=1$
$\displaystyle\Longrightarrow\mathcal{D}_{2}=\bigg{\langle}\dfrac{\partial}{\partial
x^{1}}-\dfrac{\partial}{\partial x^{4}}\bigg{\rangle},$
$\displaystyle\lambda_{3}=f_{2}-1,\quad\rho_{3}=1$
$\displaystyle\Longrightarrow\mathcal{D}_{3}=\bigg{\langle}f_{3}\dfrac{\partial}{\partial
x^{2}}-f_{3}\dfrac{\partial}{\partial x^{4}}+\dfrac{\partial}{\partial
x^{5}}\bigg{\rangle},$ $\displaystyle\lambda_{4}=f_{2},\quad\rho_{4}=2$
$\displaystyle\Longrightarrow\mathcal{D}_{4}=\bigg{\langle}\dfrac{\partial}{\partial
x^{3}}-\dfrac{\partial}{\partial x^{5}},\dfrac{\partial}{\partial
x^{2}}-\dfrac{\partial}{\partial x^{4}}\bigg{\rangle}.$

From these eigen-distributions, we can compute their characteristic
distributions and consequently their annihilators:

(6.4) $\displaystyle\mathcal{E}_{1}=\oplus_{i\neq
1}\mathcal{D}_{i}=\bigg{\langle}\dfrac{\partial}{\partial
x^{1}}-\dfrac{\partial}{\partial x^{4}},f_{3}\dfrac{\partial}{\partial
x^{2}}-f_{3}\dfrac{\partial}{\partial x^{4}}+\dfrac{\partial}{\partial
x^{5}},\dfrac{\partial}{\partial x^{2}}-\dfrac{\partial}{\partial
x^{4}},\dfrac{\partial}{\partial x^{3}}-\dfrac{\partial}{\partial
x^{5}}\bigg{\rangle}$ $\displaystyle\hskip
56.9055pt\Longrightarrow\mathcal{E}_{1}^{\circ}=\langle
dx^{1}+dx^{2}+dx^{4}\rangle$ $\displaystyle\mathcal{E}_{2}=\oplus_{i\neq
2}\mathcal{D}_{i}=\bigg{\langle}\dfrac{\partial}{\partial
x^{1}}+2\dfrac{\partial}{\partial x^{2}}-\dfrac{\partial}{\partial
x^{4}},f_{3}\dfrac{\partial}{\partial x^{2}}-f_{3}\dfrac{\partial}{\partial
x^{4}}+\dfrac{\partial}{\partial x^{5}},\dfrac{\partial}{\partial
x^{2}}-\dfrac{\partial}{\partial x^{4}},\dfrac{\partial}{\partial
x^{3}}-\dfrac{\partial}{\partial x^{5}}\bigg{\rangle}$ $\displaystyle\hskip
56.9055pt\Longrightarrow\mathcal{E}_{2}^{\circ}=\langle
dx^{1}-dx^{2}-dx^{4}\rangle$ $\displaystyle\mathcal{E}_{3}=\oplus_{i\neq
3}\mathcal{D}_{i}=\bigg{\langle}\dfrac{\partial}{\partial
x^{1}}+2\dfrac{\partial}{\partial x^{2}}-\dfrac{\partial}{\partial
x^{4}},\dfrac{\partial}{\partial x^{1}}-\dfrac{\partial}{\partial
x^{4}},\dfrac{\partial}{\partial x^{2}}-\dfrac{\partial}{\partial
x^{4}},\dfrac{\partial}{\partial x^{3}}-\dfrac{\partial}{\partial
x^{5}}\bigg{\rangle}$ $\displaystyle\hskip
56.9055pt\Longrightarrow\mathcal{E}_{3}^{\circ}=\langle dx^{3}+dx^{5}\rangle,$
$\displaystyle\mathcal{E}_{4}=\oplus_{i\neq
4}\mathcal{D}_{i}=\bigg{\langle}\dfrac{\partial}{\partial
x^{1}}+2\dfrac{\partial}{\partial x^{2}}-\dfrac{\partial}{\partial
x^{4}},\dfrac{\partial}{\partial x^{1}}-\dfrac{\partial}{\partial
x^{4}},f_{3}\dfrac{\partial}{\partial x^{2}}-f_{3}\dfrac{\partial}{\partial
x^{4}}+\dfrac{\partial}{\partial x^{5}}\bigg{\rangle}$ $\displaystyle\hskip
56.9055pt\Longrightarrow\mathcal{E}_{4}^{\circ}=\langle
dx^{3},dx^{1}+dx^{4}+f_{3}dx^{5}\rangle.$

######  Remark 3.

We can easily extract an Abelian cyclic algebra $\mathscr{H}^{(3)}$ from the
class (6.1). In order to achieve that, we set $f_{3}=x^{3}$ for all the
operators of the algebra and vary the other arbitrary functions. Observe that
all the operators of the family obtained in this way will share both the same
eigen-distributions and the degree of the minimal polynomial. Thus, any of
them generate the full algebra $\mathscr{H}^{(3)}$ with its independent
powers. Notice that for this algebra the hypotheses of Theorem 5 are all
satisfied.

By way of an example of simultaneous diagonalization, we can consider the
three commuting operators

(6.5)
$\boldsymbol{L}_{1}^{(\boldsymbol{x})}=\left[\begin{array}[]{ccccc}x^{1}&1&0&1&0\\\
x^{1}+1&x^{1}+1&-x^{3}&x^{1}+1&-x^{3}\\\ 1&0&x^{3}&1&x^{3}\\\
-x^{1}&-1&x^{3}&-1&x^{3}\\\ -1&0&-x^{3}-1&-1&-x^{3}-1\\\ \end{array}\right],$
(6.6)
$\boldsymbol{L}_{2}^{(\boldsymbol{x})}=\left[\begin{array}[]{ccccc}x^{2}&1&0&1&0\\\
x^{2}-x^{5}+1&x^{2}+1&-x^{3}&x^{2}-x^{5}+1&-x^{3}\\\
1&0&x^{3}+x^{5}&1&x^{3}\\\ -x^{2}+x^{5}&-1&x^{3}&x^{5}-1&x^{3}\\\
-1&0&-x^{3}-1&-1&-x^{3}+x^{5}-1\\\ \end{array}\right]\ ,$ (6.7)
$\boldsymbol{L}_{3}^{(\boldsymbol{x})}=\left[\begin{array}[]{ccccc}0&1&0&1&0\\\
-x^{4}-x^{3}x^{5}+1&1&-x^{3}&-x^{4}-x^{3}x^{5}+1&-x^{3}\\\
1&0&x^{4}+x^{3}\left(x^{5}+1\right)&1&x^{3}\\\
x^{4}+x^{3}x^{5}&-1&x^{3}&x^{4}+x^{3}x^{5}-1&x^{3}\\\
-1&0&-x^{3}-1&-1&x^{4}+x^{3}\left(x^{5}-1\right)-1\\\ \end{array}\right]\ .$

These operators are $C^{\infty}(\mathbb{R})$-linearly independent. Any of
these three operators can be chosen to be the generator of a level-three
Haantjes cyclic algebra $\mathscr{H}^{(3)}$ of rank $5$. A basis of this
algebra is, for instance,
$\mathscr{B}=\\{\boldsymbol{I},\boldsymbol{L}_{1},\boldsymbol{L}_{1}^{2},\boldsymbol{L}_{1}^{3},\boldsymbol{L}_{1}^{4}\\}$.
In the following subsection, we shall determine a set of coordinates which
block-diagonalize the full algebra $\mathscr{H}^{(3)}$.

###  6.2. Block-diagonalization

By integrating the annihilators (6.4) with the choice $f_{3}=x^{3}$ of the
eigen-distributions of these operators, we obtain a set of coordinates which
block-diagonalized the full algebra $\mathscr{H}^{(3)}$. We have explicitly:

(6.8) $\displaystyle y^{1}=x^{1}+x^{2}+x^{4},$ $\displaystyle
y^{2}=x^{1}-x^{2}-x^{4},$ $\displaystyle y^{3}=x^{3}+x^{5},$ $\displaystyle
y^{4}=x^{3},$ $\displaystyle y^{5}=x^{1}+x^{4}+x^{3}x^{5}.$

In particular, in the block-separation coordinates, the operators
$\boldsymbol{L}_{1},\boldsymbol{L}_{2},\boldsymbol{L}_{3}$ read:

(6.9)
$\boldsymbol{L}_{1}^{(\boldsymbol{y})}=\left[\begin{array}[]{ccccc}\frac{1}{2}\left(y^{1}+y^{2}\right)+1&\vrule\lx@intercol\hfil
0\hfil\lx@intercol&0&0&0\\\ \cline{1-2}\cr\lx@intercol\hfil
0\hfil\lx@intercol\vrule\lx@intercol&\frac{1}{2}\left(y^{1}+y^{2}\right)-1&\vrule\lx@intercol\hfil
0\hfil\lx@intercol&0&0\\\ \cline{2-3}\cr 0&\lx@intercol\hfil
0\hfil\lx@intercol\vrule\lx@intercol&-1&\vrule\lx@intercol\hfil
0\hfil\lx@intercol&0\\\ \cline{3-5}\cr 0&0&\lx@intercol\hfil
0\hfil\lx@intercol\vrule\lx@intercol&-y^{3}+2y^{4}&1\\\ 0&0&\lx@intercol\hfil
0\hfil\lx@intercol\vrule\lx@intercol&-\left(y^{3}-2y^{4}\right)^{2}&y^{3}-2y^{4}\\\
\end{array}\right],$ (6.10)
$\boldsymbol{L}_{2}^{(\boldsymbol{y})}=\left[\begin{array}[]{ccccc}y^{1}+(y^{3}-y^{4})y^{4}-y^{5}+1&\vrule\lx@intercol\hfil
0\hfil\lx@intercol&0&0&0\\\ \cline{1-2}\cr\lx@intercol\hfil
0\hfil\lx@intercol\vrule\lx@intercol&y^{1}+(y^{3}-y^{4})y^{4}-y^{5}-1&\vrule\lx@intercol\hfil
0\hfil\lx@intercol&0&0\\\ \cline{2-3}\cr 0&\lx@intercol\hfil
0\hfil\lx@intercol\vrule\lx@intercol&y^{3}-y^{4}-1&\vrule\lx@intercol\hfil
0\hfil\lx@intercol&0\\\ \cline{3-5}\cr 0&0&\lx@intercol\hfil
0\hfil\lx@intercol\vrule\lx@intercol&y^{4}&1\\\ 0&0&\lx@intercol\hfil
0\hfil\lx@intercol\vrule\lx@intercol&-\left(y^{3}-2y^{4}\right)^{2}&2y^{3}-3y^{4}\\\
\end{array}\right]\ ,$ (6.11)
$\boldsymbol{L}_{3}^{(\boldsymbol{y})}=\left[\begin{array}[]{ccccc}1&\vrule\lx@intercol\hfil
0\hfil\lx@intercol&0&0&0\\\ \cline{1-2}\cr\lx@intercol\hfil
0\hfil\lx@intercol\vrule\lx@intercol&-1&\vrule\lx@intercol\hfil
0\hfil\lx@intercol&0&0\\\ \cline{2-3}\cr 0&\lx@intercol\hfil
0\hfil\lx@intercol\vrule\lx@intercol&-\frac{1}{2}(y^{1}+y^{2})+y^{5}-1&\vrule\lx@intercol\hfil
0\hfil\lx@intercol&0\\\ \cline{3-5}\cr 0&0&\lx@intercol\hfil
0\hfil\lx@intercol\vrule\lx@intercol&-\frac{1}{2}(y^{1}+y^{2})-y^{3}+2y^{4}+y^{5}&1\\\
0&0&\lx@intercol\hfil
0\hfil\lx@intercol\vrule\lx@intercol&-\left(y^{3}-2y^{4}\right)^{2}&-\frac{1}{2}(y^{1}+y^{2})+y^{3}-2y^{4}+y^{5}\\\
\end{array}\right]\ .$

Observe that more generally, in these coordinates, the full class (6.1) takes
the block-diagonal form

(6.12)
$\boldsymbol{L}^{(\boldsymbol{y})}=\left[\begin{array}[]{ccccc}f_{1}+1&\vrule\lx@intercol\hfil
0\hfil\lx@intercol&0&0&0\\\ \cline{1-2}\cr\lx@intercol\hfil
0\hfil\lx@intercol\vrule\lx@intercol&f_{1}-1&\vrule\lx@intercol\hfil
0\hfil\lx@intercol&0&0\\\ \cline{2-3}\cr 0&\lx@intercol\hfil
0\hfil\lx@intercol\vrule\lx@intercol&f_{2}-1&\vrule\lx@intercol\hfil
0\hfil\lx@intercol&0\\\ \cline{3-5}\cr 0&0&\lx@intercol\hfil
0\hfil\lx@intercol\vrule\lx@intercol&f_{2}+f_{3}-(y^{3}-y^{4})f^{\prime}_{3}&1\\\
0&0&\lx@intercol\hfil
0\hfil\lx@intercol\vrule\lx@intercol&-\left(f_{3}-(y^{3}-y^{4})f^{\prime}_{3}\right)^{2}&f_{2}-f_{3}+(y^{3}-y^{4})f^{\prime}_{3}\\\
\end{array}\right],$

where
$f_{1}=f_{1}\left(\frac{1}{2}(y^{1}+y^{2}),y^{1}+(y^{3}-y^{4})f_{3}-y^{5}\right)$,
$f_{2}=f_{2}\left(y^{4},-\frac{1}{2}(y^{1}+y^{2})-(y^{3}-y^{4})f_{3}+y^{5},y^{3}-y^{4}\right)$
and $f_{3}=f_{3}(y^{4})$ are arbitrary functions (with $f_{3}^{\prime}\neq
0$).

##  7\. A level-four generalized Haantjes algebra

Let us consider the following fourth-level generalized Nijenhuis family of
operator fields defined over a seven-dimensional differentiable manifold $M$,
which is obtained with the same procedure described in the previous section:

(7.1)
$\boldsymbol{K}^{(\boldsymbol{x})}=\left[\begin{array}[]{ccccccc}g_{1}&x^{1}&x^{1}+g_{2}&-x^{1}&x^{1}&-x^{1}&x^{1}\\\
1&g_{5}&-g_{1}+g_{5}+x^{1}&g_{4}-g_{5}&0&g_{1}-g_{5}-x^{1}&-1-g_{4}+g_{5}\\\
0&0&g_{1}&0&0&0&0\\\
1+g_{1}-g_{4}&x^{1}&x^{1}+g_{2}&-1-x^{1}+g_{4}&x^{1}&-x^{1}&x^{1}\\\
0&g_{1}-g_{5}&g_{1}+g_{3}-g_{5}+\frac{1}{x^{1}}&-g_{1}+g_{5}&g_{1}&-g_{1}+g_{5}-\frac{1}{x^{1}}&g_{1}-g_{5}\\\
0&0&x^{1}&0&0&g_{1}-x^{1}&0\\\
g_{1}-g_{4}&x^{1}&x^{1}+g_{2}&-1-x^{1}&x^{1}&-x^{1}&1+x^{1}+g_{4}\\\
\end{array}\right].$

Here $g_{1}=g_{1}(x^{1},x^{2},x^{3},x^{4})$,
$g_{2}=g_{2}(x^{1},x^{2},x^{3},x^{4})$,
$g_{3}=g_{3}(x^{1},x^{2},x^{3},x^{4})$, $g_{4}=g_{4}(x^{5},x^{6})$,
$g_{5}=g_{5}(x^{7})$.

As a result of a direct computation, one can prove the following

######  Proposition 4.

The operator family (7.1) satisfies the property:

$\tau^{(4)}(\boldsymbol{K})=0\ .$

In addition, its third level torsion does not vanish whenever $g_{3}\neq 0$.

###  7.1. Spectral analysis

Its minimal polynomial is of seventh degree,

(7.2) $m_{\boldsymbol{K}}(\boldsymbol{x},\lambda)=(\lambda-g_{5})(\lambda-
g_{4}-1)(\lambda-g_{4}+1)(\lambda-g_{1}+x^{1})(\lambda-g_{1})^{3},$

and its eigenvalues and generalized eigenvectors are

(7.3) $\displaystyle\lambda_{1}=g_{5},\quad\rho_{1}=1$
$\displaystyle\Longrightarrow\mathcal{D}_{1}=\bigg{\langle}\dfrac{\partial}{\partial
x^{2}}-\dfrac{\partial}{\partial x^{5}}\bigg{\rangle},$
$\displaystyle\lambda_{2}=g_{4}+1,\quad\rho_{2}=1$
$\displaystyle\Longrightarrow\mathcal{D}_{2}=\bigg{\langle}\dfrac{\partial}{\partial
x^{2}}-\dfrac{\partial}{\partial x^{7}}\bigg{\rangle},$
$\displaystyle\lambda_{3}=g_{4}-1,\quad\rho_{3}=1$
$\displaystyle\Longrightarrow\mathcal{D}_{3}=\bigg{\langle}\dfrac{\partial}{\partial
x^{2}}+2\dfrac{\partial}{\partial x^{4}}+\dfrac{\partial}{\partial
x^{7}}\bigg{\rangle},$ $\displaystyle\lambda_{4}=g_{1}-x^{1},\quad\rho_{4}=1$
$\displaystyle\Longrightarrow\mathcal{D}_{4}=\bigg{\langle}\dfrac{\partial}{\partial
x^{1}}-(x^{1})^{2}\dfrac{\partial}{\partial x^{2}}+\dfrac{\partial}{\partial
x^{4}}-\dfrac{\partial}{\partial x^{5}}-(x^{1})^{2}\dfrac{\partial}{\partial
x^{6}}+\dfrac{\partial}{\partial x^{7}}\bigg{\rangle},$
$\displaystyle\lambda_{5}=g_{1},\quad\rho_{5}=3$
$\displaystyle\Longrightarrow\mathcal{D}_{5}=\bigg{\langle}\dfrac{\partial}{\partial
x^{1}}+\dfrac{\partial}{\partial x^{4}}+\dfrac{\partial}{\partial
x^{7}},\dfrac{\partial}{\partial x^{5}},\dfrac{\partial}{\partial
x^{3}}+\dfrac{\partial}{\partial x^{6}}\bigg{\rangle}.$

The characteristic distributions and the corresponding annihilators are

(7.4) $\displaystyle\mathcal{E}_{1}=\oplus_{i\neq
1}\mathcal{D}_{i}=\bigg{\langle}\dfrac{\partial}{\partial
x^{2}}-\dfrac{\partial}{\partial x^{7}},\dfrac{\partial}{\partial
x^{2}}+2\dfrac{\partial}{\partial x^{4}}+\dfrac{\partial}{\partial
x^{7}},\dfrac{\partial}{\partial x^{1}}-(x^{1})^{2}\dfrac{\partial}{\partial
x^{2}}+\dfrac{\partial}{\partial x^{4}}-\dfrac{\partial}{\partial
x^{5}}-(x^{1})^{2}\dfrac{\partial}{\partial x^{6}}+\dfrac{\partial}{\partial
x^{7}},$ $\displaystyle\hskip 256.0748pt\dfrac{\partial}{\partial
x^{1}}+\dfrac{\partial}{\partial x^{4}}+\dfrac{\partial}{\partial
x^{7}},\dfrac{\partial}{\partial x^{3}}+\dfrac{\partial}{\partial
x^{6}},\dfrac{\partial}{\partial x^{5}}\bigg{\rangle}$ $\displaystyle\hskip
56.9055pt\Longrightarrow\mathcal{E}_{1}^{\circ}=\langle
dx^{2}+dx^{3}-dx^{4}-dx^{6}+dx^{7}\rangle$
$\displaystyle\mathcal{E}_{2}=\oplus_{i\neq
2}D_{i}=\bigg{\langle}\dfrac{\partial}{\partial
x^{2}}-\dfrac{\partial}{\partial x^{5}},\dfrac{\partial}{\partial
x^{2}}+2\dfrac{\partial}{\partial x^{4}}+\dfrac{\partial}{\partial
x^{7}},\dfrac{\partial}{\partial x^{1}}-(x^{1})^{2}\dfrac{\partial}{\partial
x^{2}}+\dfrac{\partial}{\partial x^{4}}-\dfrac{\partial}{\partial
x^{5}}-(x^{1})^{2}\dfrac{\partial}{\partial x^{6}}+\dfrac{\partial}{\partial
x^{7}},$ $\displaystyle\hskip 256.0748pt\dfrac{\partial}{\partial
x^{1}}+\dfrac{\partial}{\partial x^{4}}+\dfrac{\partial}{\partial
x^{7}},\dfrac{\partial}{\partial x^{3}}+\dfrac{\partial}{\partial
x^{6}},\dfrac{\partial}{\partial x^{5}}\bigg{\rangle}$ $\displaystyle\hskip
56.9055pt\Longrightarrow\mathcal{E}_{2}^{\circ}=\langle
dx^{1}+dx^{4}-2dx^{7}\rangle$ $\displaystyle\mathcal{E}_{3}=\oplus_{i\neq
3}\mathcal{D}_{i}=\bigg{\langle}\dfrac{\partial}{\partial
x^{2}}-\dfrac{\partial}{\partial x^{5}},\dfrac{\partial}{\partial
x^{2}}-\dfrac{\partial}{\partial x^{7}},\dfrac{\partial}{\partial
x^{1}}-(x^{1})^{2}\dfrac{\partial}{\partial x^{2}}+\dfrac{\partial}{\partial
x^{4}}-\dfrac{\partial}{\partial x^{5}}-(x^{1})^{2}\dfrac{\partial}{\partial
x^{6}}+\dfrac{\partial}{\partial x^{7}},$ $\displaystyle\hskip
256.0748pt\dfrac{\partial}{\partial x^{1}}+\dfrac{\partial}{\partial
x^{4}}+\dfrac{\partial}{\partial x^{7}},\dfrac{\partial}{\partial
x^{3}}+\dfrac{\partial}{\partial x^{6}},\dfrac{\partial}{\partial
x^{5}}\bigg{\rangle}$ $\displaystyle\hskip
56.9055pt\Longrightarrow\mathcal{E}_{3}^{\circ}=\langle dx^{1}-dx^{4}\rangle,$
$\displaystyle\mathcal{E}_{4}=\oplus_{i\neq
4}\mathcal{D}_{i}=\bigg{\langle}\dfrac{\partial}{\partial
x^{2}}-\dfrac{\partial}{\partial x^{5}},\dfrac{\partial}{\partial
x^{2}}-\dfrac{\partial}{\partial x^{7}},\dfrac{\partial}{\partial
x^{2}}+2\dfrac{\partial}{\partial x^{4}}+\dfrac{\partial}{\partial
x^{7}},\dfrac{\partial}{\partial x^{1}}+\dfrac{\partial}{\partial
x^{4}}+\dfrac{\partial}{\partial x^{7}},\dfrac{\partial}{\partial
x^{3}}+\dfrac{\partial}{\partial x^{6}},\dfrac{\partial}{\partial
x^{5}}\bigg{\rangle}$ $\displaystyle\hskip
56.9055pt\Longrightarrow\mathcal{E}_{4}^{\circ}=\langle dx^{3}-dx^{6}\rangle$
$\displaystyle\mathcal{E}_{5}=\oplus_{i\neq
5}\mathcal{D}_{i}=\bigg{\langle}\dfrac{\partial}{\partial
x^{2}}-\dfrac{\partial}{\partial x^{5}},\dfrac{\partial}{\partial
x^{2}}-\dfrac{\partial}{\partial x^{7}},\dfrac{\partial}{\partial
x^{2}}+2\dfrac{\partial}{\partial x^{4}}+\dfrac{\partial}{\partial x^{7}},$
$\displaystyle\hskip 142.26378pt\dfrac{\partial}{\partial
x^{1}}-(x^{1})^{2}\dfrac{\partial}{\partial x^{2}}+\dfrac{\partial}{\partial
x^{4}}-\dfrac{\partial}{\partial x^{5}}-(x^{1})^{2}\dfrac{\partial}{\partial
x^{6}}+\dfrac{\partial}{\partial x^{7}}\bigg{\rangle}$ $\displaystyle\hskip
56.9055pt\Longrightarrow\mathcal{E}_{5}^{\circ}=\langle
dx^{3},(x^{1})^{2}dx^{1}+dx^{6},dx^{1}+dx^{2}-dx^{4}+dx^{5}-dx^{6}+dx^{7}\rangle;$

By specializing adequately the arbitrary functions, and maintaining the same
choice for the function $g_{3}$, we easily obtain an Abelian, cyclic and four-
level Haantjes algebra. As in the previous example, all the operators of the
family obtained in this way will share both the same eigen-distributions and
the degree of the minimal polynomial. Thus, any of them generates the full
algebra.

To illustrate the simultaneous diagonalization procedure, by way of an example
we can consider the three commuting operators

(7.5)
$\boldsymbol{K}_{1}^{(\boldsymbol{x})}=\left[\begin{array}[]{ccccccc}x^{2}&x^{1}&x^{1}&-x^{1}&x^{1}&-x^{1}&x^{1}\\\
1&0&x^{1}-x^{2}&0&0&-x^{1}+x^{2}&-1\\\ 0&0&x^{2}&0&0&0&0\\\
1+x^{2}&x^{1}&x^{1}&-1-x^{1}&x^{1}&-x^{1}&x^{1}\\\
0&x^{2}&x^{2}&-x^{2}&x^{2}&-\frac{1}{x^{1}}-x^{2}&x^{2}\\\
0&0&x^{1}&0&0&-x^{1}+x^{2}&0\\\
x^{2}&x^{1}&x^{1}&-1-x^{1}&x^{1}&-x^{1}&1+x^{1}\\\ \end{array}\right]\ ,$
(7.6)
$\boldsymbol{K}_{2}^{(\boldsymbol{x})}=\left[\begin{array}[]{ccccccc}x^{3}&x^{1}&-\frac{1}{x^{1}}&-x^{1}&x^{1}&-x^{1}&x^{1}\\\
1&x^{7}&x^{1}-x^{3}+x^{7}&x^{6}-x^{7}&0&-x^{1}+x^{3}-x^{7}&-1-x^{6}+x^{7}\\\
0&0&x^{3}&0&0&0&0\\\
1+x^{3}-x^{6}&x^{1}&-\frac{1}{x^{1}}&-1-x^{1}+x^{6}&x^{1}&-x^{1}&x^{1}\\\
0&x^{3}-x^{7}&x^{3}-x^{7}&-x^{3}+x^{7}&x^{3}&-\frac{1}{x^{1}}-x^{3}+x^{7}&x^{3}-x^{7}\\\
0&0&x^{1}&0&0&-x^{1}+x^{3}&0\\\
x^{3}-x^{6}&x^{1}&-\frac{1}{x^{1}}&-1-x^{1}&x^{1}&-x^{1}&1+x^{1}+x^{6}\\\
\end{array}\right]\ ,$ (7.7)
$\boldsymbol{K}_{3}^{(\boldsymbol{x})}=\left[\begin{array}[]{ccccccc}x^{4}&x^{1}&0&-x^{1}&x^{1}&-x^{1}&x^{1}\\\
1&1&1+x^{1}-x^{4}&-1+x^{5}&0&-1-x^{1}+x^{4}&-x^{5}\\\ 0&0&x^{4}&0&0&0&0\\\
1+x^{4}-x^{5}&x^{1}&0&-1-x^{1}+x^{5}&x^{1}&-x^{1}&x^{1}\\\
0&-1+x^{4}&-1+x^{4}&1-x^{4}&x^{4}&1-\frac{1}{x^{1}}-x^{4}&-1+x^{4}\\\
0&0&x^{1}&0&0&-x^{1}+x^{4}&0\\\
x^{4}-x^{5}&x^{1}&0&-1-x^{1}&x^{1}&-x^{1}&1+x^{1}+x^{5}\\\
\end{array}\right].$

These operators are $C^{\infty}(\mathbb{R})$-linearly independent. Each of
them generates cyclically a level-four Haantjes algebra $\mathscr{H}^{(4)}$ of
rank 7. A basis of this algebra is given, for instance, by
$\mathscr{B}=\\{\boldsymbol{I},\boldsymbol{K}_{1},\boldsymbol{K}_{1}^{2},\boldsymbol{K}_{1}^{3},\boldsymbol{K}_{1}^{4},\boldsymbol{K}_{1}^{5},\boldsymbol{K}_{1}^{6}\\}$.
In the following subsection, we shall determine a set of coordinates which
block-diagonalize the full algebra $\mathscr{H}^{(4)}$.

###  7.2. Block-diagonalization

By integrating the annihilators (7.4) of the eigendistributions of these
operators, we get the block-separating coordinates:

(7.8) $\displaystyle y^{1}=x^{2}+x^{3}-x^{4}-x^{6}+x^{7},$ $\displaystyle
y^{2}=x^{1}+x^{4}-2x^{7},$ $\displaystyle y^{3}=x^{1}-x^{4},$ $\displaystyle
y^{4}=x^{3}-x^{6},$ $\displaystyle y^{5}=x^{3},$ $\displaystyle
y^{6}=\frac{(x^{1})^{3}}{3}-x^{6},$ $\displaystyle
y^{7}=x^{1}+x^{2}-x^{4}+x^{5}-x^{6}+x^{7}.$

In these coordinates, all the operators of the algebra $\mathscr{H}^{(4)}$
block-diagonalize simultaneously. In particular, by defining
$\chi=\sqrt[3]{3(y^{4}-y^{5}+y^{6})}$, the operators $\boldsymbol{K}_{1}$,
$\boldsymbol{K}_{2}$, $\boldsymbol{K}_{3}$ take the form:

(7.9)
$\boldsymbol{K}_{1}^{(\boldsymbol{y})}=\left[\begin{array}[]{ccccccc}0&\vrule\lx@intercol\hfil
0\hfil\lx@intercol&0&0&0&0&0\\\ \cline{1-2}\cr\lx@intercol\hfil
0\hfil\lx@intercol\vrule\lx@intercol&1&\vrule\lx@intercol\hfil
0\hfil\lx@intercol&0&0&0&0\\\ \cline{2-3}\cr 0&\lx@intercol\hfil
0\hfil\lx@intercol\vrule\lx@intercol&-1&\vrule\lx@intercol\hfil
0\hfil\lx@intercol&0&0&0\\\ \cline{3-4}\cr 0&0&\lx@intercol\hfil
0\hfil\lx@intercol\vrule\lx@intercol&y^{1}+\frac{1}{2}(y^{2}-y^{3})-y^{4}-\chi&\vrule\lx@intercol\hfil
0\hfil\lx@intercol&0&0\\\ \cline{4-7}\cr 0&0&0&\lx@intercol\hfil
0\hfil\lx@intercol\vrule\lx@intercol&y^{1}+\frac{1}{2}(y^{2}-y^{3})-y^{4}&0&0\\\
0&0&0&\lx@intercol\hfil
0\hfil\lx@intercol\vrule\lx@intercol&\chi+\chi^{3}&y^{1}+\frac{1}{2}(y^{2}-y^{3})-y^{4}-\chi&\chi^{3}\\\
0&0&0&\lx@intercol\hfil
0\hfil\lx@intercol\vrule\lx@intercol&\chi&-\chi^{-1}&y^{1}+\frac{1}{2}(y^{2}-y^{3})-y^{4}+\chi\\\
\end{array}\right]\ ,$ (7.10)
$\boldsymbol{K}_{2}^{(\boldsymbol{y})}=\left[\begin{array}[]{ccccccc}-\frac{1}{2}(y^{2}+y^{3})+\chi&\vrule\lx@intercol\hfil
0\hfil\lx@intercol&0&0&0&0&0\\\ \cline{1-2}\cr\lx@intercol\hfil
0\hfil\lx@intercol\vrule\lx@intercol&1-y^{4}+y^{5}&\vrule\lx@intercol\hfil
0\hfil\lx@intercol&0&0&0&0\\\ \cline{2-3}\cr 0&\lx@intercol\hfil
0\hfil\lx@intercol\vrule\lx@intercol&-1-y^{4}+y^{5}&\vrule\lx@intercol\hfil
0\hfil\lx@intercol&0&0&0\\\ \cline{3-4}\cr 0&0&\lx@intercol\hfil
0\hfil\lx@intercol\vrule\lx@intercol&y^{5}-\chi&\vrule\lx@intercol\hfil
0\hfil\lx@intercol&0&0\\\ \cline{4-7}\cr 0&0&0&\lx@intercol\hfil
0\hfil\lx@intercol\vrule\lx@intercol&y^{5}&0&0\\\ 0&0&0&\lx@intercol\hfil
0\hfil\lx@intercol\vrule\lx@intercol&0&y^{5}-\chi&\chi^{3}\\\
0&0&0&\lx@intercol\hfil
0\hfil\lx@intercol\vrule\lx@intercol&-\chi^{-1}&-\chi^{-1}&y^{5}+\chi\\\
\end{array}\right]\ ,$ (7.11)
$\boldsymbol{K}_{3}^{(\boldsymbol{y})}=\left[\begin{array}[]{ccccccc}1&\vrule\lx@intercol\hfil
0\hfil\lx@intercol&0&0&0&0&0\\\ \cline{1-2}\cr\lx@intercol\hfil
0\hfil\lx@intercol\vrule\lx@intercol&1-y^{1}+y^{5}-\chi+y^{7}&\vrule\lx@intercol\hfil
0\hfil\lx@intercol&0&0&0&0\\\ \cline{2-3}\cr 0&\lx@intercol\hfil
0\hfil\lx@intercol\vrule\lx@intercol&-1-y^{1}+y^{5}-\chi+y^{7}&\vrule\lx@intercol\hfil
0\hfil\lx@intercol&0&0&0\\\ \cline{3-4}\cr 0&0&\lx@intercol\hfil
0\hfil\lx@intercol\vrule\lx@intercol&-y^{3}&\vrule\lx@intercol\hfil
0\hfil\lx@intercol&0&0\\\ \cline{4-7}\cr 0&0&0&\lx@intercol\hfil
0\hfil\lx@intercol\vrule\lx@intercol&-y^{3}+\chi&0&0\\\
0&0&0&\lx@intercol\hfil
0\hfil\lx@intercol\vrule\lx@intercol&\chi&-y^{3}&\chi^{3}\\\
0&0&0&\lx@intercol\hfil
0\hfil\lx@intercol\vrule\lx@intercol&0&-\chi^{-1}&-y^{3}+2\chi\\\
\end{array}\right].$

Also, in the block-separation coordinates, the full family (7.1) takes a block
diagonal form:

(7.12)
$\boldsymbol{K}^{(\boldsymbol{y})}=\left[\begin{array}[]{ccccccc}g_{5}&\vrule\lx@intercol\hfil
0\hfil\lx@intercol&0&0&0&0&0\\\ \cline{1-2}\cr\lx@intercol\hfil
0\hfil\lx@intercol\vrule\lx@intercol&g_{4}+1&\vrule\lx@intercol\hfil
0\hfil\lx@intercol&0&0&0&0\\\ \cline{2-3}\cr 0&\lx@intercol\hfil
0\hfil\lx@intercol\vrule\lx@intercol&g_{4}-1&\vrule\lx@intercol\hfil
0\hfil\lx@intercol&0&0&0\\\ \cline{3-4}\cr 0&0&\lx@intercol\hfil
0\hfil\lx@intercol\vrule\lx@intercol&g_{1}-\chi&\vrule\lx@intercol\hfil
0\hfil\lx@intercol&0&0\\\ \cline{4-7}\cr 0&0&0&\lx@intercol\hfil
0\hfil\lx@intercol\vrule\lx@intercol&g_{1}&0&0\\\ 0&0&0&\lx@intercol\hfil
0\hfil\lx@intercol\vrule\lx@intercol&g_{2}\chi^{2}+\chi\left(1+\chi^{2}\right)&g_{1}-\chi&\chi^{3}\\\
0&0&0&\lx@intercol\hfil
0\hfil\lx@intercol\vrule\lx@intercol&g_{3}+g_{2}+\chi^{-1}\left(1+\chi^{2}\right)&-\chi^{-1}&g_{1}+\chi\\\
\end{array}\right].$

Here $g_{1}=g_{1}(\chi,y^{1}+\frac{1}{2}(y^{2}-y^{3})-y^{4},y^{5})$,
$g_{2}=g_{2}(\chi,y^{1}+\frac{1}{2}(y^{2}-y^{3})-y^{4},y^{5})$,
$g_{3}=g_{3}(\chi,y^{1}+\frac{1}{2}(y^{2}-y^{3})-y^{4},y^{5})$,
$g_{4}=g_{4}(-y^{1}-\chi+y^{5}+y^{7},-y^{4}+y^{5})$ and
$g_{5}=g_{5}(-\frac{1}{2}(y^{2}+y^{3})+\chi)$ are arbitrary functions.

## Acknowledgement

This work has been partly supported by the research project
PGC2018-094898-B-I00, MICINN, Spain, and by the ICMAT Severo Ochoa project
SEV-2015-0554 (MICINN). D. R. N. acknowledges the financial support of EXINA
S.L.

P. T. and G. T. are members of the Gruppo Nazionale di Fisica Matematica
(GNFM).

## References

  * [1] O. I. Bogoyavlenskij. Necessary conditions for existence of non-degenerate Hamiltonian structures. Commun. Math. Phys. 182, 253–290 (1996).
  * [2] O.I. Bogoyavlenskij. General algebraic identities for the Nijenhuis and Haantjes torsions. Izvestya Mathematics 68, 1129-1141 (2004).
  * [3] O. I. Bogoyavlenskij. Block-diagonalizability problem for hydrodynamic type systems. J. Math. Phys. 47, paper 063502, 9 pages (2006).
  * [4] O. I. Bogoyavlenskij. Decoupling problem for systems of quasi-linear PDE’s. Commun. Math. Phys. 269, 545-556 (2007).
  * [5] A. V. Bolsinov, A. Yu. Konyaev, V. S. Matveev. Nijenhuis geometry. Adv. Math. 394, 108001 (2022).
  * [6] C. M. Chanu, G. Rastelli. Block-separation of variables: a form of partial separation for natural Hamiltonians. SIGMA, 15, paper 013, 22 pages (2019)
  * [7] R. Courant and D. Hilbert. Methods of Mathematical Physics, II, Interscience Publishers, New York, 1962.
  * [8] E. V. Ferapontov and D. G. Marshall. Differential-geometric approach to the integrability of hydrodynamic chains: the Haantjes tensor. Math. Annalen 339, 61–99 (2007).
  * [9] E. Ferapontov, K. Khusnutdinova. The Haantjes tensor and double waves for multi-dimensional systems of hydrodynamic type: a necessary condition for integrability. Proc. Royal Soc. A, 462, 1197-1219 (2006).
  * [10] E. V. Ferapontov and M. V. Pavlov. Kinetic equation for soliton gas: integrable reductions, arxiv:2109.11962v1 (2021).
  * [11] A. Frölicher, A. Nijenhuis. Theory of vector valued differential forms. Part I. Indag. Math. 18, 338–359 (1956).
  * [12] J. Haantjes. On $X_{n-1}$-forming sets of eigenvectors. Indag. Math. 17, 158–162 (1955).
  * [13] Y. Kosmann-Schwarzbach. Beyond recursion operators, Preprint arXiv:1712.08908, 2017, in Proceedings of the XXXVI Workshop on Geometric Methods in Physics, Białowieża, Poland, July 2017, Birkhauser, (2019).
  * [14] F. Magri. Haantjes manifolds with symmetries. Theor. Math. Phys., 196, 1217–1229 (2018).
  * [15] A. Newlander, L. Nirenberg. Complex analytic coordinates in almost complex manifolds. Ann. of Math. 65, 391–404 (1957).
  * [16] A. Nijenhuis. $X_{n-1}$-forming sets of eigenvectors. Indag. Math. 54, 200–212 (1951).
  * [17] A. Nijenhuis. Jacobi-type identities for bilinear differential concomitants of certain tensor fields. I, Indag. Math. 17, 390–397 (1955).
  * [18] A. Nijenhuis. Jacobi-type identities for bilinear differential concomitants of certain tensor fields. II. Indag. Math. 17, 398–403 (1955).
  * [19] D. Reyes Nozaleda, P. Tempesta and G. Tondo. Classical multiseparable Hamiltonian systems, superintegrability and Haantjes geometry. Commun. Nonl. Sci. Num. Sim. 104, 106021 (2022).
  * [20] P. Tempesta, G. Tondo. Haantjes Algebras and diagonalization (preprint, arXiv:1710.04522) J. Geom. Phys. 160, paper 103968, 21 pages (2021).
  * [21] P. Tempesta, G. Tondo. Haantjes manifolds and classical integrable systems. Ann. Mat. Pura Appl. 201, 57–90 (2022).
  * [22] P. Tempesta, G. Tondo. Higher Haantjes brackets and integrability. Commun. Math. Phys. 389, 1647–1671 (2022).
  * [23] G. Tondo, P. Tempesta. Haantjes structures for the Jacobi-Calogero model and the Benenti Systems. SIGMA 12, paper 023, 18 pages (2016).
  * [24] G. Tondo. Haantjes algebras of the Lagrange top. Theor. Math. Phys. 196, 1366–1379 (2018).


# Scroll ring chimera states in oscillatory networks

Volodymyr Maistrenko1, Oleksandr Sudakov1,2, Ievgen Sliusar1,2 1Scientific
Centre for Medical and Biotechnical Research, National Academy of Sciences of
Ukraine, Volodymyrska Str. 54, 01030 Kyiv, Ukraine
2 Taras Shevchenko National University of Kyiv, Volodymyrska Str. 60, 01030
Kyiv, Ukraine<EMAIL_ADDRESS>
###### Abstract

We report the appearance of a scroll ring and scroll toroid chimera states
from the proposed initial conditions for the Kuramoto model of coupled phase
oscillators in the 3D grid topology with inertia. The proposed initial
conditions provide an opportunity to obtain as single as well as multiple
scroll ring and toroid chimeras with different major and minor diameters. We
analyze their properties and demonstrate, in particular, the patterns of
coherent, partially coherent, and incoherent scroll ring chimera states with
different structures of filaments and chaotic oscillators. Those patterns can
coexist with solitary states and solitary patterns in the oscillatory
networks.

Keywords: scroll rings, chimera states, oscillatory networks

Scroll rings appear in the models of various real physical systems and are
among the most paradigmatic examples of spatio-temporal self-organizing
structures in excitable media. Scroll rings occur in three-dimensional media
and are formed from a scroll wave. In general, scroll rings are three-
dimensional spiral waves rotating around closed one-dimensional space curves.
A scroll wave is usually characterized by its filament that can be considered
as the line connecting the rotation centers of the spirals in the two-
dimensional cross-sections of the scroll wave. Then this filament can be
closed into a ring structure called the scroll ring [1].

Scroll rings have been observed in a variety of systems including the chemical
Belousov–Zhabotinsky reaction [2], in a fibrillating cardiac tissue [3], etc.
The existence of a scroll ring in the media of various nature was reported as
the results of numerous simulations and experiments in the fields of physics,
chemistry, biology, etc. (see, e.g., [4, 5, 6, 7, 8, 9, 10]).

Chimera states [11, 12, 13] as a phenomenon of coexistence of coherent and
incoherent patterns in the arrays of nonlocally coupled oscillators were
investigated in a wide range of systems. For these special spatio-temporal
patterns, some network’s elements oscillate synchronously with unique
frequency, and the others behave themselves asynchronously. A number of
articles are devoted to the theoretical and experimental studies of chimera
states. Most of them deal with the models of one- and two-dimensional networks
of oscillators [14].

Recently, the chimera states in a three-dimensional grid topology were
investigated within various models of coupled phase oscillators [15, 16, 17,
18, 20, 19, 21]. The first report of a stable scroll ring chimera state was
done in 2020 in [21] for the Kuramoto model of coupled phase oscillators in a
3D grid topology with inertia. This pattern was obtained from random initial
conditions for the following equation:

$\displaystyle
m\ddot{\varphi}_{ijk}+\epsilon\dot{\varphi}_{ijk}=\frac{\mu}{|B_{P}(i,j,k)|}\sum\limits_{(i^{\prime},j^{\prime},k^{\prime})\in
B_{P}(i,j,k)}\sin(\varphi_{i^{\prime}j^{\prime}k^{\prime}}-\varphi_{ijk}-\alpha),$
(1)

where $i,j,k=1,...,N$, $\varphi_{ijk}$ are phase variables, and the indices
$i,j,k$ are periodic modulo $N$. The coupling is assumed long-ranged and
isotropic: each oscillator $\varphi_{ijk}$ is coupled with equal strength
$\mu$ to all its nearest neighbors $\varphi_{i^{\prime}j^{\prime}k^{\prime}}$
in a ball of radius $P$, i.e., to those falling in the neighborhood

$B_{P}(i,j,k):=\\{(i^{\prime},j^{\prime},k^{\prime}){:}(i^{\prime}-i)^{2}+(j^{\prime}-j)^{2}+(k^{\prime}-k)^{2}\leq
P^{2}\\},$

where the distances $|i^{\prime}-i|$, $|j^{\prime}-j|$ and $|k^{\prime}-k|$
are calculated regarding the periodic boundary conditions of the network.
$|B_{P}(i,j,k)|$ denotes the cardinality of $B_{P}(i,j,k)$. The phase lag
parameter $\alpha$ is selected from the segment $[0,\pi/2]$. The relative
coupling radius $r=P/N$ varies from $1/N$ (local coupling) to $0.5$ (close to
the global coupling).

The parameter $\mu$ is the oscillator coupling strength, and $\epsilon$ is the
damping coefficient. The parameter $m$ is the mass. In the case $m=0,$ Eq. (1)
is transformed into the 3D Kuramoto model without inertia. We put $m=1$
without any loss of generality.

In the case of scroll ring chimera states, a filament that consists of
oscillators and connects scroll waves’ rotation centers has a shape of ring or
toroid with different major and minor diameters in the 3D oscillatory network.

In the present paper, we propose the initial conditions for the generation of
scroll ring chimera states in the Kuramoto model of coupled phase oscillators
in the 3D grid topology with inertia (1). They can generate a variety of
scroll ring and scroll toroid chimeras with different shapes and inner
structures of their filaments. We study the properties of scroll ring chimeras
and will demonstrate that they can be coherent, partially coherent, or
incoherent with a hyper-chaotic behavior characterized by a number of positive
Lyapunov exponents.

Scroll ring chimeras in the Kuramoto model (1) can be surrounded by solitary
oscillators and can coexist with other patterns in the 3D oscillatory network.
The solitary state behavior means that some number of oscillators start to
rotate with a different time-averaged frequency as compared to the
synchronized oscillators [22, 23, 24, 21].

Numerical simulations were performed on the base of the Runge–Kutta solver
DOPRI5 on the Ukrainian Grid Infrastructure with graphics processing units
[25, 26]. In total, more than 10000 network trajectories were computed and
analyzed for system (1) with $N=100$ and $200$ (1 and 8 millions of
oscillators, respectively).

To obtain the scroll ring chimeras in system (1), we propose two types of the
following initial conditions:

$\displaystyle\varphi_{xyz}=-\theta\exp{\left(-\frac{2r_{rot}}{D}\right)},$
(2)
$\displaystyle\varphi_{xyz}=-\theta\exp{\left(-\frac{|r_{xy}|+|2z-1|}{2D}\right)},$
(3) $\dot{\varphi}_{xyz}=\left\\{\begin{array}[]{ll}\omega_{max}\ \textrm{if}\
(2x-d_{x})^{2}+(2y-d_{y})^{2}+(2z-1)^{2}\leq d^{2},\\\ \omega_{min}\
\textrm{if}\
(2x-d_{x})^{2}+(2y-d_{y})^{2}+(2z-1)^{2}>d^{2},\end{array}\right.$

where

$\displaystyle r_{rot}=\sqrt{(r_{xy})^{2}+(z-0.5)^{2}},\
r_{xy}=(x-0.5)\cos(\psi)+(y-0.5)\sin(\psi)+\frac{D}{2},$ $\displaystyle
d_{x}=1+D\cos(\psi),d_{y}=1+D\sin(\psi),\
\theta={\textrm{atan2}}({z-0.5},r_{xy}),$
$\displaystyle\psi={\textrm{atan2}}({y-0.5,x-0.5}),\ x=i/N,y=j/N,z=k/N.$

The parameters $D,d\in(0,1]$. $x,y,z\in[1/N,1],\psi\in[-\pi,\pi]$.

Figure 1: Schematic view of the construction of initial conditions (a). Phase
cross-sections of initial conditions $\varphi_{xyz}$ modulo $2\pi$ along
y=0.5: (b) - type (2), (c) - type (3). Parameters $D=0.5,d=0.04,N=100$.

The initial conditions (2), (3) describe the rotation of the phase
$\varphi_{xyz}$ around all axes tangent to the circle with diameter $D$ in the
plane $z=0.5$. The schematic illustration of their construction is presented
in Fig. 1(a). To satisfy the boundary conditions, the phase exponentially
damps from the rotation center to the boundary. The fast damping to zero
provides the same phase at large distances from the rotation center. The
damping laws are different for (2) and (3). The analytic expression (2) yields
a slower damping in the $(x,y)$ plane and a faster damping in the $z$
direction. The analytic expression (3) describes the symmetric damping around
the rotation axis. A single scroll ring without phase damping satisfies
periodic boundary conditions in the plane $z=0.5$. The symmetric phase damping
(3) is considered as the most simple case of damping law for the scroll rings.
Figures 1(b) and 1(c) demonstrate the difference between the initial
conditions (2) and (3) by the presentation of the cross-sections of the phase
$\varphi_{xyz}$ modulo $2\pi$ along $y=0.5$ at the parameter values
$D=0.5,d=0.04,N=100$.

The value of the parameter $\omega_{max}$ is the maximal frequency of
chimera’s oscillators, $\omega_{min}$ is the frequency of the rest
synchronized oscillators for any chimera states (not necessarily a scroll
ring) obtained for fixed parameter values $\alpha,\epsilon,\mu,r,N$ of model
(1), which can be obtained from the random initial condition at these
parameter values. These parameters must be determined before starting the
simulation procedure.

Figure 2: Stability regions for scroll ring chimeras in the parameter planes
$(\alpha,\mu)$. Blue - scroll rings with the incoherent or partially coherent
inner part, red - coherent rings. Snapshots of typical scroll ring chimeras
and cross-sections of their phases $\varphi_{xyz}$ along y=0.5 are shown in
the insets. Parameters $\epsilon=0.05,r=0.01,N=200$.

In this way, the initial conditions (2), (3) guarantee the generation of
single scroll ring or toroid chimeras, as well as multiple scroll rings or
other scroll wave chimeras. The scroll toroid chimeras must have minor
diameter more than $1/N$.

Nevertheless, although the minor diameter of resulting scroll ring chimeras is
determined largely by the parameter $d$, the value of the relative coupling
radius $r=P/N$ has also a great influence on it.

Before starting the simulation of scroll ring and scroll toroid chimeras with
expected major and minor diameters, we must preliminary calculate the
parameter domains of model (1), where the patterns exist and are stable, by
using proposed initial conditions. Such regions for the scroll ring chimeras
in the parameter plane $(\alpha,\mu)$ at $\epsilon=0.05,r=0.01,N=200$ is
presented in Fig. 2. These regions are similar to the stability regions for
scroll ring chimeras obtained from the random initial conditions, but for
another value of $r=0.04$ (see Fig. 3(a)) in [21]).

Our simulation shows that the scroll ring chimera states exist for any
infinitely small coupling strength $\mu>0$. Crossing the left and left bottom
sides of the scroll ring stability region, all oscillators are synchronized.
After crossing the right side of the region, the rings are destroyed with the
generation of the chaotic oscillatory mode or their total synchronization.

The examples of typical scroll ring chimeras and their cross-sections in the
insets of Fig. 2 at parameter point a) - c) demonstrate that the scroll waves
are more densely twisted around the scroll ring chimeras with decreasing the
parameter $\mu$.

Figure 3:  Phase snapshots of scroll ring and scroll toroid chimeras obtained
from the initial conditions (2): (a) - scroll ring chimera
($\mu=0.016,r=0.01,d=0.005$), (b) - scroll toroid chimera
($\mu=0.01,r=0.03,d=0.08$). Common parameters
$\alpha=0.4,\epsilon=0.05,D=0.5,N=200$. Simulation time $t=10^{5}$. Half of
the wave has been removed to permit the view of the scroll chimeras.
Coordinates $x=i/N,y=j/N,z=k/N$. The space–time dynamics of the scroll rind
and scroll toroid chimera states are illustrated by videos in supplemental
data.

Figure 3 illustrates the results of simulation with the use of the initial
condition (2) for scroll ring chimera (a) and for scroll toroid chimera (b)
with major diameter of $0.5$. In the case of scroll toroid chimera, its minor
diameter is far more than $1/N$, which clearly seen in Fig. 3(b). In
particular, the projection of the pattern onto the $(x,y)$ plane confirms it.
Here, we take $\omega_{max}=0.06$, $\omega_{min}=-0.19$. In these figures, a
half of the wave has been removed to permit one to better see the chimeras.

Figure 4: Examples of coherent, partially coherent, and incoherent scroll ring
chimeras. Phase snapshots $\varphi_{xyz}$ (upper row), their phase cross-
sections along y=0.5, the average frequencies $\bar{\omega}_{xyz}$ (next two
rows), ordered oscillator index (bottom row): (a) - coherent scroll ring
chimera ($\alpha=0.35,\mu=0.02$), (b) - partially coherent scroll ring chimera
($\alpha=0.4,\mu=0.01$); (c) - incoherent scroll ring chimera
($\alpha=0.5,\mu=0.007$). Common parameters $r=0.01,\epsilon=0.05,N=200$.
Simulation time $t=5\times 10^{4}$, frequency averaging interval $\Delta
T=10^{4}$.

Scroll ring chimeras can have different inner structures of filaments and
different dynamics of their oscillators that are characterized by the time-
averaged frequencies $\bar{\omega}_{xyz}$ of their oscillators. The examples
of scroll ring chimeras with coherent, partially coherent, and incoherent
inner parts with the major diameter of 0.5 are presented in Fig. 4. Location
of their parameter values in the parameter plane $(\alpha,\mu)$ for these
examples are indicated by black points a) - c) in Fig. 2.

In the case of a completely coherent scroll ring (Fig. 4(a)) all oscillators
of the scroll ring chimera rotate with the same time-averaged frequency
$\bar{\omega}_{xyz}$. The scroll ring chimera with partially coherent
oscillatory organization has a segments of the same average frequency of
scroll rings oscillators (green color) (Fig. 4(b)). Its ordered oscillator
index $I$ has a long band for oscillators with the same average frequency
$\bar{\omega}_{xyz}$. In the case of a incoherent scroll ring chimera, its
average frequencies profile is chaotic (Fig. 4(c)). The dynamical complexity
of the chimera state can be characterized by a number of positive Lyapunov
exponents.

Lyapunov exponents $\lambda_{n}$ characterize how the initial perturbations
behave themselves along the whole trajectory and are defined as the
eigenvalues of a matrix

$\Lambda=\lim\limits_{t\to\infty}\frac{1}{2t}\log(Y(t)Y^{T}(t)),$

where the matrix $Y$ is a solution of the linearized differential equation
$\dot{Y}(t)=J(t)Y(t)$ with the identity matrix as initial conditions, and
$J(t)$ is a Jacobi matrix of system (1). Lyapunov exponents were estimated
using the QR approach [27].

We suggest that, in the considered example (Fig. 4(c)), the scroll ring
chimera has hyper-chaotic character with the huge number, more than one
hundred, of positive Lyapunov exponents.

Figure 5:  Lyapunov spectrum for 20 largest Lyapunov exponents (a) of the
scroll ring chimera (b) and its average frequencies $\bar{\omega}_{xyz}$ (c)
for the parameters
$\alpha=0.45,\mu=0.01,\epsilon=0.05,r=0.02,D=0.5,d=0.01,N=100$. Simulation
time $t=5\times 10^{4}$, frequency averaging interval $\Delta T=10^{4}$.

Figure 6:  Examples of scroll ring chimeras generated by the initial
conditions (2), (3). Phase distributions $\varphi_{xyz}$ and their cross-
sections along $y=0.5$: (a) - scroll ring with a major diameter of $0.3$
($r=0.02,D=0.3,d=0.01,N=100$), (b) - scroll ring with a major diameter of
$0.7$ ($r=0.02,D=0.7,d=0.01,N=100$), (c) - double concentric scroll rings
($r=0.02,D=0.9,d=0.005,N=200$), (d) - double scroll rings with wave profile
($r=0.03,D=0.88,d=0.1,N=100$), (e) - double scroll rings with wave profile
($r=0.03,D=0.58,d=0.2,N=100$), (f) - three scroll rings
($r=0.03,D=0.66,d=0.2,N=100$), (g) - three scroll rings
($r=0.03,D=0.64,d=0.22,N=100$), (h) - three scroll rings with wave profile
$(r=0.03,D=0.74,d=0.24,N=100$). Common parameters
$\alpha=0.38,\mu=0.02,\epsilon=0.05$. Simulation time $t=5\times 10^{4}$.
Coordinates $x=i/N,y=j/N,z=k/N$.

Figure 5 illustrates the Lyapunov spectrum of 20 largest Lyapunov exponents
(a) of the scroll ring chimera (b) and its average frequencies
$\bar{\omega}_{xyz}$ (c). At the time $t\approx 3\times 10^{4}$, all 20
largest Lyapunov exponents become positive, confirming the hyper-chaotic
character of this scroll ring chimera state.

We believe that if system (1) with dimension $N$ increases, then it will be
possible, using the initial conditions (2) and (3), to obtain different scroll
ring chimeras including such ones with a stepwise profile with the average
frequencies $\bar{\omega}_{xyz}$ similarly to the structure of spiral cores in
the 2D Kuramoto model with inertia [24].

The initial conditions (2) and (3) can generate single scroll ring or scroll
toroid chimeras, as well as multiple scroll pattens with different major and
minor diameters. Some filaments of them are transformed into polygons or can
have a wave profile in the $z$ direction.

In Fig. 6, a few examples of scroll ring chimeras, as a result of the
simulation with the use of the initial conditions (2) and (3), are presented
for the fixed parameters $\alpha=0.38,\mu=0.02,\epsilon=0.05$. The initial
conditions are taken in the form of (2) for Fig. 6(a, b, e, h) and in the form
of (3) for Fig. 6(c, d, f, g).

We note that the proposed initial conditions (2), (3) can be used for any
values of $N$ of system (1). We present the results of their application in
the cases of $N=100$ and $N=200$ in Fig. 6(a, b, d-h) and Fig. 6(c),
respectively.

In addition to the example of a scroll ring chimera with the major diameter of
0.5 (Fig 3(a)), the examples of scroll ring chimeras with the major diameter
of 0.3 and 0.7 are presented in Fig. 6(a) ($D=0.3,d=0.01$) and Fig. 6(b)
($D=0.7,d=0.01$), respectively.

The example of a multiple scroll ring chimera obtained from the proposed
initial conditions is given in Fig. 6(c-h): (c) - double concentric scroll
rings ($D=0.9,d=0.005$), (d) - double scroll rings with wave profile
($D=0.88,d=0.1$), (e) - double scroll rings with wave profile
($D=0.58,d=0.2$), (f) - three scroll rings ($D=0.66,d=0.2$), (g) - three
scroll rings with polygon profile ($D=0.64,d=0.22$), (h) - three scroll rings
with wave profile ($D=0.74,d=0.24$). Multiple scroll pattens in the form of
polygons or with wave profile can be generated by initial conditions (2), (3)
with parameter $d>0.09$. So, using the initial conditions (2), (3) with
different parameters $D$ and $d$, we can obtain as many multiple scroll wave
ring and toroid chimeras as we simulate.

Our simulation shows that all scroll ring chimeras obtained in this way are
stable with respect to perturbations of the phase variables $\varphi_{xyz}$
and frequencies $\omega_{xyz}$ by a uniformly distributed noise. The perturbed
scroll ring chimeras still exist and retain their shapes under a perturbation
of the amplitude less than 0.5 at the parameters
$\alpha=0.38,r=0.02,\mu=0.1,\epsilon=0.05$, and $N=200$. Stronger
perturbations lead to changing the shapes of scroll ring chimeras, their
destruction with complete oscillatory synchronization, or the creation of
different types of scroll wave chimera states. Moreover, if the disturbed
scroll ring lies in the parameter region, where solitary states exist, the
perturbation can give rise to another scroll ring chimera with solitary clouds
[21]. Due to these properties, we can build complex patterns in the system
(1), for example, consisting of scroll ring chimeras and other patterns.

Figure 7: Phase snapshot of the 3D image of the Eiffel Tower with a scroll
ring chimera. The parameters
$\alpha=0.4,r=0.04,\mu=0.1,\epsilon=0.05,D=0.5,d=0.005,N=200$. A half of the
wave has been removed to permit the view of the scroll ring chimera and the
Eiffel Tower image. The space–time dynamics of the chimera states are
illustrated by video in supplemental data.

Finally, to illustrate the possibility of the coexistence of the scroll rings
with other patterns in the 3D oscillatory network, we present the Eiffel tower
image surrounded by scroll ring chimeras in Fig. 7. The initial conditions for
the patterns were built, by using the Eiffel tower solitary oscillators and
the initial conditions (2) for a scroll ring chimera with the parameters
$D=0.5$ and $d=0.005$ placed in a 3D cube at $z=0.6$. Distribution of scroll
ring chimera oscillators was taken as the base for initial conditions. Then
the model of Eiffel tower for the 3D printing was used as a spatial template
for the placement of solitary oscillators inside a scroll ring chimera. Model
for 3D printing contains spatial coordinates of voxels that should be filled
with plastic. These voxel coordinates were scaled to the size of the scroll
ring chimera system and used for placement of solitary oscillators inside a
system with a scroll ring chimera. The average frequency of solitary
oscillators was selected to be close to the average frequency of scroll ring
oscillators. Due to properties of solitary states [21], there is no
interference between solitary oscillators of the Eiffel tower and the
oscillators of a scroll ring chimera after several hundreds time units
transient. This image is a stable solution of the 3D Kuramoto model with
inertia (1).

The scroll ring has a ring-like filament and is related to a scroll wave and a
toroidal vortex/swirl as a special case. Scroll rings also appear in a variety
of experimentally observed effects and as solutions of the equations of
realistic mathematical models. Equations similar to (1) appear in different
problems of hydrodynamics, aerodynamics, and plasma dynamics and are related
to flows, flames, superfluidity etc. (see, e.g., [28, 29, 30]). The
fibrillation of cardiac tissue is described by scroll ring dynamics, and some
chemical reactions lead to the propagation of scroll ring reagents’
concentration waves. The appearance of scroll rings and other related
phenomena are sometimes considered as harmful (flow turbulence, fibrillation)
and, sometimes, as useful (combustion chambers, chemical reactions). Knowledge
of conditions, where scroll rings can and cannot exist, may be inculcated in
practice: design of new devices, therapies, technologies.

## Acknowledgments

The authors are grateful to the Ukrainian Grid Infrastructure for providing
the computing cluster resources and the parallel and distributed software used
during this work.

## References

## References

  * [1] A. Winfree. Science 181, pp. 937–939, (1973).
  * [2] A. Winfree. Science 175, pp. 634–636, (1972).
  * [3] A. Medvinsky, A. Panfilov, A. Pertsov. Springer, Heidelberg, p. 195, (1984).
  * [4] A. Winfree, S. Strogatz. Physica D, pp. 35–49, (1983).
  * [5] A. Panfilov, A. Rudenko, A. N. Physica D, 28, 215, (1987).
  * [6] A. Winfree. The Geometry of Biological Time, Springer, Berlin, (2001).
  * [7] E. Cherry, F. Fenton. New J. Phys. 10 125016 (2008).
  * [8] Z. Qua, G. Hub, A, Garfinkelac, J. Weissad. Physics Reports 543, Issue 2, 10, pp. 61–162, (2014).
  * [9] V. Biktashev, I. Biktasheva. Engineering of Chemical Complexity II, pp. 221 –- 238, (2014).
  * [10] J. Totz, H. Engel, O. Steinbock. New J. Phys. 17, 093043, (2015).
  * [11] Y. Kuramoto. Nonl. Dyn. and Chaos, CRC Press, pp. 209 – 227, (2002).
  * [12] Y. Kuramoto, D. Battogtokh. Nonlinear Phenom. Complex Syst. 5, pp. 380 – 385, (2002).
  * [13] D. Abrams, S. Strogatz. Phys. Rev. Lett., 93, 174102, (2004).
  * [14] F. Parastesh, S. Jafari, H. Azarnoush, Z. Shahriari, Z. Wang, S. Boccaletti, M. Perc. Physics Reports, 898, pp. 1–114 (2021).
  * [15] Yu. Maistrenko, O. Sudakov, O. Osiv, V. Maistrenko. New Journal of Physics, 17, 073037, (2015).
  * [16] H. Lau, J. Davidsen. Phys. Rev.E, 94, 010204(R), (2016).
  * [17] V. Maistrenko, O. Sudakov, O. Osiv, Yu. Maistrenko. EPJ ST, V. 226, Issue 9, (2017).
  * [18] T. Kasimatis, J. Hizanidis, A. Provata. Phys. Rev. E 97, 052213, (2018).
  * [19] S. Kundu, B.K. Bera, B, D. Ghosh, M. Lakshmanan. Phys. Rev. E, 99(2), (2019).
  * [20] O.Omel’chenko, E.Knobloch. New Journal of Physics, 21, (2019).
  * [21] V. Maistrenko, O. Sudakov, O. Osiv. Chaos 30, 063113 (2020).
  * [22] P. Jaros, Yu. Maistrenko, T. Kapitaniak. Phys. Rev. E 91, 022907 (2015).
  * [23] P. Jaros, S. Brezetsky, R. Levchenko, D. Dudkowski, T. Kapitaniak, Yu. Maistrenko. Chaos 28, 011103 (2018).
  * [24] V. Maistrenko, O. Sudakov, Yu. Maistrenko. Eur. Phys. J. Spec. Top. 229, 2327–2340 (2020).
  * [25] A. Salnikov, R. Levchenko, O. Sudakov. Proc. 6th IEEE (IDAACS), pp. 198 – 202 (2011).
  * [26] O. Sudakov, A. Cherederchuk, V. Maistrenko. Proc. 9th IEEE (IDAACS), pp. 311 - 316 (2017).
  * [27] L. Dieci, M. Jolly, E. Vleck. J. Comput. Nonlinear Dynam. Jan 2011, 6(1), 011003 (2011).
  * [28] D. Dennis, C. Seraudie, R. Poole. Phys. of fluids 26, 053602 (2014).
  * [29] Y. Huang, V. Yang, V. Proc. Combustion Institute 30, 1775-1782 (2005).
  * [30] N. Guenther, P. Massignan, A. Fetter. Phys. Rev. A, 101, 053606 (2020).