[current]500pt680pt

Racah Algebra $R(n)$ from Coalgebraic Structures and Chains of $R(3)$
Substructures

Danilo Latini, Ian Marquette and Yao-Zhong Zhang

School of Mathematics and Physics, The University of Queensland, Brisbane, QLD
4072, Australia

abstract

###### Abstract

The recent interest in the study of higher-rank polynomial algebras related to
$n$-dimensional classical and quantum superintegrable systems with coalgebra
symmetry and their connection with the generalised Racah algebra $R(n)$, a
higher-rank generalisation of the rank one Racah algebra $R(3)$, raises the
problem of understanding the role played by the $n-2$ quadratic subalgebras
generated by the left and right Casimir invariants (sometimes referred as
_universal quadratic substructures_) from this new perspective. Such
subalgebra structures play a signficant role in the algebraic derivation of
spectrum of quantum superintegrable systems. In this work, we tackle this
problem and show that the above quadratic subalgebra structures can be
understood, at a fixed $n>3$, as the images of $n-2$ injective morphisms of
$R(3)$ into $R(n)$. We show that each of the $n-2$ quadratic subalgebras is
isomorphic to the rank one Racah algebra $R(3)$. As a byproduct, we also
obtain an equivalent presentation for the universal quadratic subtructures
generated by the partial Casimir invariants of the coalgebra. The
construction, which relies on explicit (symplectic or differential)
realisations of the generators, is performed in both the classical and the
quantum cases.

##  1 Introduction

Superintegrable systems represent an exceptional subset of integrable
Hamiltonian systems. For a classical Hamiltonian system
$H=H(x_{1},\dots,x_{n},p_{1},\dots,p_{n})$ with $n$ degrees of freedom, the
notion of integrability requires the existence of $n$ integrals of motion,
i.e. $n$ well-defined functionally independent functions on the phase-space,
say $\\{X_{1},\dots,X_{n}\\}$, that Poisson commute with the Hamiltonian
$X_{1}:=H$, i.e. $\\{X_{1},X_{i}\\}=0$ for $i=1,\dots,n$. Moreover, the above
constants have to be in involution, so that $\\{X_{i},X_{j}\\}=0$ for $1\leq
i,j\leq n$. The quantum counterpart of these models also exists. In this case,
it is required the existence of $n$ “quantum integrals of motion”, i.e. $n$
well-defined algebraically independent Hermitian operators
$\\{\hat{X}_{1},\dots,\hat{X}_{n}\\}$ on a given Hilbert space, which commute
with the quantum Hamiltonian
$\hat{H}=\hat{H}(\hat{x}_{1},\dots,\hat{x}_{n},\hat{p}_{1},\dots,\hat{p}_{n})$,
i.e. $[\hat{X}_{1},\hat{X}_{i}]=0$, with $\hat{X}_{1}:=\hat{H}$. Moreover,
they have to commute pair-wise, meaning that $[\hat{X}_{i},\hat{X}_{j}]=0$ for
$1\leq i,j\leq n$.

When $k$ functionally/algebraically independent additional classical/quantum
integrals of motion arise, i.e. there exists another set
$\\{Y_{1},\dots,Y_{k}\\}$/$\\{\hat{Y}_{1},\dots,\hat{Y}_{k}\\}$, with $1\leq
k\leq n-1$, such as $\\{Y_{i},H\\}=0$/$[\hat{Y}_{i},\hat{H}]=0$ for
$i=1,\dots,k$ we come to the definition of classical/quantum
superintegrability. In particular, for $k=1$ we deal with _minimal
superintegrability_ whereas if $k=n-1$ with _maximal superintegrability_
111Here, and throughout the paper, we shall assume the classical/quantum
integrals to be polynomials in the momenta/finite-order differential
operators, so that we are implicitly defining, following the terminology used
in [1], polynomial superintegrability/superintegrability of finite-order. In
this perspective, the order as a polynomial in the momenta/as a linear
differential operator defines the order of the classical/quantum integral of
motion.. It is also common to call _quasi-maximally superintegrable_ those
systems for which just one classical/quantum integral is missing, so that one
has a total number of $2n-2$ functionally/algebraically independent
classical/quantum integrals (the Hamiltonian included). In the maximally
superintegrable case, it is not difficult to imagine that the existence of
such a large number of symmetries for a given Hamiltonian system will make it
very special, as many unique properties will arise. For example, finite closed
trajectories and periodic motion in classical mechanics or accidental
degeneracy of the energy spectrum in quantum mechanics, just to cite a few of
them. For a state-of-the-art perspective on superintegrability we refer the
reader to [1] where an exhaustive description (mostly focused on the algebraic
side) of superintegrable models and their properties can be found, also in
connection to special functions theory, together with several explicit
examples. For a more geometrical point of view we refer the reader to [2].

From the algebraic perspective, the integrals of motion of a $n$D
superintegrable system usually close to higher rank polynomial algebras (see
for example [3, 4, 5, 6, 7, 8]). This is contrast to the purely integrable
case for which all the integrals of motion just define abelian algebras. The
richness of unique properties these classical and quantum models show
justified the beginning of their classification [9, 10, 11, 12, 13].
Remarkably, for the two-dimensional case, it has been proved that all second-
order superintegrable systems can be obtained from a unique model known as
_the generic superintegrable system on the 2-sphere_ [14]. A higher
dimensional extension of this model, _the generic superintergrable system on
the $(n-1)$-sphere_, has been recently studied in a series of papers (see [15,
16, 17, 18, 19] and references therein) devoted to the so-called generalised
Racah algebra $R(n)$, a higher rank generalisation of the rank one Racah
algebra $R(3)$. Interestingly, the authors proved that the above algebraic
structure is naturally realised as the symmetry algebra of the above-mentioned
superintegrable model.

In a recent paper [20], we have shown that the above algebraic structure also
appears inside a larger symmetry algebra (with additional generators besides
the ones associated to the Racah sector) related to two specific Euclidean
models with non-central terms, the Smorodinsky-Winternitz system [21, 22, 23,
24, 25, 26] and the generalized Kepler-Coulomb system [27, 28, 29]. More
generally, it is realised in classical/quantum mechanics as the quadratic
Poisson/associative algebra of classical/quantum integrals of motion arising
for a quasi-maximally superintegrable family of $n$D Euclidean
classical/quantum Hamiltonian systems characterised by a radial potential
$V=V(r)$ and $n$ additional non-central terms of the type $a_{i}/x_{i}^{2}$
($i=1,\dots,n$) which break the radial symmetry. Such a family of Hamiltonians
is included in the subset of $n$D superintegrable systems that are endowed
with an $\mathfrak{sl}(2,\mathbb{R})$ _coalgebra symmetry_ [30, 31, 32]. For
these models, two sets composed by $n-1$ functionally independent integrals of
motion can be extracted from the so-called “left”and “right”coproducts of the
non-linear Casimir of the $\mathfrak{sl}(2,\mathbb{R})$ coalgebra. One of the
integrals is in common, so that one has a total number of $(n-1)+(n-1)-1=2n-3$
classical/quantum integrals of motion besides the Hamiltonian (one constant
left for maximal superintegrability). These classical/quantum integrals, which
arise under a given symplectic/differential realisation, turn out to be
functionally/algebraically independent by construction. Such an approach to
superintegrability, which holds for both classical and quantum systems because
of its algebraic nature, has found application to a variety of problems, and
many superintegrable systems have been understood in this framework, such as
for example systems defined on non Euclidean spaces [33, 34, 35, 36, 37],
models with spin-orbital interactions [38] and discrete quantum mechanical
models [39].

A more recent application, deeply connected to the aim of the present work,
can be found in [40]. In that work, the chain structure of quadratic algebras
with three generators discovered in [41] has been rephrased in the framework
of coalgebra symmetry. Such algebraic structure has been used to construct
algebraically the energy spectrum of a second-order $n$D Euclidean maximally
superintegrable system with $n-1$ non-central terms (a quasi-generalised KC
system) through deformed oscillator realisations of the chain structures and
set of cubic Casimirs. Such idea of substructures was originally introduced
for a three-dimensional system, the nondegenerate (sometimes called
generalised) Kepler-Coulomb model [29]. The link with coalgebra symmetry led
to the identification of the generators of part of the above chain of
quadratic algebras ($n-2$ of the $n-1$ quadratic substructures forming the
entire chain) with the left and right Casimir invariants of the coalgebra. As
a consequence, the result has been generalised to all $n$D superintegrable
models (both classical and quantum) possessing the same
$\mathfrak{sl}(2,\mathbb{R})$ coalgebra symmetry. For this reason, we referred
to the above chain of quadratic algebras as being _universal_. The full chain
of $n-1$ quadratic algebras used in [41] to construct the spectrum of the $n$D
model algebraically also include an additional constant besides the left and
right Casimirs. This is related to the specific model investigated. In fact,
it was second-order maximally superintegrable and, because of this, admitted
an additional integral of motion (a generalisation of the Laplace-Runge-Lenz
(LRL) vector [42, 43, 44]). However, here we are only interested to
investigate, at a fixed $n\geq 3$, the $n-2$ quadratic structures generated by
the above left and right constants. Additional commutation relations appear
when maximally superintegrable subcases are considered.

The connection observed between the coalgebra symmetry and the substructures
needs to be further explained. In this paper we plan to connect explicitly
these quadratic chain structures of three generators to the Racah algebra.

Specifically, the aim of this work is to show that the above quadratic
substructures can be understood as being the subalgebras of $R(n)$ arising as
the images of $n-2$ injective morphisms of $R(3)$ into $R(n)$ and that, as a
direct consequence, they turn out to be isomorphic to (a classical/quantum
realisation of) the rank one Racah algebra $R(3)$, thus elucidating the deeper
algebraic meaning behind the above substructures. We organize the paper as
follows:

  * •

In Section 2 we recall briefly the general idea behind the coalgebra symmetry
approach to superintegrable systems and, in particular, we focus on the
definition of the left and right Casimir invariants. This is in order to
introduce some background that will be useful to follow the developments of
the work. We present the construction for Poisson coalgebras (the extension to
noncommutative coalgebras is direct).

  * •

In Section 3 we specialise to the Lie-Poisson (co)algebra
$\mathfrak{sl}(2,\mathbb{R})$ endowed with the (primitive) coassociative
coproduct $\Delta$. After obtaining the corresponding left and right Casimir
invariants, we elucidate their connection with the generators of (a Poisson
analog) of the generalised Racah algebra $R(n)$ [45, 20]. Finally, we proceed
towards the main goal of the work and show that the quadratic substructures
mentioned above can in fact be interpreted as a classical (Poisson)
realisation of some specific Racah subalgebras of rank one. The latter can be
understood, at a fixed $n>3$, as the result of the application of a family of
$n-2$ injective morphisms of $R(3)$ into $R(n)$. As a consequence, we show
that each of the $n-2$ quadratic subalgebras is isomorphic to a Poisson analog
of the rank one Racah algebra $R(3)$. We conclude the Section by providing the
classical Casimir functions associated to each of the $n-2$ classical
quadratic algebras.

  * •

In Section 4 we deal with the quantum case. The starting point, to construct
the left and right quantum integrals, is the Lie algebra
$\mathfrak{sl}(2,\mathbb{R})$ endowed with the (primitive) coassociative
coproduct $\Delta$. In this case, the realisation is given in terms of linear
differential operators. We rely on injective morphisms to establish the
explicit commutation relations of these substructures and extend the results
to the quantum case. In particular, we construct the quantum analog of the
substructures and show that they give rise to a quantum realisation of the
rank one embedded Racah algebras. Again, we conclude the Section by providing
the quantum Casimir operators associated to each of the above-mentioned
quadratic algebras. We mention that the classical case can be obtained from
the quantum case by performing the appropriate $\hbar\to 0$ limit, so that all
the formulas appearing in the paper can be easily compared.

  * •

Section 5 is devoted to the concluding remarks.

##  2 Coalgebra symmetry approach to superintegrability: left and right
Casimir invariants

We start by recalling some definitions following [30, 31, 32].

######  Definition 2.1.

A coalgebra $(\mathfrak{A},\Delta)$ is a (unital, associative) algebra
$\mathfrak{A}$ endowed with a coproduct map:

$\Delta:\mathfrak{A}\to\mathfrak{A}\otimes\mathfrak{A}$

which is coassociative:

$(\Delta\otimes\text{id})\circ\Delta=(\text{id}\otimes\Delta)\circ\Delta\,$

meaning that the following diagram is commutative:

${\mathfrak{A}}$${\mathfrak{A}\otimes\mathfrak{A}}$${\mathfrak{A}\otimes\mathfrak{A}}$${\mathfrak{A}\otimes\mathfrak{A}\otimes\mathfrak{A}}$$\scriptstyle{\Delta}$$\scriptstyle{\Delta}$$\scriptstyle{\Delta\otimes\text{id}}$$\scriptstyle{\text{id}\otimes\Delta}$

and it is an algebra homomorphism from $\mathfrak{A}$ to
$\mathfrak{A}\otimes\mathfrak{A}$:

$\Delta(X\cdot Y)=\Delta(X)\cdot\Delta(Y)\qquad\forall\,X,Y\in\mathfrak{A}\,.$

######  Definition 2.2.

$(\mathfrak{A},\Delta)$ is a Poisson coalgebra if $\mathfrak{A}$ is a Poisson
algebra and $\Delta$ is a Poisson homomorphism between $\mathfrak{A}$ and
$\mathfrak{A}\otimes\mathfrak{A}$, i.e.:

$\Delta\bigl{(}\\{X,Y\\}_{\mathfrak{A}}\bigl{)}=\\{\Delta(X),\Delta(Y)\\}_{\mathfrak{A}\otimes\mathfrak{A}}\quad\forall\,X,Y\in\mathfrak{A}\,,$

with respect to the standard Poisson structure on
$\mathfrak{A}\otimes\mathfrak{A}$:

$\\{X\otimes Y,W\otimes
Z\\}_{\mathfrak{A}\otimes\mathfrak{A}}:=\\{X,W\\}_{\mathfrak{A}}\otimes
YZ+XW\otimes\\{Y,Z\\}_{\mathfrak{A}}\,,\quad X,Y,Z,W\in\mathfrak{A}\,.$

Let us consider a Poisson coalgebra with $\ell$ generators $J_{i}$
($i=1,\dots,\ell$) and a non-linear Casimir $C(J_{1},\dots,J_{\ell})$.
Starting from the coproduct $\Delta=\Delta^{[2]}$ it is possible to define the
$m$-th coproduct map, with $2<m\leq n$ in two possible ways. The first one is
through the following recursive application of $\Delta^{[2]}$ (which is both a
Poisson and an algebra homomorphism):

$\Delta_{L}^{[m]}:=(id\otimes\dots^{m-2)}\otimes
id\otimes\Delta^{[2]})\circ\Delta_{L}^{[m-1]}\,,$ (2.1)

whereas the second one, that comes from a generalization of the
coassociativity property to an arbitrary number of tensor product of
$\mathfrak{A}$, by the following recursive one:

$\Delta^{[m]}_{R}:=(\Delta^{[2]}\otimes id\otimes\dots^{m-2)}\otimes
id)\circ\Delta_{R}^{[m-1]}\,.$ (2.2)

The main point now resides in the fact that, although the $n$-th coproduct of
any generator of the algebra is the same due to the coassociativity of the
$n$-th coproduct ($\Delta^{[n]}_{L}(J_{i})=\Delta_{R}^{[n]}(J_{i})$), lower
dimensional ones with $2<m<n$ will not be equivalent. This is because, by
labelling with $1\otimes 2\otimes\dots\otimes n$ the sites of the chain
$\mathfrak{A}\otimes\mathfrak{A}\otimes\dots^{n)}\otimes\mathfrak{A}$, the
_left_ and _right_ coproducts will be defined on the tensor product space
$1\otimes 2\otimes\dots\otimes m$ and
$(n-m+1)\otimes(n-m+2)\otimes\dots\otimes n$, respectively. As a consequence,
the so-called _left_ and _right_ Casimirs arise:

$\displaystyle
C^{[m]}:=\Delta_{L}^{[m]}(C(J_{1},\dots,J_{\ell}))=C(\Delta_{L}^{[m]}(J_{1}),\dots,\Delta_{L}^{[m]}(J_{\ell}))$
(2.3) $\displaystyle
C_{[m]}:=\Delta^{[m]}_{R}(C(J_{1},\dots,J_{\ell}))=C(\Delta^{[m]}_{R}(J_{1}),\dots,\Delta^{[m]}_{R}(J_{\ell}))\,,$
(2.4)

with $C^{[n]}=C_{{[n]}}$ because of the coassociativity property. Here the
main result: if we consider a $n$-site Hamiltonian $H^{[n]}$ defined on the
chain $\mathfrak{A}\otimes\mathfrak{A}\otimes\dots^{n)}\otimes\mathfrak{A}$,
i.e. the $n$-th coproduct of a (smooth/formal power series) function of the
generators $J_{i}$ (taken as the initial 1-site Hamiltonian), i.e.:

$H^{[n]}:=H(\Delta_{L}^{[n]}(J_{1}),\dots,\Delta_{L}^{[n]}(J_{\ell}))=H(\Delta^{[n]}_{R}(J_{1}),\dots,\Delta^{[n]}_{R}(J_{\ell}))\,,$
(2.5)

then, for $m,m^{\prime}=1,\dots,n$, the following Poisson commutation
relations hold:

$\displaystyle\\{C^{[m]},H^{[n]}\\}_{\mathfrak{A}\otimes\mathfrak{A}\dots^{n)}\otimes\mathfrak{A}}\,\,=0\,,$
(2.6)
$\displaystyle\\{C^{{[m]}},C^{[m^{\prime}]}\\}_{\mathfrak{A}\otimes\mathfrak{A}\dots^{n)}\otimes\mathfrak{A}}=0\,,$
(2.7)

where $C^{[1]}\equiv C$, so that the set of left Casimirs Poisson commute with
the $n$-sites Hamiltonian $H^{[n]}$. Moreover, they define a set of involutive
functions. The explicit proof can be found in [31]. The same result can be
rephrased for the set of right Casimirs. In view of this, the idea is to
introduce explicit systems for which these algebraic results apply. In
particular, the connection with classical Hamiltonian systems relies on the
use of symplectic realisations. They allow to link the abstract algebraic
construction to explicit classical mechanical models. In particular, once a
symplectic realisation $D:\mathfrak{A}\to C^{\infty}(x,p)$ is given, the
Hamiltonian $H^{[n]}$ will result in a function of the $2n$ canonical
coordinates $(x_{i},p_{i})$ and, at least in principle, will be
superintegrable by construction (since it Poisson commutes with the left and
right integrals of motion arising under the chosen symplectic realisation). We
refer the reader to [34] for a comprehensive discussion on the integrability
conditions for these models. Once the realisation $D$ is fixed, the bracket is
the canonical one, i.e.:

$\\{f,g\\}_{\mathfrak{A}\otimes\dots^{n)}\otimes\mathfrak{A}}=\sum_{j=1}^{n}\bigl{(}\partial_{x_{j}}f\partial_{p_{j}}g-\partial_{p_{j}}f\partial_{x_{j}}g\bigl{)}\,\qquad
f,g\in C^{\infty}(\boldsymbol{{x}},\boldsymbol{{p}}).$ (2.8)

Notice that the above algebraic construction also works for quantum mechanical
systems, where commutators replace Poisson brackets. Of course, usual ordering
problems have to be taken into account due to noncommutativity. Also, the
construction can be extended when we deal with a Lie algebra possessing more
than just one Casimir, say $r$ Casimirs, among which $R$ of them being non-
linear ones (they are the ones of interest in relation to integrability). For
all the details we refer the reader to [30, 31, 32], where several
applications (for deformed coalgebras also) can be found.

##  3 Quadratic substructures generated by the left and right Casimirs as rank
one Racah subalgebras: The classical case

For our purposes, we are interested in this work to the Lie-Poisson
(co)algebra
$\mathfrak{A}=\mathfrak{sl}(2,\mathbb{R})\simeq\mathfrak{su}(1,1)$. The latter
is endowed with $\ell=3$ generators $(J_{\pm},J_{3})$ and it is defined
through the following Lie-Poisson brackets:

$\\{J_{-},J_{+}\\}=2J_{3}\quad\\{J_{3},J_{\pm}\\}=\pm J_{\pm}\,.$ (3.1)

The coproduct $\Delta:\mathfrak{A}\to\mathfrak{A}\otimes\mathfrak{A}$ is
primitive:

$\Delta(J_{\sigma})=J_{\sigma}\otimes 1+1\otimes
J_{\sigma}\,,\qquad\Delta(1)=1\otimes 1\,,$ (3.2)

and the (non-linear) Casimir read:

$C(J_{+},J_{-},J_{3})=J_{3}^{2}-J_{+}J_{-}\,.$ (3.3)

A one-dimensional symplectic realisation $D$ for the generators of this Lie-
Poisson coalgebra is:

$J_{+}^{[1]}:=D(J_{+})=\frac{1}{2}\bigl{(}p_{1}^{2}+\frac{a_{1}}{x_{1}^{2}}\bigl{)}\quad
J_{-}^{[1]}:=D(J_{-})=\frac{1}{2}x_{1}^{2}\quad
J_{3}^{[1]}:=D(J_{3})=\frac{1}{2}x_{1}p_{1}\,.$ (3.4)

Thus, under the canonical Poisson brackets, we realise the initial Lie-Poisson
algebra:

$\\{J^{[1]}_{-},J^{[1]}_{+}\\}=2J^{[1]}_{3}\quad\\{J^{[1]}_{3},J^{[1]}_{\pm}\\}=\pm
J^{[1]}_{\pm}\,.$ (3.5)

The Casimir function, at a fixed realisation, reads:

$C^{[1]}:=D(C(J_{+},J_{-},J_{3}))=-a_{1}/4\,.$ (3.6)

The $n$D realisation of the generators is constructed by applying the $n$-th
coproduct
$\Delta^{[n]}:\mathfrak{A}\to\mathfrak{A}\otimes\mathfrak{A}\otimes\dots^{n)}\otimes\mathfrak{A}$
to the basis generators $J_{\sigma}$ ($\sigma=\pm,3$) and by taking the
explicit realisation, so that:

$J_{+}^{[n]}=\frac{1}{2}\bigl{(}\boldsymbol{{p}}^{2}+\sum_{j=1}^{n}\frac{a_{j}}{x_{j}^{2}}\bigl{)}\quad
J_{-}^{[n]}=\frac{1}{2}\boldsymbol{{x}}^{2}\quad
J_{3}^{[n]}=\frac{1}{2}\boldsymbol{{x}}\cdot\boldsymbol{{p}}\,,$ (3.7)

where we assume the $a_{j}$ to be arbitrary real parameters. This means that
these new $nD$ generators close:

$\\{J_{-}^{[n]},J_{+}^{[n]}\\}=2J_{3}^{[n]}\quad\\{J_{3}^{[n]},J_{\pm}^{[n]}\\}=\pm
J^{[n]}_{\pm}\,,$

through the $n$D canonical Poisson bracket (2.8). The total Casimir $C^{[n]}$
at a fixed realisation reads:

$C^{[n]}=-\frac{1}{4}\biggl{(}\sum_{1\leq
i<j}^{n}\biggl{(}L_{ij}^{2}+a_{i}\frac{x_{j}^{2}}{x_{i}^{2}}+a_{j}\frac{x_{i}^{2}}{x_{j}^{2}}\biggl{)}+\sum_{i=1}^{n}a_{i}\biggl{)}\,,$
(3.8)

with $L_{ij}=x_{i}p_{j}-x_{j}p_{i}$. This object can be seen just as the
common left-right Casimir obtained when $m=n$. In fact, the left and right
Casimirs (2.3)-(2.4) give rise, at a fixed realisation, to the following left
and right classical integrals of motion:

$\displaystyle C^{[m]}=-\frac{1}{4}\biggl{(}\hskip 11.38092pt\sum_{1\leq
i<j}^{m}\hskip
11.38092pt\biggl{(}L_{ij}^{2}+a_{i}\frac{x_{j}^{2}}{x_{i}^{2}}+a_{j}\frac{x_{i}^{2}}{x_{j}^{2}}\biggl{)}+\hskip
11.38092pt\sum_{i=1}^{m}\hskip 11.38092pta_{i}\biggl{)}$ (3.9) $\displaystyle
C_{[m]}=-\frac{1}{4}\biggl{(}\sum_{n-m+1\leq
i<j}^{n}\biggl{(}L_{ij}^{2}+a_{i}\frac{x_{j}^{2}}{x_{i}^{2}}+a_{j}\frac{x_{i}^{2}}{x_{j}^{2}}\biggl{)}+\sum_{i=n-m+1}^{n}a_{i}\biggl{)}\,,$
(3.10)

where $C^{[1]}=-a_{1}/4$, $C_{[1]}=-a_{n}/4$ are just constants and
$C^{[n]}=C_{[n]}$ as a consequence of the coassociativity property of the
coproduct $\Delta^{[n]}$ on
$\mathfrak{A}\otimes\mathfrak{A}\otimes\dots^{n)}\otimes\mathfrak{A}$.
Therefore, as a consequence of the general theory, the above $2n-3$ functions
Poisson commute with any arbitrary smooth function
$H^{[n]}=H(J_{+}^{[n]},J_{-}^{[n]},J_{3}^{[n]})$ of the generators:

$\\{C^{[m]},H^{[n]}\\}=\\{C_{{[m]}},H^{[n]}\\}=0\,.$ (3.11)

Thus, any Hamiltonian that is endowed with the above coalgebra symmetry
(meaning that can be expressed as in (2.5)) will be automatically endowed with
the $2n-3$ conserved quantities coming from the left and right coproducts.
Moreover, the left and right Casimirs define two sets composed by $n-1$
functions in involution ($m,m^{\prime}=1,\dots,n$):

$\\{C^{[m]},C^{[m^{\prime}]}\\}=\\{C_{{[m]}},C_{{[m^{\prime}]}}\\}=0\,.$
(3.12)

If we add the Hamiltonian to the above sets, they will be both composed by $n$
functions in involution.

To summarise, the construction based on coalgebras led us to define two
subsets, say $Y_{L}$ and $Y_{R}$, composed by $n-1$ left and right constants
of motion in involution, i.e.:

$Y_{L}=\\{C^{[1]},\dots,C^{[n]}\\}\,,\quad
Y_{R}=\\{C_{[1]},\dots,C_{[n]}\\}\,,$ (3.13)

the total Casimir $C^{[n]}=C_{[n]}$ being a common element of the sets and
$C^{[1]}$, $C_{[1]}$ just constants. Once arrived at this point, let us
observe that the total Casimir can be rewritten in the following way:

$\displaystyle C^{[n]}$ $\displaystyle=-\frac{1}{4}\biggl{(}\sum_{1\leq
i<j}^{n}\biggl{(}L_{ij}^{2}+a_{i}\frac{x_{j}^{2}}{x_{i}^{2}}+a_{j}\frac{x_{i}^{2}}{x_{j}^{2}}\biggl{)}+\sum_{i=1}^{n}a_{i}\biggl{)}$
$\displaystyle=-\frac{1}{4}\biggl{(}\sum_{1\leq
i<j}^{n}L_{ij}^{2}+\boldsymbol{x}^{2}\sum_{j=1}^{n}\frac{a_{j}}{x_{j}^{2}}\biggl{)}$
$\displaystyle=\sum_{1\leq i<j}^{n}C_{ij}-(n-2)\sum_{i=1}^{n}C_{i}\,,$ (3.14)

where we defined the quantities:

$C_{ij}:=-\frac{1}{4}\biggl{(}L_{ij}^{2}+a_{i}\frac{x_{j}^{2}}{x_{i}^{2}}+a_{j}\frac{x_{i}^{2}}{x_{j}^{2}}+a_{i}+a_{j}\biggl{)}\qquad
C_{i}:=-\frac{a_{i}}{4}\,.$ (3.15)

Also, the left and right Casimirs can be re-expressed as:

$C^{[m]}=\sum_{1\leq i<j}^{m}C_{ij}-(m-2)\sum_{i=1}^{m}C_{i}\,,\qquad
C_{[m]}=\sum_{n-m+1\leq i<j}^{n}C_{ij}-(m-2)\sum_{i=n-m+1}^{n}C_{i}\,.$ (3.16)

These new expressions for the left and right partial Casimirs emphasize, in a
more general perspective, their connection with the (abstract) generators of
the rank $n-2$ generalised Racah algebra $R(n)$ [17, 18], the latter generated
by the two indices generators $C_{ij}$ together with the one index generators
$C_{i}$, here appearing as the building blocks of the left and right Casimirs
in the given realisation of the Poisson-Lie algebra. Notice that the “central
elements”$C_{i}$ are realised here just as the constants appearing in (3.15),
which underline the one-particle symplectic realisation used on the generic
$i$-th site of the chain
$\mathfrak{A}\otimes\mathfrak{A}\otimes\dots^{n)}\otimes\mathfrak{A}$. At this
point, we can go further by introducing the quantities:

$P_{ij}:=C_{ij}-C_{i}-C_{j}\,,$ (3.17)

in such a way to simplify the above expressions to the following ones:

$C^{[m]}=\sum_{1\leq i<j}^{m}P_{ij}+\sum_{i=1}^{m}C_{i}\,,\qquad
C_{[m]}=\sum_{n-m+1\leq i<j}^{n}P_{ij}+\sum_{i=n-m+1}^{n}C_{i}\,.$ (3.18)

The defining relations from $P_{ij}$ are those of the classical analog of the
Racah algebra $R(n)$ [45, 20]:

$\displaystyle\\{P_{ij},P_{jk}\\}=:2F_{ijk}\hskip
147.95424pt(F_{ijk}=-{F_{jik}}=-F_{ikj})$ (3.19)
$\displaystyle\\{P_{jk},F_{ijk}\\}=P_{ik}P_{jk}-P_{jk}P_{ij}+2P_{ik}C_{j}-2P_{ij}C_{k}$
(3.20) $\displaystyle\\{P_{kl},F_{ijk}\\}=P_{ik}P_{jl}-P_{il}P_{jk}$ (3.21)
$\displaystyle\\{F_{ijk},F_{jkl}\\}=F_{jkl}P_{ij}-F_{ikl}(P_{jk}+2C_{j})-F_{ijk}P_{jl}$
(3.22) $\displaystyle\\{F_{ijk},F_{klm}\\}=F_{ilm}P_{jk}-P_{ik}F_{jlm}\,.$
(3.23)

Finally, if we introduce the new definitions $\mathcal{L}_{k}:=C^{[k]}$ and
$\mathcal{R}_{k}:=C_{[n-k+1]}$, for $k=1,\dots,n$, we get:

$\mathcal{L}_{k}:=\sum_{1\leq
i<j}^{k}P_{ij}+\sum_{i=1}^{k}C_{i}\,,\qquad\mathcal{R}_{k}:=\sum_{k\leq
i<j}^{n}P_{ij}+\sum_{i=k}^{n}C_{i}\,.$ (3.24)

We thus have, at a fixed $n$ for $k=1,\dots,n$, the following identifications:

  1. $-$

$(n;\\{k\\})=(1;\\{1\\})$

$\mathcal{L}_{1}=C^{[1]}=C_{1}=C_{[1]}=\mathcal{R}_{1}\,.$ (3.25)

  2. $-$

$(n;\\{k\\})=(2;\\{1,2\\})$

$\mathcal{L}_{1}=C^{[1]}\,,\quad\mathcal{L}_{2}=C^{[2]}=C_{[2]}=\mathcal{R}_{1}\,,\quad\mathcal{R}_{2}=C_{[1]}\,,$
(3.26)

with:

$\displaystyle\mathcal{L}_{1}=C_{1}$
$\displaystyle\mathcal{L}_{2}=P_{12}+C_{1}+C_{2}=C_{12}$
$\displaystyle\mathcal{R}_{1}=P_{12}+C_{1}+C_{2}=C_{12}$
$\displaystyle\mathcal{R}_{2}=C_{2}\,.$

  3. $-$

$(n;\\{k\\})=(3;\\{1,2,3\\})$

$\mathcal{L}_{1}=C^{[1]}\,,\quad\mathcal{L}_{2}=C^{[2]}\,,\quad\mathcal{L}_{3}=C^{[3]}=C_{[3]}=\mathcal{R}_{1}\,,\quad\mathcal{R}_{2}=C_{[2]}\,,\quad\mathcal{R}_{3}=C_{[1]}\,,$
(3.27)

with:

$\displaystyle\mathcal{L}_{1}=C_{1}$
$\displaystyle\mathcal{L}_{2}=P_{12}+C_{1}+C_{2}=C_{12}$
$\displaystyle\mathcal{L}_{3}=P_{12}+P_{13}+P_{23}+C_{1}+C_{2}+C_{3}=C_{12}+C_{13}+C_{23}-C_{1}-C_{2}-C_{3}=C_{123}$
$\displaystyle\mathcal{R}_{1}=P_{12}+P_{13}+P_{23}+C_{1}+C_{2}+C_{3}=C_{12}+C_{13}+C_{23}-C_{1}-C_{2}-C_{3}=C_{123}$
$\displaystyle\mathcal{R}_{2}=P_{23}+C_{2}+C_{3}=C_{23}$
$\displaystyle\mathcal{R}_{3}=C_{3}\,.$

⋮

  4. $-$

$(n;\\{k\\})=(n;\\{1,\dots,n\\})$

$\displaystyle\mathcal{L}_{1}=C^{[1]}\,,\quad\mathcal{L}_{2}=C^{[2]}\,,\quad\,\quad\mathcal{L}_{3}=C^{[3]}\,,\qquad\dots\quad,\quad\mathcal{L}_{n}=C^{[n]}$
(3.28)
$\displaystyle\mathcal{R}_{1}=C_{[n]}\,,\quad\mathcal{R}_{2}=C_{[n-1]}\,,\quad\mathcal{R}_{3}=C_{[n-2]}\,,\quad\dots\quad\,,\quad\mathcal{R}_{n}=C_{[1]}\,,$
(3.29)

with:

$\displaystyle\mathcal{L}_{1}=C_{1}$
$\displaystyle\mathcal{L}_{2}=P_{12}+C_{1}+C_{2}=C_{12}$
$\displaystyle\mathcal{L}_{3}=P_{12}+P_{13}+P_{23}+C_{1}+C_{2}+C_{3}=C_{12}+C_{13}+C_{23}-C_{1}-C_{2}-C_{3}=C_{123}$
$\displaystyle\vdots$ $\displaystyle\mathcal{L}_{n-1}=\sum_{1\leq
i<j}^{n-1}P_{ij}+\sum_{i=1}^{n-1}C_{i}=\sum_{1\leq
i<j}^{n-1}C_{ij}-(n-3)\sum_{i=1}^{n-1}C_{i}=C_{1\dots n-1}$
$\displaystyle\mathcal{L}_{n}=\mathcal{R}_{1}=\sum_{1\leq
i<j}^{n}P_{ij}+\sum_{i=1}^{n}C_{i}=\sum_{1\leq
i<j}^{n}C_{ij}-(n-2)\sum_{i=1}^{n}C_{i}=C_{1\dots n}$
$\displaystyle\mathcal{R}_{2}=\sum_{2\leq
i<j}^{n}P_{ij}+\sum_{i=2}^{n}C_{i}=\sum_{2\leq
i<j}^{n}C_{ij}-(n-3)\sum_{i=2}^{n}C_{i}=C_{2\dots n}$
$\displaystyle\mathcal{R}_{3}=\sum_{3\leq
i<j}^{n}P_{ij}+\sum_{i=3}^{n}C_{i}=\sum_{3\leq
i<j}^{n}C_{ij}-(n-4)\sum_{i=3}^{n}C_{i}=C_{3\dots n}$ $\displaystyle\vdots$
$\displaystyle\mathcal{R}_{n-1}=P_{n-1\,n}+C_{n-1}+C_{n}=C_{n-1\,n}$
$\displaystyle\mathcal{R}_{n}=C_{n}\,.$

The left and right Casimirs (3.24) define the generators of the universal
quadratic substructures appearing in [40, 41].

Our main goal now is to show that the above-mentioned substructures can be
reinterpreted, at a fixed $n>3$, as the images of $n-2$ injective morphisms of
$R(3)$ into $R(n)$ and that, as a direct consequence, they turn out to be
isomorphic to the rank one Racah algebra $R(3)$. As a first step we rephrase
this in the classical framework and show that for $n=3$ the only substructure
that arises from the left and right Casimir invariants is a Poisson analog of
the Racah algebra $R(3)$.

###  3.1 The $\boldsymbol{n=3}$ case: Poisson analog of the rank one Racah
algebra $\boldsymbol{R(3)}$

The generators of the Poisson analog of the rank one Racah algebra are:
$\\{C_{1},C_{2},C_{3},C_{12},C_{13},C_{23},C_{123}\\}$ with:

$C_{123}=C_{12}+C_{13}+C_{23}-C_{1}-C_{2}-C_{3}\,,$ (3.30)

where $C_{1}$, $C_{2}$, $C_{3}$ and $C_{123}$ are central elements. Now, by
defining:

$\mathcal{F}:=\frac{1}{2}\\{C_{12},C_{23}\\}=\frac{1}{2}\\{C_{23},C_{13}\\}=\frac{1}{2}\\{C_{13},C_{12}\\}$
(3.31)

we got the following defining Poisson brackets:

$\displaystyle\\{C_{12},\mathcal{F}\\}=C_{23}C_{12}-C_{12}C_{13}+(C_{2}-C_{1})(C_{3}-C_{123})$
(3.32)
$\displaystyle\\{C_{23},\mathcal{F}\\}=C_{13}C_{23}-C_{23}C_{12}+(C_{3}-C_{2})(C_{1}-C_{123})$
(3.33)
$\displaystyle\\{C_{13},\mathcal{F}\\}=C_{12}C_{13}-C_{13}C_{23}+(C_{1}-C_{3})(C_{2}-C_{123})\,.$
(3.34)

From (3.30), using the properties of the Poisson brackets, we obtain the
relations:

$\\{C_{23},C_{12}+C_{13}\\}=0\qquad\\{C_{12},C_{13}+C_{23}\\}=0\qquad\\{C_{13},C_{12}+C_{23}\\}=0\,.$
(3.35)

Also, the following sum is zero:

$\\{C_{12},\mathcal{F}\\}+\\{C_{23},\mathcal{F}\\}+\\{C_{13},\mathcal{F}\\}=0\,.$
(3.36)

Now, if we use (3.30) to explicitate the $C_{13}$ generator, i.e.:

$C_{13}=C_{123}-C_{12}-C_{23}+C_{1}+C_{2}+C_{3}\,$ (3.37)

and we substitute this expressions into (3.32)-(3.33), taking into account
(3.36), we get:

$\displaystyle\\{C_{12},\mathcal{F}\\}=+C_{12}^{2}+2C_{23}C_{12}-(C_{123}+C_{1}+C_{2}+C_{3})C_{12}+(C_{2}-C_{1})(C_{3}-C_{123})$
(3.38)
$\displaystyle\\{C_{23},\mathcal{F}\\}=-C_{23}^{2}-2C_{12}C_{23}+(C_{123}+C_{1}+C_{2}+C_{3})C_{23}+(C_{3}-C_{2})(C_{1}-C_{123})\,.$
(3.39)

We recognize this quadratic Poisson algebra to be nothing but the substructure
$(n,{k})=(3,2)$, which we indicate as222From now on we use the notation
$S_{(n,k)}$ to indicate the generic $k$-th substructure at fixed $n$.
$S_{(3,2)}$ as reported in [40]. Notice that in this presentation the operator
$C_{13}$, being a linear combination of the others, does not appear
explicitly. Now, if we consider the relations among the generators of the
Racah algebra and the left and right Casimirs in dimension $n=3$, as we
reported previously, this connection appears more evident. In fact, if
substitute the expressions we get:

$\displaystyle\\{\mathcal{L}_{2},\mathcal{F}\\}=+\mathcal{L}_{2}^{2}+2\mathcal{R}_{2}\mathcal{L}_{2}-(\mathcal{L}_{3}+\mathcal{L}_{1}+C_{2}+\mathcal{R}_{3})\mathcal{L}_{2}+(C_{2}-\mathcal{L}_{1})(\mathcal{R}_{3}-\mathcal{L}_{3})$
(3.40)
$\displaystyle\\{\mathcal{R}_{2},\mathcal{F}\\}=-\mathcal{R}_{2}^{2}-2\mathcal{L}_{2}\mathcal{R}_{2}+(\mathcal{L}_{3}+\mathcal{L}_{1}+C_{2}+\mathcal{R}_{3})\mathcal{R}_{2}+(\mathcal{R}_{3}-C_{2})(\mathcal{L}_{1}-\mathcal{L}_{3})\,,$
(3.41)

with:

$\mathcal{F}:=\frac{1}{2}\\{\mathcal{L}_{2},\mathcal{R}_{2}\\}=\frac{1}{2}\\{\mathcal{R}_{2},\mathcal{M}_{2}\\}=\frac{1}{2}\\{\mathcal{M}_{2},\mathcal{L}_{2}\\}\,,$
(3.42)

where we introduced the new definition
$\mathcal{M}_{2}:=\mathcal{L}_{3}-\mathcal{L}_{2}-\mathcal{R}_{2}+\mathcal{L}_{1}+C_{2}+\mathcal{R}_{3}$
for the generator $C_{13}$, the latter will be useful later in the
generalization to higher dimensions. Thus, on the basis of these results, we
can provide an equivalent presentation for the substructure $S_{(3,2)}$, which
reads:

$\displaystyle\\{\mathcal{L}_{2},\mathcal{F}\\}\,=\mathcal{R}_{2}\mathcal{L}_{2}-\mathcal{L}_{2}\mathcal{M}_{2}+(C_{2}-\mathcal{L}_{1})(\mathcal{R}_{3}-\mathcal{L}_{3})$
(3.43)
$\displaystyle\\{\mathcal{R}_{2},\mathcal{F}\\}\,=\mathcal{M}_{2}\mathcal{R}_{2}-\mathcal{R}_{2}\mathcal{L}_{2}+(\mathcal{R}_{3}-C_{2})(\mathcal{L}_{1}-\mathcal{L}_{3})$
(3.44)
$\displaystyle\\{\mathcal{M}_{2},\mathcal{F}\\}=\mathcal{L}_{2}\mathcal{M}_{2}-\mathcal{M}_{2}\mathcal{R}_{2}+(\mathcal{L}_{1}-\mathcal{R}_{3})(C_{2}-\mathcal{L}_{3})\,.$
(3.45)

In conclusion, we observed that the substructure arising in dimension $n=3$ in
[40] represents a classical realisation (given in terms of Poisson brackets)
of the rank one Racah algebra $R(3)$, the latter being generated by the set
$\\{\mathcal{L}_{1},C_{2},\mathcal{R}_{3},\mathcal{L}_{2},\mathcal{M}_{2},\mathcal{R}_{2},\mathcal{L}_{3}=\mathcal{R}_{1}\\}$,
where $C_{2}=-a_{2}/4$ in the given realisation and $\mathcal{M}_{2}$ appears
as a linear combination of the other generators.

###  3.2 Higher dimensional classical substructures, injective morphisms and
(Poisson) Racah algebras $\boldsymbol{R^{K_{1},K_{2},K_{3}}(3)}$

The main aim of this section is to provide a characterisation of the
substructures $S_{(n,k)}$, at a fixed $n>3$ for $k=2,\dots,n-1$ with the help
of the following [15, 17, 19, 46]:

######  Lemma 3.1.

Let $\\{K_{1},K_{2},K_{3}\\}$ be a set composed by three disjoint subsets of
the set $[n]:=\\{1,\dots,n\\}$. Define $K_{B}:=\cup_{q\in B}K_{q}$ with
$B\subset[3]$. Then, the following map:

$\displaystyle\theta:\,$ $\displaystyle R(3)\to R(n)$
$\displaystyle\,C_{B}\,\,\,\to C_{K_{B}}$

is an injective morphism. Its image is denoted by $R^{K_{1},K_{2},K_{3}}(3)$
and it is isomorphic to the rank one Racah algebra.

This result appears to be a specific case of the more general one appearing in
[17, 46], where the authors deal with a general number k of disjoint subsets
$\\{K_{p}\\}_{p=1,\dots,\textsf{k}}$ of $[n]$. We refer to [15] for the proof
of this Lemma (see also [19] for the generalisation to the k-fold tensor
product). So, the idea is to make use of the above result paraphrasing it in
this classical context. In particular, if we consider the generators of the
Racah algebra $R(3)$, i.e. the set
$\\{C_{1},C_{2},C_{3},C_{12},C_{13},C_{23},C_{123}\\}$, then their images
under the map $\theta$:

$\small\\{\theta(C_{1}),\theta(C_{2}),\theta(C_{3}),\theta(C_{12}),\theta(C_{13}),\theta(C_{23}),\theta(C_{123})\\}=\\{C_{K_{1}},C_{K_{2}},C_{K_{3}},C_{K_{1}\cup
K_{2}},C_{K_{1}\cup K_{3}},C_{K_{2}\cup K_{3}},C_{K_{1}\cup K_{2}\cup
K_{3}}\\}\,,$ (3.46)

define a new set composed by seven generators that closes an algebra
isomorphic to the rank one Racah algebra $R(3)$. In our classical setting, in
terms of Poisson brackets, the algebra $R^{K_{1},K_{2},K_{3}}(3)$ is defined
as:

$\displaystyle\\{C_{K_{1}K_{2}},\mathcal{F}\\}=C_{K_{2}K_{3}}C_{K_{1}K_{2}}-C_{K_{1}K_{2}}C_{K_{1}K_{3}}+(C_{K_{2}}-C_{K_{1}})(C_{K_{3}}-C_{K_{1}K_{2}K_{3}})$
(3.47)
$\displaystyle\\{C_{K_{2}K_{3}},\mathcal{F}\\}=C_{K_{1}K_{3}}C_{K_{2}K_{3}}-C_{K_{2}K_{3}}C_{K_{1}K_{2}}+(C_{K_{3}}-C_{K_{2}})(C_{K_{1}}-C_{K_{1}K_{2}K_{3}})$
(3.48)
$\displaystyle\\{C_{K_{1}K_{3}},\mathcal{F}\\}=C_{K_{1}K_{2}}C_{K_{1}K_{3}}-C_{K_{1}K_{3}}C_{K_{2}K_{3}}+(C_{K_{1}}-C_{K_{3}})(C_{K_{2}}-C_{K_{1}K_{2}K_{3}})\,,$
(3.49)

with:

$\mathcal{F}:=\frac{1}{2}\\{C_{K_{1}K_{2}},C_{K_{2}K_{3}}\\}=\frac{1}{2}\\{C_{K_{2}K_{3}},C_{K_{1}K_{3}}\\}=\frac{1}{2}\\{C_{K_{1}K_{3}},C_{K_{1}K_{2}}\\}\,.$
(3.50)

Here, we used the shorthand notation $C_{K_{i}K_{j}}:=C_{K_{i}\cup K_{j}}$ for
$1\leq i<j\leq 3$ and $C_{K_{1}K_{2}K_{3}}:=C_{K_{1}\cup K_{2}\cup K_{3}}$.
Notice that the linear relation (3.30) is rephrased here as:

$C_{K_{1}K_{2}K_{3}}=C_{K_{1}K_{2}}+C_{K_{1}K_{3}}+C_{K_{2}K_{3}}-C_{K_{1}}-C_{K_{2}}-C_{K_{3}}\,.$
(3.51)

What we are interested to show in the following is that the substructures
$S_{(n,k)}$ studied in [40, 41], at a fixed $n>3$, can be interpreted as a
realisation of the algebras arising as the images of $n-2$ injective
morphisms:

$\theta_{k}:R(3)\to R(n)\quad(k=2,\dots n-1)\,,$

for specific choices of the three disjoint subset $\\{K_{i}\\}_{i=1,2,3}$ as
defined above. Thus, each substructure turns out to be isomorphic to a
classical realisation of the rank one Racah algebra $R(3)$. Let us present
this result as the following:

######  Proposition 1.

Let $[n]:=\\{1,\dots,n\\}$ be a set with $n>3$ fixed. For each $k=2,\dots,n-1$
define $K_{1}^{(k)}:=\\{1,\dots,k-1\\}$, $K_{2}^{(k):}=\\{k\\}$ and
$K_{3}^{(k)}:=\\{k+1,\dots,n\\}$ to be three disjoint subsets of the set
$[n]$. Then, the images of the $n-2$ injective morphisms $\theta_{k}:$
$R(3)\to R(n)$ $(k=2,\dots,n-1)$:

$\displaystyle\theta_{k}(C_{1})=C_{K_{1}^{(k)}}\equiv C_{1\dots
k-1}=\mathcal{L}_{k-1}$ $\displaystyle\theta_{k}(C_{2})=C_{K_{2}^{(k)}}\equiv
C_{k}$ $\displaystyle\theta_{k}(C_{3})=C_{K_{3}^{(k)}}\equiv C_{k+1\dots
n}=\mathcal{R}_{k+1}$
$\displaystyle\theta_{k}(C_{12})=C_{K_{1}^{(k)}K_{2}^{(k)}}\equiv C_{1\dots
k}=\mathcal{L}_{k}$
$\displaystyle\theta_{k}(C_{23})=C_{K_{2}^{(k)}K_{3}^{(k)}}\equiv C_{k\dots
n}=\mathcal{R}_{k}$
$\displaystyle\theta_{k}(C_{13})=C_{K_{1}^{(k)}K_{3}^{(k)}}\equiv C_{1\dots
k-1k+1\dots n}=\mathcal{M}_{k}$
$\displaystyle\theta_{k}(C_{123})=C_{K_{1}^{(k)}K_{2}^{(k)}K_{3}^{(k)}}\equiv
C_{1\dots n}=\mathcal{L}_{n}=\mathcal{R}_{1}$

with:

$\mathcal{M}_{k}:=\mathcal{L}_{n}-\mathcal{L}_{k}-\mathcal{R}_{k}+\mathcal{L}_{k-1}+C_{k}+\mathcal{R}_{k+1}\,,$

result in the universal classical substructures $S_{(n,k)}$. Moreover, as a
direct consequence of Lemma 3.1 they are isomorphic to the Poisson analog of
the Racah algebra $R(3)$.

###### Proof.

At a fixed $n>3$, for each $k=2,\dots,n-1$, we can define the algebras
$R^{K_{1}^{(k)},K_{2}^{(k)},K_{3}^{(k)}}(3)$ as the images of the $n-2$
morphisms defined above and lift the relations valid for $R(3)$ into $R(n)$
(we can think the procedure as $n-2$ applications of Lemma 3.1). This implies
that each of the $n-2$ subalgebras are generated by the elements:

$\\{C_{K_{1}^{(k)}},C_{K_{2}^{(k)}},C_{K_{3}^{(k)}},C_{K_{1}^{(k)}K_{2}^{(k)}},C_{K_{1}^{(k)}K_{3}^{(k)}},C_{K_{1}^{(k)}K_{2}^{(k)}K_{3}^{(k)}}\\}$
(3.52)

$C_{K_{1}^{(k)}},C_{K_{2}^{(k)}},C_{K_{3}^{(k)}}$ and
$C_{K_{1}^{(k)}K_{2}^{(k)}K_{3}^{(k)}}$ playing the role of central elements
for each of the $n-2$ substructures. The linear relation (3.30) is lifted to
the following $n-2$ linear relations (one relation for each $k=2,\dots,n-1$):

$C_{K_{1}^{(k)}K_{2}^{(k)}K_{3}^{(k)}}=C_{K_{1}^{(k)}K_{2}^{(k)}}+C_{K_{1}^{(k)}K_{3}^{(k)}}+C_{K_{2}^{(k)}K_{3}^{(k)}}-C_{K_{1}^{(k)}}-C_{K_{2}^{(k)}}-C_{K_{3}^{(k)}}\,.$
(3.53)

So, we have:

$\mathcal{F}_{k}:=\frac{1}{2}\\{C_{K_{1}^{(k)}K_{2}^{(k)}},C_{K_{2}^{(k)}K_{3}^{(k)}}\\}=\frac{1}{2}\\{C_{K_{2}^{(k)}K_{3}^{(k)}},C_{K_{1}^{(k)}K_{3}^{(k)}}\\}=\frac{1}{2}\\{C_{K_{1}^{(k)}K_{3}^{(k)}},C_{K_{1}^{(k)}K_{2}^{(k)}}\\}\,,$
(3.54)

together with:

$\displaystyle\\{C_{K_{1}^{(k)}K_{2}^{(k)}},\mathcal{F}_{k}\\}=C_{K_{2}^{(k)}K_{3}^{(k)}}C_{K_{1}^{(k)}K_{2}^{(k)}}-C_{K_{1}^{(k)}K_{2}^{(k)}}C_{K_{1}^{(k)}K_{3}^{(k)}}+(C_{K_{2}^{(k)}}-C_{K_{1}^{(k)}})(C_{K_{3}^{(k)}}-C_{K_{1}^{(k)}K_{2}^{(k)}K_{3}^{(k)}})$
(3.55)
$\displaystyle\\{C_{K_{2}^{(k)}K_{3}^{(k)}},\mathcal{F}_{k}\\}=C_{K_{1}^{(k)}K_{3}^{(k)}}C_{K_{2}^{(k)}K_{3}^{(k)}}-C_{K_{2}^{(k)}K_{3}^{(k)}}C_{K_{1}^{(k)}K_{2}^{(k)}}+(C_{K_{3}^{(k)}}-C_{K_{2}^{(k)}})(C_{K_{1}^{(k)}}-C_{K_{1}^{(k)}K_{2}^{(k)}K_{3}^{(k)}})$
(3.56)
$\displaystyle\\{C_{K_{1}^{(k)}K_{3}^{(k)}},\mathcal{F}_{k}\\}=C_{K_{1}^{(k)}K_{2}^{(k)}}C_{K_{1}^{(k)}K_{3}^{(k)}}-C_{K_{1}^{(k)}K_{3}^{(k)}}C_{K_{2}^{(k)}K_{3}^{(k)}}+(C_{K_{1}^{(k)}}-C_{K_{3}^{(k)}})(C_{K_{2}^{(k)}}-C_{K_{1}^{(k)}K_{2}^{(k)}K_{3}^{(k)}})\,.$
(3.57)

The following relations, the equivalent of (3.35), also holds:

$\displaystyle\\{C_{K_{2}^{(k)}K_{3}^{(k)}},C_{K_{1}^{(k)}K_{2}^{(k)}}+C_{K_{1}^{(k)}K_{3}^{(k)}}\\}=0$
(3.58)
$\displaystyle\\{C_{K_{1}^{(k)}K_{2}^{(k)}},C_{K_{1}^{(k)}K_{3}^{(k)}}+C_{K_{2}^{(k)}K_{3}^{(k)}}\\}=0$
(3.59)
$\displaystyle\\{C_{K_{1}^{(k)}K_{3}^{(k)}},C_{K_{1}^{(k)}K_{2}^{(k)}}+C_{K_{2}^{(k)}K_{3}^{(k)}}\\}=0\,,$
(3.60)

and the following sum is still zero when we lift it into $R(n)$:

$\\{C_{K_{1}^{(k)}K_{2}^{(k)}},\mathcal{F}_{k}\\}+\\{C_{K_{2}^{(k)}K_{3}^{(k)}},\mathcal{F}_{k}\\}+\\{C_{K_{1}^{(k)}K_{3}^{(k)}},\mathcal{F}_{k}\\}=0\,.$
(3.61)

What we need to show is the equivalence with the substructures $S_{(n,k)}$. To
do this, we can proceed as we have done for the $n=3$ case in Subsection 3.1,
as we are dealing with $n-2$ algebras isomorphic to $R(3)$.

For each $k=2,\dots,n-1$, let $K_{1}^{(k)}:=\\{1,\dots,k-1\\}$,
$K_{2}^{(k):}=\\{k\\}$ and $K_{3}^{(k)}:=\\{k+1,\dots,n\\}$ be three specific
disjoint subsets of $[n]$. With this choice, the generators of
$R^{K_{1}^{(k)},K_{2}^{(k)},K_{3}^{(k)}}(3)$ are:

$\\{\mathcal{L}_{k-1},C_{k},\mathcal{R}_{k+1},\mathcal{L}_{k},\mathcal{R}_{k},\mathcal{M}_{k},\mathcal{L}_{n}=\mathcal{R}_{1}\\},$
(3.62)

$\mathcal{L}_{k-1}$, $C_{k}$, $\mathcal{R}_{k+1}$ and
$\mathcal{L}_{n}=\mathcal{R}_{1}$ being central elements associated to the
$k$-th substructure. The linear relation (3.30) is lifted to the $n-2$ linear
relations:

$\mathcal{L}_{n}=\mathcal{L}_{k}+\mathcal{R}_{k}+\mathcal{M}_{k}-\mathcal{L}_{k-1}-C_{k}-\mathcal{R}_{k+1}=\mathcal{R}_{1}\,.$
(3.63)

and the Poisson algebras $R^{K_{1}^{(k)},K_{2}^{(k)},K_{3}^{(k)}}(3)$ turn out
to be:

$\displaystyle\\{\mathcal{L}_{k},\mathcal{F}_{k}\\}=\mathcal{R}_{k}\mathcal{L}_{k}-\mathcal{L}_{k}\mathcal{M}_{k}+(C_{k}-\mathcal{L}_{k-1})(\mathcal{R}_{k+1}-\mathcal{L}_{n})$
(3.64)
$\displaystyle\\{\mathcal{R}_{k},\mathcal{F}_{k}\\}=\mathcal{M}_{k}\mathcal{R}_{k}\,-\mathcal{R}_{k}\mathcal{L}_{k}\,+(\mathcal{R}_{k+1}-C_{k})(\mathcal{L}_{k-1}-\mathcal{L}_{n})$
(3.65)
$\displaystyle\\{\mathcal{M}_{k},\mathcal{F}_{k}\\}=\mathcal{L}_{k}\mathcal{M}_{k}-\mathcal{M}_{k}\mathcal{R}_{k}+(\mathcal{L}_{k-1}-\mathcal{R}_{k+1})(C_{k}-\mathcal{L}_{n})\,,$
(3.66)

where:

$\mathcal{F}_{k}:=\frac{1}{2}\\{\mathcal{L}_{k},\mathcal{R}_{k}\\}=\frac{1}{2}\\{\mathcal{R}_{k},\mathcal{M}_{k}\\}=\frac{1}{2}\\{\mathcal{M}_{k},\mathcal{L}_{k}\\}\,.$
(3.67)

The relations given in (3.35), are lifted to the following ones:

$\\{\mathcal{R}_{k},\mathcal{L}_{k}+\mathcal{M}_{k}\\}=0\,,\qquad\\{\mathcal{L}_{k},\mathcal{M}_{k}+\mathcal{R}_{k}\\}=0\,,\qquad\\{\mathcal{M}_{k},\mathcal{L}_{k}+\mathcal{R}_{k}\\}=0\,.$
(3.68)

Moreover, relation (3.36), is now lifted to:

$\\{\mathcal{L}_{k},\mathcal{F}_{k}\\}+\\{\mathcal{R}_{k},\mathcal{F}_{k}\\}+\\{\mathcal{M}_{k},\mathcal{F}_{k}\\}=0\,.$
(3.69)

In total analogy to the three dimensional case, we use (3.63) to explicitate
the $\mathcal{M}_{k}$ generators, i.e.:

$\mathcal{M}_{k}=\mathcal{L}_{n}-\mathcal{L}_{k}-\mathcal{R}_{k}+\mathcal{L}_{k-1}+C_{k}+\mathcal{R}_{k+1}\,,$
(3.70)

and we substitute these expressions into (3.64)-(3.65), taking into account
(3.69). In this way, we get:

$\displaystyle\\{\mathcal{L}_{k},\mathcal{F}_{k}\\}=+\mathcal{L}_{k}^{2}+2\mathcal{R}_{k}\mathcal{L}_{k}-(\mathcal{L}_{n}+\mathcal{L}_{k-1}+C_{k}+\mathcal{R}_{k+1})\mathcal{L}_{k}+(C_{k}-\mathcal{L}_{k-1})(\mathcal{R}_{k+1}-\mathcal{L}_{n})$
(3.71)
$\displaystyle\\{\mathcal{R}_{k},\mathcal{F}_{k}\\}=-\mathcal{R}_{k}^{2}-2\mathcal{L}_{k}\mathcal{R}_{k}+(\mathcal{L}_{n}+\mathcal{L}_{k-1}+C_{k}+\mathcal{R}_{k+1})\mathcal{R}_{k}+(\mathcal{R}_{k+1}-C_{k})(\mathcal{L}_{k-1}-\mathcal{L}_{n})\,,$
(3.72)

where $\mathcal{L}_{n}=C^{[n]}=C_{[n]}=\mathcal{R}_{1}$. At a fixed $n$, for
each $k=2,\dots,n-1$ these Poisson commutation relations define the universal
quadratic substructures $S_{(n,k)}$ generated by the classical left and right
Casimir invariants that can be found in [40].
∎

######  Remark 1.

If we let $n$ to be equal three in the construction, so that $k=2$, we obtain
the three disjoint subsets $K_{1}^{(2)}=\\{1\\}$, $K_{2}^{(2)}=\\{2\\}$ and
$K_{3}^{(2)}=\\{3\\}$. In this case, the morphism $\theta_{2}:R(3)\to R(3)$
acts as follows:

$\displaystyle\theta_{2}(C_{1})=C_{K_{1}^{(2)}}\equiv C_{1}=\mathcal{L}_{1}$
$\displaystyle\theta_{2}(C_{2})=C_{K_{2}^{(2)}}\equiv C_{2}$
$\displaystyle\theta_{2}(C_{3})=C_{K_{3}^{(2)}}\equiv C_{3}=\mathcal{R}_{3}$
$\displaystyle\theta_{2}(C_{12})=C_{K_{1}^{(2)}K_{2}^{(2)}}\equiv
C_{12}=\mathcal{L}_{2}$
$\displaystyle\theta_{2}(C_{23})=C_{K_{2}^{(2)}K_{3}^{(2)}}\equiv
C_{23}=\mathcal{R}_{2}$
$\displaystyle\theta_{2}(C_{13})=C_{K_{1}^{(2)}K_{3}^{(2)}}\equiv
C_{13}=\mathcal{M}_{2}$
$\displaystyle\theta_{2}(C_{123})=C_{K_{1}^{(2)}K_{2}^{(2)}K_{3}^{(2)}}\equiv
C_{123}=\mathcal{L}_{3}=\mathcal{R}_{1}\,,$

with:

$\mathcal{M}_{2}:=\mathcal{L}_{3}-\mathcal{L}_{2}-\mathcal{R}_{2}+\mathcal{L}_{1}+C_{2}+\mathcal{R}_{3}\,.$

and the construction collapses to the one we performed in Subsection 3.1 for
$S_{(3,2)}=R^{1,2,3}(3)$.

$\mathcal{L}_{k}$$\mathcal{R}_{k+1}$$\mathcal{L}_{k-1}$$\mathcal{R}_{k}$
Figure 1: the generic $k$-th substructure $S_{(n,k)}$. We have not reported
$\mathcal{L}_{n}=\mathcal{R}_{1}$, which is central element of all $n-2$
substructures, as well as the one index central element $C_{k}$. Remember also
that the generator $\mathcal{M}_{k}$ is a linear combination of the others (by
virtue of (3.70)).

Just as an explicit example, let us consider the $n=4$ case, so that two
substructures arise: $S_{(4,2)}$ and $S_{(4,3)}$. The former, which emerges
with the choice of the three disjoint subsets $K_{1}^{(2)}=\\{1\\}$,
$K_{2}^{(2)}=\\{2\\}$, $K_{3}^{(2)}=\\{3,4\\}$, is generated by the elements:

$\displaystyle\\{\theta(C_{1}),\theta(C_{2}),\theta(C_{3}),\theta(C_{12}),\theta(C_{13}),\theta(C_{23}),\theta(C_{123})\\}$
$\displaystyle\equiv\\{C_{1},C_{2},C_{34},C_{12},C_{134},C_{234},C_{1234}\\}$
$\displaystyle=\\{\mathcal{L}_{1},C_{2},\mathcal{R}_{3},\mathcal{L}_{2},\mathcal{M}_{2},\mathcal{R}_{2},\mathcal{L}_{4}\\}\,,$
(3.73)

with $\mathcal{L}_{4}=\mathcal{R}_{1}$ and
$\mathcal{M}_{2}=\mathcal{L}_{4}-\mathcal{L}_{2}-\mathcal{R}_{2}+\mathcal{L}_{1}+C_{2}+\mathcal{R}_{3}$
by virtue of (3.70).

The latter, which emerges instead by considering the disjoint subsets
$K_{1}^{(3)}=\\{1,2\\}$, $K_{2}^{(3)}=\\{3\\}$, $K_{3}^{(3)}=\\{4\\}$, is
generated by the elements:

$\displaystyle\\{\theta(C_{1}),\theta(C_{2}),\theta(C_{3}),\theta(C_{12}),\theta(C_{13}),\theta(C_{23}),\theta(C_{123})\\}$
$\displaystyle\equiv\\{C_{12},C_{3},C_{4},C_{123},C_{124},C_{34},C_{1234}\\}$
$\displaystyle=\\{\mathcal{L}_{2},C_{3},\mathcal{R}_{4},\mathcal{L}_{3},\mathcal{M}_{3},\mathcal{R}_{3},\mathcal{L}_{4}\\}\,,$
(3.74)

where
$\mathcal{M}_{3}=\mathcal{L}_{4}-\mathcal{L}_{3}-\mathcal{R}_{3}+\mathcal{L}_{2}+C_{3}+\mathcal{R}_{4}$.
Observing the two sets, we can see that the element $\mathcal{R}_{3}$, which
appears as a central element of $S_{(4,2)}$, appears at the same time as a
generator of $S_{(4,3)}$. The same happens for $\mathcal{L}_{2}$, which is
central of $S_{(4,3)}$ and a generator of $S_{(4,2)}$.

$\mathcal{L}_{3}$$\mathcal{R}_{4}$$\mathcal{L}_{2}$$\mathcal{R}_{3}$$\mathcal{R}_{2}$$\mathcal{L}_{1}$
Figure 2: chain of quadratic algebras generated by the substructures
$S_{(4,2)}$ and $S_{(4,3)}$ in dimension $n=4$. We have not reported
$\mathcal{L}_{4}=\mathcal{R}_{1}$, which is central element of both
substructures, as well as the one index central elements $C_{2},C_{3}$.

Observing Figure 2 we can appreciate that at a fixed $n=4$, the element
$\mathcal{R}_{i+1}$ (for a fixed $i=1,2,3$) Poisson commute with all the
elements $\mathcal{L}_{j}$ with $j=1,\dots,i$. This is a general feature in
the chain of these quadratic algebras and holds for any $n\geq 3$. The reason
behind this is that the $(i+1)$-th right Casimir element has no common indices
with the $i$ left Casimirs elements $\mathcal{L}_{j}$. This is a direct
consequence of their definition on
$\mathfrak{A}\otimes\mathfrak{A}\otimes\dots^{n)}\otimes\mathfrak{A}$.

$\mathcal{L}_{1}$$\mathcal{R}_{2}$$\mathcal{L}_{2}$$\mathcal{R}_{3}$$\mathcal{L}_{3}$$\mathcal{R}_{4}$$\mathcal{L}_{n-2}$$\mathcal{R}_{n-1}$$\mathcal{L}_{n-1}$$\mathcal{R}_{n}$$\small{{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}b}^{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\ddots}}}$$\small{{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}a}^{\color[rgb]{0,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,1}\pgfsys@color@cmyk@stroke{1}{0}{0}{0}\pgfsys@color@cmyk@fill{1}{0}{0}{0}{\ddots}}}$
Figure 3: chain of quadratic algebras composed by the $n-2$ substructures
$S_{(n,k)}$. Again, the elements $\mathcal{L}_{n}=\mathcal{R}_{1}$ and
$C_{k}$, with $k=2,\dots,n-1$, have been omitted.

If we focus on the generic $k$-th substructure in the chain, we can conclude
that two elements, say $\\{\mathcal{R}_{k}$, $\mathcal{R}_{k+1}\\}$, Poisson
commute each other because belong to the set of right Casimirs (indicated in
red), two other elements, $\\{\mathcal{L}_{k},\mathcal{L}_{k-1}\\}$, Poisson
commute each other because belong to the set of left Casimirs (indicated in
blue) and the right element $R_{k+1}$ Poisson commute with the left ones in
the $k$-th substructure because they have no common indices (this extends also
to the left elements $\\{\mathcal{L}_{1},\dots,\mathcal{L}_{k-2}\\}$ outside
the $k$-th substructure).

To summarise, this is the chain of quadratic algebras generated by the left
and right Casimir invariants. The main point of this result concerns the fact
that behind the above chain structure, which we recall has been crucial in the
quantum case to construct the spectrum of $n$D quantum superintegrable models
algebraically [41], there is in fact a chain of (Poisson) Racah algebras in
disguise. They are generated by the elements in the set (3.62). We conclude
the discussion about the classical case observing that each of the $n-2$
substructures is endowed with a Casimir function. Specifically, at fixed
$n\geq 3$, for each $k=2,\dots,n-1$ we can define the $k$-th Casimir
associated to the $k$-th embedded Poisson algebra
$R^{\\{1,\dots,k-1\\},\\{k\\},\\{k+1,\dots,n\\}}(3)$, which turns out to be:

$\mathcal{K}_{k}=\mathcal{F}^{2}_{k}+\mathcal{L}_{k}\mathcal{M}_{k}\mathcal{R}_{k}+(\mathcal{R}_{k+1}-C_{k})(\mathcal{L}_{k-1}-\mathcal{L}_{n})\mathcal{L}_{k}-(C_{k}-\mathcal{L}_{k-1})(\mathcal{R}_{k+1}-\mathcal{L}_{n})\mathcal{R}_{k}\,.$
(3.75)

By direct computation, at a fixed $n$ and $k$, it is verified that each
Casimir Poisson commute with the corresponding elements in the set (3.62).

##  4 Quadratic substructures generated by the left and right Casimirs as rank
one Racah subalgebras: The quantum case

Our starting point for the quantum case is the Lie coalgebra $\mathfrak{A}$ in
the basis $(J_{\pm},J_{3})$ with commutation rules:

$[J_{-},J_{+}]=2\imath\hbar J_{3}\quad[J_{3},J_{\pm}]=\pm\imath\hbar
J_{\pm}\,,$ (4.1)

endowed with the usual primitive coproduct
$\Delta:\mathfrak{A}\to\mathfrak{A}\otimes\mathfrak{A}$ and non-linear
Casimir:

$C(J_{+},J_{-},J_{3})=J_{3}^{2}-\frac{1}{2}(J_{+}J_{-}+J_{-}J_{+})\,.$ (4.2)

A one-dimensional differential realisation $\bar{D}$ for this Lie coalgebra
is:

$\hat{J}_{+}^{[1]}:=\bar{D}(J_{+})=\frac{1}{2}\bigl{(}-\hbar^{2}\partial_{x_{1}}^{2}+\frac{a_{1}}{x_{1}^{2}}\bigl{)}\quad\hat{J}_{-}^{[1]}:=\bar{D}(J_{-})=\frac{1}{2}x_{1}^{2}\quad\hat{J}_{3}^{[1]}:=\bar{D}(J_{3})=-\frac{\imath\hbar}{2}(x_{1}\cdot\partial_{x_{1}}+1/2)\,.$
(4.3)

In the given realisation, the Casimir operator reads:

$\hat{C}^{[1]}:=\bar{D}(C(J_{+},J_{-},J_{3}))=\frac{1}{16}(3\hbar^{2}-4a_{1})\,.$
(4.4)

A $n$D realisation is then given by $n$ copies of it:

$\hat{J}_{+}^{[n]}=\frac{1}{2}\biggl{(}-\hbar^{2}\Delta+\sum_{j=1}^{n}\frac{a_{j}}{x_{j}^{2}}\biggl{)}\quad\hat{J}_{-}^{[n]}=\frac{1}{2}\boldsymbol{{x}}^{2}\quad\hat{J}_{3}^{[n]}=-\frac{\imath\hbar}{2}(\boldsymbol{{x}}\cdot\nabla+n/2)\,,$
(4.5)

and, in total analogy to the classical case, the left and right partial
Casimirs read:

$\hat{C}^{[m]}=\sum_{1\leq
i<j}^{m}\hat{C}_{ij}-(m-2)\sum_{i=1}^{m}\hat{C}_{i}\qquad\hat{C}_{[m]}=\sum_{n-m+1\leq
i<j}^{n}\hat{C}_{ij}-(m-2)\sum_{i=n-m+1}^{n}\hat{C}_{i}\,,$ (4.6)

where we introduced the quantities (the quantum analog of (3.15)):

$\hat{C}_{ij}:=-\frac{1}{4}\biggl{(}\hat{L}_{ij}^{2}+a_{i}\frac{\hat{x}_{j}^{2}}{\hat{x}_{i}^{2}}+a_{j}\frac{\hat{x}_{i}^{2}}{\hat{x}_{j}^{2}}+a_{i}+a_{j}-\hbar^{2}\biggl{)}\qquad\hat{C}_{i}:=\frac{1}{16}(3\hbar^{2}-4a_{i})\,,$
(4.7)

with $\hat{L}_{ij}=\hat{x}_{i}\hat{p}_{j}-\hat{x}_{j}\hat{p}_{i}$
($\hat{p}_{i}\equiv-\imath\hbar\partial_{x_{i}}$). As expected, we see that
the two expressions (4.6) are formally equivalent to the ones in (3.16), what
changes is just the different realisation used in the quantum case for the
generators. Again, if we introduce the operators:
$\hat{P}_{ij}:=\hat{C}_{ij}-\hat{C}_{i}-\hat{C}_{j}$ the Casimirs can be
expressed as:

$\hat{C}^{[m]}=\sum_{1\leq
i<j}^{m}\hat{P}_{ij}+\sum_{i=1}^{m}\hat{C}_{i}\,,\qquad\hat{C}_{[m]}=\sum_{n-m+1\leq
i<j}^{n}\hat{P}_{ij}+\sum_{i=n-m+1}^{n}\hat{C}_{i}$ (4.8)

and, with the new definitions $\hat{\mathcal{L}}_{k}:=\hat{C}^{[k]}$ and
$\hat{\mathcal{R}}_{k}:=\hat{C}_{[n-k+1]}$, for $k=1,\dots,n$, we can
construct the quantum analog of (3.24), which read:

$\hat{\mathcal{L}}_{k}=\sum_{1\leq
i<j}^{k}\hat{P}_{ij}+\sum_{i=1}^{k}\hat{C}_{i}\,,\qquad\hat{\mathcal{R}}_{k}:=\sum_{k\leq
i<j}^{n}\hat{P}_{ij}+\sum_{i=k}^{n}\hat{C}_{i}\,.$ (4.9)

Clearly, because of its underlying algebraic nature, the construction is very
much the same as the one we performed for the classical case. We need to take
into account, besides the different realisation, the noncommutativity of the
product of operators. In what follow, we limit ourselves to review the
construction for the $n=3$ case and to rephrase Proposition 1 in the quantum
framework.

###  4.1 The $\boldsymbol{n=3}$ case: the rank one Racah algebra
$\boldsymbol{R(3)}$

The generators of the rank one Racah algebra are:
$\\{\hat{C}_{1},\hat{C}_{2},\hat{C}_{3},\hat{C}_{12},\hat{C}_{13},\hat{C}_{23},\hat{C}_{123}\\}$
with:

$\hat{C}_{123}=\hat{C}_{12}+\hat{C}_{13}+\hat{C}_{23}-\hat{C}_{1}-\hat{C}_{2}-\hat{C}_{3}\,,$
(4.10)

where, as asual, $\hat{C}_{1}$, $\hat{C}_{2}$, $\hat{C}_{3}$ and
$\hat{C}_{123}$ are central elements. Now, by defining:

$\hat{\mathcal{F}}:=\frac{1}{2\imath\hbar}[\hat{C}_{12},\hat{C}_{23}]=\frac{1}{2\imath\hbar}[\hat{C}_{23},\hat{C}_{13}]=\frac{1}{2\imath\hbar}[\hat{C}_{13},\hat{C}_{12}]$
(4.11)

we get the following defining commutation relations:

$\displaystyle[\hat{C}_{12},\hat{\mathcal{F}}]/\imath\hbar=\hat{C}_{23}\hat{C}_{12}-\hat{C}_{12}\hat{C}_{13}+(\hat{C}_{2}-\hat{C}_{1})(\hat{C}_{3}-\hat{C}_{123})$
(4.12)
$\displaystyle[\hat{C}_{23},\hat{\mathcal{F}}]/\imath\hbar=\hat{C}_{13}\hat{C}_{23}-\hat{C}_{23}\hat{C}_{12}+(\hat{C}_{3}-\hat{C}_{2})(\hat{C}_{1}-\hat{C}_{123})$
(4.13)
$\displaystyle[\hat{C}_{13},\hat{\mathcal{F}}]/\imath\hbar=\hat{C}_{12}\hat{C}_{13}-\hat{C}_{13}\hat{C}_{23}+(\hat{C}_{1}-\hat{C}_{3})(\hat{C}_{2}-\hat{C}_{123})\,.$
(4.14)

From (4.10), using the properties of the commutators, we obtain the relations:

$[\hat{C}_{23},\hat{C}_{12}+\hat{C}_{13}]=0\qquad[\hat{C}_{12},\hat{C}_{13}+\hat{C}_{23}]=0\qquad[\hat{C}_{13},\hat{C}_{12}+\hat{C}_{23}]=0\,.$
(4.15)

Also, the following sum is easily checked to be zero:

$[\hat{C}_{12},\hat{\mathcal{F}}]+[\hat{C}_{23},\hat{\mathcal{F}}]+[\hat{C}_{13},\hat{\mathcal{F}}]=0\,.$
(4.16)

Again, if we use (4.10) to explicitate the $\hat{C}_{13}$ generator, i.e.:

$\hat{C}_{13}=\hat{C}_{123}-\hat{C}_{12}-\hat{C}_{23}+\hat{C}_{1}+\hat{C}_{2}+\hat{C}_{3}\,$
(4.17)

and we substitute this expressions into (4.12)-(4.13), taking into account
(4.16), we get:

$\displaystyle[\hat{C}_{12},\hat{\mathcal{F}}]/\imath\hbar=+\hat{C}_{12}^{2}+\\{\hat{C}_{23},\hat{C}_{12}\\}-(\hat{C}_{123}+\hat{C}_{1}+\hat{C}_{2}+\hat{C}_{3})\hat{C}_{12}+(\hat{C}_{2}-\hat{C}_{1})(\hat{C}_{3}-\hat{C}_{123})$
(4.18)
$\displaystyle[\hat{C}_{23},\hat{\mathcal{F}}]/\imath\hbar=-\hat{C}_{23}^{2}-\\{\hat{C}_{12},\hat{C}_{23}\\}+(\hat{C}_{123}+\hat{C}_{1}+\hat{C}_{2}+\hat{C}_{3})\hat{C}_{23}+(\hat{C}_{3}-\hat{C}_{2})(\hat{C}_{1}-\hat{C}_{123})\,,$
(4.19)

which represents the quantum analog of (3.38)-(3.39). Here
$\\{\hat{A},\hat{B}\\}:=\hat{A}\hat{B}+\hat{B}\hat{A}$ is the anti-commutator.
In total analogy to the classical case, we can then rewrite the substructure
$(n,k)=(3,2)$, which we indicate as $S^{(\hbar)}_{(3,2)}$, as:

$\displaystyle[\hat{\mathcal{L}}_{2},\hat{\mathcal{F}}]/\imath\hbar=+\hat{\mathcal{L}}_{2}^{2}+\\{\hat{\mathcal{R}}_{2},\mathcal{\hat{L}}_{2}\\}-(\mathcal{\hat{L}}_{3}+\hat{\mathcal{L}}_{1}+\hat{C}_{2}+\hat{\mathcal{R}}_{3})\hat{\mathcal{L}}_{2}+(\hat{C}_{2}-\hat{\mathcal{L}}_{1})(\hat{\mathcal{R}}_{3}-\hat{\mathcal{L}}_{3})$
(4.20)
$\displaystyle[\hat{\mathcal{R}}_{2},\hat{\mathcal{F}}]/\imath\hbar=-\hat{\mathcal{R}}_{2}^{2}-\\{\hat{\mathcal{L}}_{2},\hat{\mathcal{R}}_{2}\\}+(\hat{\mathcal{L}}_{3}+\hat{\mathcal{L}}_{1}+\hat{C}_{2}+\hat{\mathcal{R}}_{3})\hat{\mathcal{R}}_{2}+(\hat{\mathcal{R}}_{3}-\hat{C}_{2})(\hat{\mathcal{L}}_{1}-\hat{\mathcal{L}}_{3})\,,$
(4.21)

with:

$\hat{\mathcal{F}}:=\frac{1}{2\imath\hbar}[\hat{\mathcal{L}}_{2},\hat{\mathcal{R}}_{2}]=\frac{1}{2\imath\hbar}[\hat{\mathcal{R}}_{2},\hat{\mathcal{M}}_{2}]=\frac{1}{2\imath\hbar}[\hat{\mathcal{M}}_{2},\hat{\mathcal{L}}_{2}]\,,$
(4.22)

and the usual definition
$\hat{\mathcal{M}}_{2}:=\hat{\mathcal{L}}_{3}-\hat{\mathcal{L}}_{2}-\hat{\mathcal{R}}_{2}+\hat{\mathcal{L}}_{1}+\hat{C}_{2}+\hat{\mathcal{R}}_{3}$
for the generator $\hat{C}_{13}$. The relations (4.20)-(4.21)-(4.22) represent
the quantum analog of (3.40)-(3.41)-(3.42).

###  4.2 Higher dimensional quantum substructures, injective morphisms and
rank one Racah algebras $\boldsymbol{R^{K_{1},K_{2},K_{3}}(3)}$

Once the link with the Racah algebra $R(3)$ has been found, we can proceed
with an analogous construction for the quantum substructures. In particular,
we can formulate the following:

######  Proposition 2.

Let $[n]:=\\{1,\dots,n\\}$ be a set with $n>3$ fixed. For each $k=2,\dots,n-1$
define $K_{1}^{(k)}:=\\{1,\dots,k-1\\}$, $K_{2}^{(k):}=\\{k\\}$ and
$K_{3}^{(k)}:=\\{k+1,\dots,n\\}$ to be three disjoint subsets of the set
$[n]$. Then, the images of the $n-2$ injective morphisms $\theta_{k}:$
$R(3)\to R(n)$ $(k=2,\dots,n-1)$:

$\displaystyle\theta_{k}(C_{1})=C_{K_{1}^{(k)}}\equiv\hat{C}_{1\dots
k-1}=\hat{\mathcal{L}}_{k-1}$
$\displaystyle\theta_{k}(C_{2})=C_{K_{2}^{(k)}}\equiv\hat{C}_{k}\ $
$\displaystyle\theta_{k}(C_{3})=C_{K_{3}^{(k)}}\equiv\hat{C}_{k+1\dots
n}=\hat{\mathcal{R}}_{k+1}$
$\displaystyle\theta_{k}(C_{12})=C_{K_{1}^{(k)}K_{2}^{(k)}}\equiv\hat{C}_{1\dots
k}=\hat{\mathcal{L}}_{k}$
$\displaystyle\theta_{k}(C_{23})=C_{K_{2}^{(k)}K_{3}^{(k)}}\equiv\hat{C}_{k\dots
n}=\hat{\mathcal{R}}_{k}$
$\displaystyle\theta_{k}(C_{13})=C_{K_{1}^{(k)}K_{3}^{(k)}}\equiv\hat{C}_{1\dots
k-1k+1\dots n}=\hat{\mathcal{M}}_{k}$
$\displaystyle\theta_{k}(C_{123})=C_{K_{1}^{(k)}K_{2}^{(k)}K_{3}^{(k)}}\equiv\hat{C}_{1\dots
n}=\hat{\mathcal{L}}_{n}=\hat{\mathcal{R}}_{1}$

with:

$\hat{\mathcal{M}}_{k}:=\hat{\mathcal{L}}_{n}-\hat{\mathcal{L}}_{k}-\hat{\mathcal{R}}_{k}+\hat{\mathcal{L}}_{k-1}+\hat{C}_{k}+\hat{\mathcal{R}}_{k+1}\,,$

result in the universal quantum substructures $S^{(\hbar)}_{(n,k)}$. Moreover,
as a direct consequence of Lemma 3.1 they are isomorphic to the rank one Racah
algebra $R(3)$.

###### Proof.

For each $k=2,\dots,n-1$, let $K_{1}^{(k)}:=\\{1,\dots,k-1\\}$,
$K_{2}^{(k):}=\\{k\\}$ and $K_{3}^{(k)}:=\\{k+1,\dots,n\\}$ be three specific
disjoint subsets of $[n]$. With this choice, the quantum generators are:

$\\{\hat{\mathcal{L}}_{k-1},\hat{C}_{k},\hat{\mathcal{R}}_{k+1},\hat{\mathcal{L}}_{k},\hat{\mathcal{R}}_{k},\hat{\mathcal{M}}_{k},\hat{\mathcal{L}}_{n}=\hat{\mathcal{R}}_{1}\\},$
(4.23)

$\hat{\mathcal{L}}_{k-1}$, $\hat{C}_{k}$, $\hat{\mathcal{R}}_{k+1}$ and
$\hat{\mathcal{L}}_{n}=\hat{\mathcal{R}}_{1}$ being central elements
associated to the $k$-th substructure. The linear relation (4.17) is lifted to
the $n-2$ linear relations:

$\hat{\mathcal{L}}_{n}=\hat{\mathcal{L}}_{k}+\hat{\mathcal{R}}_{k}+\hat{\mathcal{M}}_{k}-\hat{\mathcal{L}}_{k-1}-\hat{C}_{k}-\hat{\mathcal{R}}_{k+1}=\hat{\mathcal{R}}_{1}\,,$
(4.24)

and the rank one Racah subalgebras turn out to be:

$\displaystyle[\hat{\mathcal{L}}_{k},\hat{\mathcal{F}}_{k}]/\imath\hbar\,=\hat{\mathcal{R}}_{k}\hat{\mathcal{L}}_{k}-\hat{\mathcal{L}}_{k}\hat{\mathcal{M}}_{k}\,+(\hat{C}_{k}-\hat{\mathcal{L}}_{k-1})(\hat{\mathcal{R}}_{k+1}-\hat{\mathcal{L}}_{n})$
(4.25)
$\displaystyle[\hat{\mathcal{R}}_{k},\hat{\mathcal{F}}_{k}]/\imath\hbar\,=\hat{\mathcal{M}}_{k}\hat{\mathcal{R}}_{k}-\hat{\mathcal{R}}_{k}\hat{\mathcal{L}}_{k}\,+(\hat{\mathcal{R}}_{k+1}-\hat{C}_{k})(\hat{\mathcal{L}}_{k-1}-\hat{\mathcal{L}}_{n})$
(4.26)
$\displaystyle[\hat{\mathcal{M}}_{k},\hat{\mathcal{F}}_{k}]/\imath\hbar=\hat{\mathcal{L}}_{k}\hat{\mathcal{M}}_{k}-\hat{\mathcal{M}}_{k}\hat{\mathcal{R}}_{k}+(\hat{\mathcal{L}}_{k-1}-\hat{\mathcal{R}}_{k+1})(\hat{C}_{k}-\hat{\mathcal{L}}_{n})\,,$
(4.27)

where:

$\hat{\mathcal{F}}_{k}:=\frac{1}{2\imath\hbar}[\hat{\mathcal{L}}_{k},\hat{\mathcal{R}}_{k}]=\frac{1}{2\imath\hbar}[\hat{\mathcal{R}}_{k},\hat{\mathcal{M}}_{k}]=\frac{1}{2\imath\hbar}[\hat{\mathcal{M}}_{k},\hat{\mathcal{L}}_{k}]\,.$
(4.28)

The relations given in (4.15), are lifted to the following ones:

$[\hat{\mathcal{R}}_{k},\hat{\mathcal{L}}_{k}+\hat{\mathcal{M}}_{k}]=0\,,\qquad[\hat{\mathcal{L}}_{k},\hat{\mathcal{M}}_{k}+\hat{\mathcal{R}}_{k}]=0\,,\qquad[\hat{\mathcal{M}}_{k},\hat{\mathcal{L}}_{k}+\hat{\mathcal{R}}_{k}]=0\,.$
(4.29)

Moreover, relation (4.16), is now rephrased as:

$[\hat{\mathcal{L}}_{k},\hat{\mathcal{F}}_{k}]+[\hat{\mathcal{R}}_{k},\hat{\mathcal{F}}_{k}]+[\hat{\mathcal{M}}_{k},\hat{\mathcal{F}}_{k}]=0\,.$
(4.30)

In total analogy to the three dimensional case, we use (4.24) to explicitate
the $\hat{\mathcal{M}}_{k}$ generators, i.e.:

$\hat{\mathcal{M}}_{k}=\hat{\mathcal{L}}_{n}-\hat{\mathcal{L}}_{k}-\hat{\mathcal{R}}_{k}+\hat{\mathcal{L}}_{k-1}+\hat{C}_{k}+\hat{\mathcal{R}}_{k+1}\,$
(4.31)

and we substitute these expressions into (4.25)-(4.26), taking into account
(4.30). In this way we get:

$\displaystyle[\hat{\mathcal{L}}_{k},\hat{\mathcal{F}}_{k}]/\imath\hbar=+\hat{\mathcal{L}}_{k}^{2}+\\{\hat{\mathcal{R}}_{k},\hat{\mathcal{L}}_{k}\\}-(\hat{\mathcal{L}}_{n}+\hat{\mathcal{L}}_{k-1}+\hat{C}_{k}+\hat{\mathcal{R}}_{k+1})\hat{\mathcal{L}}_{k}+(\hat{C}_{k}-\hat{\mathcal{L}}_{k-1})(\hat{\mathcal{R}}_{k+1}-\hat{\mathcal{L}}_{n})$
(4.32)
$\displaystyle[\mathcal{R}_{k},\hat{\mathcal{F}}_{k}]/\imath\hbar=-\hat{\mathcal{R}}_{k}^{2}-\\{\hat{\mathcal{L}}_{k},\hat{\mathcal{R}}_{k}\\}+(\hat{\mathcal{L}}_{n}+\hat{\mathcal{L}}_{k-1}+\hat{C}_{k}+\hat{\mathcal{R}}_{k+1})\hat{\mathcal{R}}_{k}+(\hat{\mathcal{R}}_{k+1}-\hat{C}_{k})(\hat{\mathcal{L}}_{k-1}-\hat{\mathcal{L}}_{n})\,,$
(4.33)

where
$\hat{\mathcal{L}}_{n}=\hat{C}^{[n]}=\hat{C}_{[n]}=\hat{\mathcal{R}}_{1}$. At
a fixed $n$, for each $k=2,\dots,n-1$, these commutation relations define the
universal quadratic substructures $S^{(\hbar)}_{(n,k)}$ generated by the
quantum left and right Casimir invariants. ∎

######  Remark 2.

The result obtained in the classical case is recovered by performing the
appropriate $\hbar\to 0$ limit. In this limit, in fact, (3.71)-(3.72) arise
from (4.32)-(4.33) by replacing the commutator $[\hat{A},\hat{B}]/\imath\hbar$
with the Poisson bracket $\\{A,B\\}$ and by taking into account that
$\\{\hat{A},\hat{B}\\}\to 2AB$. In short, $\lim_{\hbar\to
0}S_{(n,k)}^{(\hbar)}=S_{(n,k)}$. This holds true also for all the other
commutation relations appearing in the construction.

Also in the quantum setting it is possible to construct the $k$-th Casimir
associated to the $k$-th substructure. It can be cast in the form:

$\displaystyle\hat{\mathcal{K}}_{k}=\hat{\mathcal{F}}^{2}_{k}$
$\displaystyle+\frac{1}{6}\\{\hat{\mathcal{L}}_{k},\hat{\mathcal{M}}_{k},\hat{\mathcal{R}}_{k}\\}+(\hat{\mathcal{R}}_{k+1}-\hat{C}_{k})(\hat{\mathcal{L}}_{k-1}-\hat{\mathcal{L}}_{n})\hat{\mathcal{L}}_{k}-(\hat{C}_{k}-\hat{\mathcal{L}}_{k-1})(\hat{\mathcal{R}}_{k+1}-\hat{\mathcal{L}}_{n})\hat{\mathcal{R}}_{k}$
$\displaystyle+\frac{\hbar^{2}}{3}(\\{\hat{\mathcal{L}}_{k},\hat{\mathcal{M}}_{k}\\}+\\{\hat{\mathcal{L}}_{k},\hat{\mathcal{R}}_{k}\\}+\\{\hat{\mathcal{M}}_{k},\hat{\mathcal{R}}_{k}\\}+(\hat{C}_{k}-\hat{\mathcal{L}}_{k-1})(\hat{\mathcal{R}}_{k+1}-\hat{\mathcal{L}}_{n})-(\hat{\mathcal{R}}_{k+1}-\hat{C}_{k})(\hat{\mathcal{L}}_{k-1}-\hat{\mathcal{L}}_{n}))\,,$
(4.34)

where we introduced the symmetrizer of three operators
$\\{\hat{A},\hat{B},\hat{C}\\}:=\hat{A}\hat{B}\hat{C}+\hat{A}\hat{C}\hat{B}+\hat{B}\hat{A}\hat{C}+\hat{B}\hat{C}\hat{A}+\hat{C}\hat{A}\hat{B}+\hat{C}\hat{B}\hat{A}$.
By direct computation, at a fixed $n$ and $k$, it is verified that each of the
above $n-2$ Casimir invariants commutes with the corresponding elements in the
set (4.23).

##  5 Concluding Remarks

In this paper we have reexamined the Racah algebra $R(n)$, a rank-($n-2)$
quadratic algebra, from the perspective of the subalgebra structures which are
useful in the algebraic derivation of spectrum for $n$-dimensional
superintegrable systems [41, 40, 20]. We have determined the commutation
relations of multi-indexed generators of the Racah algebra and related these
generators of substructures to the left and right Casimirs which arise from
coalgebra symmetry. In particular, we have showed that the $n-2$
classical/quantum quadratic substructures generated by the left and right
classical/quantum integrals, coming from the application of the left and right
$m$-th coproduct maps to the “seed”Casimir of the coalgebra, can be understood
as a classical/quantum realisation of $n-2$ rank one Racah algebras. We
explicitly shown the role played by the family of injective morphisms in the
construction of the universal substructures. As a byproduct, we have obtained
a new presentation for each of the $n-2$ quadratic algebras generated by the
left and right integrals.

Our results shed new light on the idea of substructures which was originally
introduced by Daskaloyannis in [29] for a $3$-dimensional system and then
applied to models on $n$-dimensional Euclidean spaces as well as on spaces of
constant curvatures [40, 41]. The advantage of this approach for calculating
spectrum is that it relies only on certain sets of constraints. The present
work allows us to understand more generally how to construct substructures and
algebraically derive the spectra for other superintegrable systems in
dimension greater than 2, which is a difficult problem as our knowledge to the
representation theory of higher rank quadratic algebras is still rather
limited.

These ideas would also be applicable with certain modifications to models
obtained from block separation of variables [7]. Such models possess quadratic
algebra structures with structure constants depending on additional central
elements given by Casimirs of some higher rank Lie algebras.

## Acknowledgement

IM was supported by Australian Research Council Future Fellowship FT180100099.
YZZ was supported by Australian Research Council Discovery Project DP190101529
and National Natural Science Foundation of China (Grant No. 11775177).

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