# Infinite affine, hyperbolic and Lorentzian Weyl groups with their associated
Calogero models

Francisco Correa∘, Andreas Fring∙ and Octavio Quintana⋆
$\circ$ Instituto de Ciencias Físicas y Matemáticas, Universidad Austral de
Chile,
Casilla 567, Valdivia, Chile
$\bullet$ Department of Mathematics, City, University of London,
Northampton Square, London EC1V 0HB, UK
$\star$ Facultad de Física, Pontificia Universidad Católica de Chile, Vicuña
Mackenna 4860,
Santiago, Chile<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract:

We propose generalizations of Calogero models that exhibit invariance with
respect to the infinite Weyl groups of affine, hyperbolic, and Lorentzian
types. Our approach involves deriving closed analytic formulas for the action
of the associated Coxeter elements of infinite order acting on arbitrary roots
within their respective root spaces. These formulas are then utilized in
formulating the new type of Calogero models.

††conference: Infinite Weyl groups with their associated Calogero models

##  1 Introduction

Calogero models [1, 2, 3], along with their closely related generalisations to
Calogero-Moser-Sutherland models [4, 5, 6] and their $\cal{PT}$-symmetric
extensions [7] are known to be invariant under the Weyl group related to
finite semi-simple Lie algebras. This symmetry property, combined with their
Liouville integrability, is largely responsible for the exact solvability of
these systems in the classical as well as in their quantum theoretical
versions. One may for instance utilise directly the Weyl invariant polynomials
as a starting point to set up exactly solvable models [8, 9, 10] or in turn
exploit the Weyl symmetry in the construction of differential-difference
(Dunkl) operators [11, 12] that can be used in the construction of the
orthogonal eigenstates in the quantum theory.

Here, we propose generalisations of these models to make them invariant under
the action of Weyl groups of infinite order such as the affine, hyperbolic and
Lorentzian generalisations of the finite groups. Thus, we are seeking
generalisations of the standard Calogero Hamiltonians

$H=\frac{1}{2}p^{2}+\sum_{\alpha\in\Delta_{\mathbf{g}}}\frac{c_{\alpha}}{\left(\alpha\cdot
q\right)^{2}}=\frac{1}{2}p^{2}+\sum_{i=1}^{r}\sum_{n=1}^{h}\frac{c_{in}}{\left[\sigma^{n}(\gamma_{i})\cdot
q\right]^{2}},$ (1)

where $\alpha$ are the roots in the root space
$\Delta_{\mathbf{g}}\in\mathbb{R}^{\ell}$ associated to the finite Lie algebra
$\mathbf{g}$ of rank $r$, $c_{\alpha}$ and $c_{in}$ are some real coupling
constants, $\sigma$ is a Coxeter element, $\gamma_{i}$ is a representative of
the $i$th orbit and $q=(q_{1},\ldots,q_{\ell})$, $p=(p_{1},\ldots,p_{\ell})$
are the $\ell$ coordinates and momenta of the system, respectively. Since the
sum in the potential extends over the entire root space, the invariance of the
Hamiltonian under the Weyl group associated to $\mathbf{g}$ is built-in by
construction.

Both variants of the Hamiltonian in (1) are equivalent. The first version is
more formal as it still involves the task of identifying the entire root
system of ${\mathbf{g}}$. This is made explicit in the second formulation in
which the entire root space of the $r\times h$ roots is systematically
generated from summing over $r$ orbits generated by the Coxeter element
$\sigma$ of order $h$ action on some well-known representatives $\gamma_{i}$.
In the finite dimensional case these representatives can be chosen to be the
simple roots dressed by plus-minus signs assigned according to a bicoloured
Dynkin diagram [13, 14, 15]. Thus, the challenge to generalise the symmetry
properties of these models to infinite Weyl groups consists in the first
version to listing systematically the entire infinite root spaces of the
respective groups or in the second version to finding closed formulae for
infinite consecutive actions of the Coxeter element on arbitrary roots
together with the identification of the representatives $\gamma_{i}$ of the
respective orbits. As we shall discuss below, the first version involved as
many infinite sums as the rank of the algebra over integers constrained by a
Diophantine equation. The second formulation only consists of a possibly
finite sum over the orbit representatives and only one infinite sum over
powers of the Coxeter element.

So far only one attempt has been made to formulate such theories, namely for
the special case of the hyperbolic $AE_{3}$-algebra [16], where the left
variant of the Hamiltonian in (1) was explored for that particular symmetry.
Here we will explore both version and also elaborate on their correspondence.
In the first instance we will extend the action of the repeated action of
Coxeter elements to infinite order and derive closed analytical formulas for
some specific algebra, which can not be found in the literature. Subsequently
we attempt to carry out the sum over the entire orbits of these Coxeter
elements to find simpler versions of the invariant potentials. For Toda field
theories, that require only simple roots of the respective root spaces in
their formulation, generalisations to Lorentzian and hyperbolic types have
recently been explored in [17]. Here we will largely adopt the notation from
there and [18, 19, 20]. In that context, the study of models based on extended
Kac-Moody algebras is partly motivated by the success of the systematic
description of conformal and massive Toda field theories in terms of finite
dimensional and affine Kac-Moody algebras, respectively. The extended versions
of hyperbolic and Lorentzian type algebra appear to provide a natural
framework for a large class of nonintegrable theories. The particular
extensions to $E_{10}$ and $E_{11}$ ($(E_{8})_{-1}$ and $(E_{8})_{-2}$ in our
notation) play a prominent role in string/M-theory, see e.g. [21, 22, 23, 24].
The mathematics developed here may also turn out to be useful in that context.

Our manuscript is organised as follows: In section 2 we discuss in detail the
infinite dimensional Weyl group associated to the
$\mathbf{(A}_{2}\mathbf{)}_{-2}$-Lorentzian Kac-Moody algebra, together with
their hyperbolic, affine and finite subalgebras. In particular we
systematically derive closed analytic formulas for the action of any power of
the Coxeter elements of infinite order on arbitrary roots. In section 3 we
discuss how invariants for these type of infinite Weyl groups can be
constructed by exploiting the possibility of the bicolouration, exemplified
for the $\mathbf{(A}_{2}\mathbf{)}_{-3}$-Lorentzian Kac-Moody algebra. We
prove that the Kostant identity also holds for these algebras, but integer
exponents do not exist, although their analogues still add up pairwise to
$\pi$. In section 4 we use the properties of the infinite Weyl groups to
propose generalised versions of Calogero models invariant under Weyl groups of
infinite order of affine, hyperbolic and Lorentzian type. Our conclusions are
stated in section 5.

##  2 Weyl groups of the $\mathbf{(A}_{2}\mathbf{)}_{-2}$-Lorentzian Kac-Moody
algebra

The essential tool in the study and classification of Lie algebras is the
Cartan matrix $K$. It encodes the structure of the algebras by capturing the
information about their weight and root systems, see e.g. [25, 26]. In turn,
this structures are inscribed into the corresponding Dynkin diagrams. The
Cartan matrix for the $\mathbf{(A}_{2}\mathbf{)}_{-2}$-Lorentzian Kac-Moody
algebra and the corresponding Dynkin diagram drawn in the usual conventions
are
$\mathbf{(A}_{2}\mathbf{)}_{-2}:$$\small{\alpha_{1}}$$\small{\alpha_{2}}$$\bullet$$\bullet$$\bullet$$\small{\alpha_{0}}$$\bullet$$\small{\alpha_{-1}}$$\bullet$$\small{\alpha_{-2}}$$\\!\\!\\!K_{ij}=2\frac{\alpha_{i}\cdot\alpha_{j}}{\alpha_{j}\cdot\alpha_{j}}=\left(\begin{array}[]{ccccc}2&-1&0&0&0\\\
-1&2&-1&0&0\\\ 0&-1&2&-1&-1\\\ 0&0&-1&2&-1\\\ 0&0&-1&-1&2\\\
\end{array}\right)_{ij},$
where we label rows and columns in the order $-2,-1,0,1,2$. Being of
Lorentzian type we expect $K$ to be nonsingular and to possess precisely one
negative eigenvalue, see [18]. Indeed, the eigenvalues are

$1,\quad 3,\quad 2+\frac{2}{\sqrt{3}}\cos\lambda\pm 2\sin\lambda,\quad
2-\frac{4}{\sqrt{3}}\cos\lambda<0,\qquad\lambda=\frac{1}{3}\arctan\left(\sqrt{\frac{37}{3}}/3\right).$
(2)

In order to reproduce $K$ from the product of roots $\alpha_{i}$ as specified,
we need to re-define the standard Euclidean inner product in the
$\Delta_{\mathbf{(A}_{2}\mathbf{)}_{-2}}$-root space. Here we choose a 3+4
dimensional representation for the $\mathbf{(A}_{2}\mathbf{)}_{-2}$-roots
$\alpha_{i}$, $i=-2,\cdots,2$, in which the positive definite part is taken in
the standard three dimensional representation, see e.g. [25, 26], and the
extensions are constructed following the prescription in [18, 19] by adding
two copies of the self-dual Lorentzian lattice $\Pi^{1,1}$-lattice

$\displaystyle\alpha_{1}$ $\displaystyle=$
$\displaystyle\left(1,-1,0;0,0,0,0\right),\quad\alpha_{2}=\left(0,1,-1;0,0,0,0\right),$
(3) $\displaystyle\alpha_{0}$ $\displaystyle=$
$\displaystyle\left(-1,0,1;1,0,0,0\right),\quad\alpha_{-1}=\left(0,0,0;-1,1,0,0\right),\quad\alpha_{-2}=\left(0,0,0;1,0,-1,1\right).$

The Lorentzian inner product for two 3+4 dimensional vectors $x=(x_{1},\ldots
x_{7})$, $y=(y_{1},\ldots y_{7})$ is then taken to be

$x\cdot
y=x_{1}y_{1}+x_{2}y_{2}+x_{3}y_{3}-x_{4}y_{5}-x_{5}y_{4}-x_{6}y_{7}-x_{7}y_{6}.$
(4)

Demanding the length of an arbitrary root

$\alpha=p\alpha_{-2}+q\alpha_{-1}+l\alpha_{0}+m\alpha_{1}+n\alpha_{2}\in\Delta_{\mathbf{(A}_{2}\mathbf{)}_{-2}},\qquad
p,q,l,m,n\in\mathbb{Z},$ (5)

to be 2, leads to the Diophantine equation

$l^{2}-l(m+n+q)+m^{2}-mn+n^{2}+p^{2}-pq+q^{2}=1\qquad\Leftrightarrow\quad\alpha\cdot\alpha=2.$
(6)

For every simple root $\alpha_{i}$ we define a Weyl reflection

$\sigma_{i}(x):=x-(\alpha_{i}\cdot x)\alpha_{i},$ (7)

about the hyperplane orthogonal to the root through the origin. Since Weyl
reflections are orthogonal transformations, the inner product,
$\alpha\cdot\alpha=\sigma_{i}(\alpha)\cdot\sigma_{i}(\alpha)$, is preserved.
The symmetries of the Diophantine equation generated by the action of the Weyl
reflections on $\alpha$ in this manner are

$\displaystyle\sigma_{-2}(\alpha):p\rightarrow
q-p,\quad\sigma_{-1}(\alpha):q\rightarrow
l+p-q,\quad\sigma_{0}(\alpha):l\rightarrow q+m+n-l,$ (8)
$\displaystyle\sigma_{1}(\alpha):m\rightarrow
l+n-m,\quad\sigma_{2}(\alpha):n\rightarrow l+m-n.$

The composition of several different Weyl reflections can be used to define
different types of Coxeter elements corresponding to the various subgroups and
orderings of the Weyl reflections.

###  2.1 Finite dimensional Coxeter orbits

Coxeter elements $\sigma$ are in general defined as products of Weyl
reflections involving all simple roots, i.e.
$\sigma:=\prod_{i=1}^{r}\sigma_{i}$ with $\alpha_{i}$ being simple and $r$
denoting the rank of the group. Since Weyl reflections do no commute in
general, different orderings in the product lead to different Coxeter
elements, although they are still in the same equivalence class and have the
same order. There are a number finite dimensional subgroups of the full
Lorentzian Coxeter group that lead to versions of Coxeter elements of finite
order that we will be briefly consider first for our example.

####  2.1.1 $A_{2}$ Coxeter orbits of order 3

Commencing with the standard definition of an $A_{2}$-Coxeter element in the
form

$\sigma_{A2}:=\sigma_{1}\sigma_{2},$ (9)

we easily calculate with (8) its action on the generic
$\mathbf{(A}_{2}\mathbf{)}_{-2}$ root $\alpha$ as defined in (5)

$\displaystyle\sigma_{A2}({\alpha})$ $\displaystyle=$ $\displaystyle
p\alpha_{-2}+q\alpha_{-1}+l\alpha_{0}+(2l-n)\alpha_{1}+(l+m-n)\alpha_{2},$
$\displaystyle\sigma_{A2}^{2}({\alpha})$ $\displaystyle=$ $\displaystyle
p\alpha_{-2}+q\alpha_{-1}+l\alpha_{0}+(l-m+n)\alpha_{1}+(2l-m)\alpha_{2},$
$\displaystyle\sigma_{A2}^{3}({\alpha})$ $\displaystyle=$
$\displaystyle\alpha.$ (10)

Orbits with similar structure and in particular the same order $3$ arise when
defining a Coxeter element by means of the product of Weyl reflections
associated to any other two roots connected on the Dynkin diagram, e.g.
$\tilde{\sigma}_{A2}:=\sigma_{-2}\sigma_{-1}$,
$\hat{\sigma}_{A2}:=\sigma_{0}\sigma_{1}$ or
$\bar{\sigma}_{A2}:=\sigma_{-1}\sigma_{0}$.

####  2.1.2 $A_{3}$ Coxeter orbits of order 4

A Coxeter element of order 4 can be constructed from three consecutive Weyl
reflections associated to roots connected on the Dynkin diagram, thus
corresponding to an $A_{3}$-diagram. For instance, defining

$\sigma_{A3}:=\sigma_{-2}\sigma_{0}\sigma_{-1},$ (11)

we compute

$\displaystyle\sigma_{A3}({\alpha})$ $\displaystyle=$
$\displaystyle(l-q)\alpha_{-2}+(l+p-q)\alpha_{-1}+(m+n+p-q)\alpha_{0}+m\alpha_{1}+n\alpha_{2},$
$\displaystyle\sigma_{A3}^{2}({\alpha})$ $\displaystyle=$
$\displaystyle(-l+m+n)\alpha_{-2}+(m+n-q)\alpha_{-1}+(m+n-p)\alpha_{0}+m\alpha_{1}+n\alpha_{2},$
$\displaystyle\sigma_{A3}^{3}({\alpha})$ $\displaystyle=$
$\displaystyle(q-p)\alpha_{-2}+(-l+m+n-p+q)\alpha_{-1}+(-l+m+n+q)\alpha_{0}+m\alpha_{1}+n\alpha_{2},$
$\displaystyle\sigma_{A3}^{4}({\alpha})$ $\displaystyle=$
$\displaystyle\alpha.$ (12)

Thus the Coxeter element $\sigma_{A3}$ has order 4.

####  2.1.3 $A_{4}$ Coxeter orbits of order 5

Similarly, when defining a Coxeter element from four consecutive Weyl
reflections associated to roots connected parts of the Dynkin diagram we
obtain an $A_{4}$-diagram. In this manner we obtain one of finite order 5. For
instance, defining

$\sigma_{A4}:=\sigma_{-2}\sigma_{0}\sigma_{-1}\sigma_{1},$ (13)

we compute

$\displaystyle\sigma_{A4}({\alpha})$ $\displaystyle=$
$\displaystyle(l-q)\alpha_{-2}+(l+p-q)\alpha_{-1}+(l-m+2n+p-q)\alpha_{0}+(l-m+n)\alpha_{1}+n\alpha_{2},$
$\displaystyle\sigma_{A4}^{2}({\alpha})$ $\displaystyle=$
$\displaystyle(2n-m)\alpha_{-2}+(l-m+2n-q)\alpha_{-1}+(3n-q)\alpha_{0}+(2n+p-q)\alpha_{1}+n\alpha_{2},$
$\displaystyle\sigma_{A4}^{3}({\alpha})$ $\displaystyle=$
$\displaystyle(m+n-l)\alpha_{-2}+(3n-l)\alpha_{-1}+(3n-p+q-l)\alpha_{0}+(2n-p)\alpha_{1}+n\alpha_{2},$
$\displaystyle\sigma_{A4}^{4}({\alpha})$ $\displaystyle=$
$\displaystyle(q-p)\alpha_{-2}+(m+n-p+q-l)\alpha_{-1}+(m+n+q-l)\alpha_{0}+(2n+q-l)\alpha_{1}+n\alpha_{2},$
$\displaystyle\sigma_{A4}^{5}({\alpha})$ $\displaystyle=$
$\displaystyle\alpha.$ (14)

Hence $\sigma_{A4}$ has order 5.

####  2.1.4 $A_{1}\otimes A_{2}$ Coxeter orbits of order 6

Naturally we may also define Coxeter elements associated to roots
corresponding to disconnected Dynkin diagrams, which lead to direct product of
the subgroups. For instance, defining

$\sigma_{A12}:=\sigma_{1}\sigma_{-2}\sigma_{-1},$ (15)

leads to orbits of order 6, associated to $A_{1}\otimes A_{2}$. We compute

$\displaystyle\sigma_{A12}({\alpha})$ $\displaystyle=$
$\displaystyle(l-q)\alpha_{-2}+(l+p-q)\alpha_{-1}+l\alpha_{0}+(l-m+n)\alpha_{1}+n\alpha_{2},$
$\displaystyle\sigma_{A12}^{2}({\alpha})$ $\displaystyle=$
$\displaystyle(q-p)\alpha_{-2}+(l-p)\alpha_{-1}+l\alpha_{0}+m\alpha_{1}+n\alpha_{2},$
$\displaystyle\sigma_{A12}^{3}({\alpha})$ $\displaystyle=$ $\displaystyle
p\alpha_{-2}+q\alpha_{-1}+l\alpha_{0}+(l-m+n)\alpha_{1}+n\alpha_{2},$
$\displaystyle\sigma_{A12}^{4}({\alpha})$ $\displaystyle=$
$\displaystyle(l-q)\alpha_{-2}+(l+p-q)\alpha_{-1}+l\alpha_{0}+m\alpha_{1}+n\alpha_{2},$
$\displaystyle\sigma_{A12}^{5}({\alpha})$ $\displaystyle=$
$\displaystyle(q-p)\alpha_{-2}+(l-p)\alpha_{-1}+l\alpha_{0}+(l-m+n)\alpha_{1}+n\alpha_{2},$
$\displaystyle\sigma_{A12}^{6}({\alpha})$ $\displaystyle=$
$\displaystyle\alpha.$ (16)

The order $6$ of this Coxeter element simply results from multiplying the two
Coxeter numbers of $A_{1}$ and $A_{2}$, i.e. $h_{A_{1}}\times
h_{A_{2}}=2\times 3$.

###  2.2 Infinite dimensional $\mathbf{(A}_{2}\mathbf{)}_{0}$-affine Kac-Moody
Coxeter orbits

We now come to the cases of our main interest, which is the construction of
infinite Weyl groups that consequently possess Coxeter elements of infinite
order. The simplest example for such a group is the affine extension of the
$A_{2}$ Lie algebra, see e.g. [27]. We will show in detail how the generic
formula for the action of an arbitrary power of the Coxeter element on a root
can be derived. Defining a specific Coxeter element for the
$\mathbf{(A}_{2}\mathbf{)}_{0}$-affine Kac-Moody algebra as

$\sigma_{a}:=\sigma_{0}\sigma_{1}\sigma_{2},$ (17)

we calculate its orbits. When acting consecutively with $\sigma_{a}$ on the
arbitrary root $\alpha$ in (5) we compute

$\displaystyle\sigma_{a}({\alpha})$ $\displaystyle=$ $\displaystyle
p\alpha_{-2}+q\alpha_{-1}+(2l+m-2n+q)\alpha_{0}+(2l-n)\alpha_{1}+(l+m-n)\alpha_{2},$
(18) $\displaystyle\sigma_{a}^{2}({\alpha})$ $\displaystyle=$ $\displaystyle
p\alpha_{-2}+q\alpha_{-1}+(4l-3n+3q)\alpha_{0}+(3l+m-3n+2q)\alpha_{1}+(3l-2n+q)\alpha_{2}.$

For arbitrary powers we obtain

$\sigma_{a}^{k}(\alpha):=p\alpha_{-2}+q\alpha_{-1}+\sum_{\nu=0}^{2}\sum_{\mu=p,q,l,m,n}a_{\nu}^{\mu}(k)\mu\alpha_{\nu},$
(19)

where the challenge is to determine the coefficients $a_{\nu}^{\mu}(k)$ for
any power $k\in\mathbb{Z}$. When acting with $\sigma$ on the generic
expressions in (19) we easily identify the recurrence relations that need to
be satisfied

$\displaystyle a_{0}^{\mu}(k+1)$ $\displaystyle=$ $\displaystyle
a_{-1}^{\mu}(k)+2a_{0}^{\mu}(k)+a_{1}^{\mu}(k)-2a_{2}^{\mu}(k),$ (20)
$\displaystyle a_{1}^{\mu}(k+1)$ $\displaystyle=$ $\displaystyle
2a_{0}^{\mu}(k)-a_{2}^{\mu}(k),$ (21) $\displaystyle a_{2}^{\mu}(k+1)$
$\displaystyle=$ $\displaystyle a_{0}^{\mu}(k)+a_{1}^{\mu}(k)-a_{2}^{\mu}(k).$
(22)

Thus we may write these equations more compactly as

$a_{\nu}^{\mu}(k+1)=\left(Ma^{\mu}(k)\right)_{\nu}=\left(M^{k+1}a^{\mu}(0)\right)_{\nu},$
(23)

involving the matrix and the vector

$M=\left(\begin{array}[]{ccccc}1&0&0&0&0\\\ 0&1&0&0&0\\\ 0&1&2&1&-2\\\
0&0&2&0&-1\\\ 0&0&1&1&-1\\\ \end{array}\right),\quad\text{and}\quad
a^{\mu}(k)=[a_{-2}^{\mu}(k),a_{-1}^{\mu}(k),a_{0}^{\mu}(k),a_{1}^{\mu}(k),a_{2}^{\mu}(k)],$
(24)

respectively. Taking as initial condition the root $\alpha$ as defined in (5),
we have

$a_{-2}^{p}(0)=a_{-1}^{q}(0)=a_{0}^{l}(0)=a_{1}^{m}(0)=a_{2}^{n}(0)=1,\quad\text{and}\quad
a_{\nu}^{\mu}(0)=0\,\,\text{otherwise},$ (25)

so that the general solution is

$a_{\nu}^{\mu}(k)=(M^{k})_{\nu\mu}.$ (26)

In order to compute the powers of the matrix $M$ we may assume that the
coefficient functions satisfy a recurrence relation of the general form

$a_{\nu}^{\mu}(k+1)=\sum_{i=1}^{N}c_{i}a_{\nu}^{\mu}(k+1-i),$ (27)

with some constants $c_{i}\in\mathbb{C}$ and $N\in\mathbb{N}$ that need to be
determined. See for instance [28] for a treatment of such type of recurrence
relations and for general techniques to solve them. Employing the matrix
representation (24) we find for $N=4$ indeed the following recurrence relation
to hold

$a_{\nu}^{\mu}(k+1)=2a_{\nu}^{\mu}(k)-2a_{\nu}^{\mu}(k-2)+b_{\nu}^{\mu}(k-3).$
(28)

The corresponding fourth order characteristic equation for this relation
resulting from $a_{\nu}^{\mu}(k)\sim x^{k}$ is

$x^{4}-2x^{3}+2x-1=0.$ (29)

The four roots of this equation are found to be

$\lambda_{1}=-1,\quad\lambda_{2/3/4}=1.$ (30)

As we have a threefold degenerate solution to the chracteristic equation, the
general solution to the recurrence relation (28) is

$a_{\nu}^{\mu}(k)=\tilde{c}_{1}(-1)^{k}+\tilde{c}_{2}+\tilde{c}_{3}k+\tilde{c}_{4}k^{2},$
(31)

where the constant coefficients $\tilde{c}_{i}$, $i=1,\ldots,4$ are determined
from the explicit evaluation of the lowest powers of the matrix in (26). In
this manner we find the generic formula

$\displaystyle\sigma_{a}^{k}(\alpha)$ $\displaystyle=$ $\displaystyle
p\alpha_{-2}+q\alpha_{-1}$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!+\left[\frac{1}{8}\left(6k^{2}+1\right)q+\frac{1}{4}(6k+3)l-\frac{1}{4}(6k+1)n+\frac{m}{2}+(-1)^{k}\frac{2l-4m+2n-q}{8}\right]\alpha_{0}$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!+\left[\frac{1}{8}\left(6k^{2}-4k-1\right)q+\frac{1}{4}(6k+1)l-\frac{1}{4}(6k-1)n+\frac{m}{2}-(-1)^{k}\frac{2l-4m+2n-q}{8}\right]\alpha_{1}$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!+\left[\frac{1}{8}\left(6k^{2}-8k+1\right)q+\frac{1}{4}(6k-1)l-\frac{1}{4}(6k-3)n+\frac{m}{2}+(-1)^{k}\frac{2l-4m+2n-q}{8}\right]\alpha_{2}.$

Thus, as expected, $\sigma_{a}$ defined in equation (17) is a Coxeter element
of infinite order.

###  2.3 Infinite dimensional $\mathbf{(A}_{2}\mathbf{)}_{-1}$-hyperbolic Kac-
Moody Coxeter orbits

Besides affine extensions of finite dimensional algebras [27] also the
hyperbolic algebras have been fully classified [29]. These algebras can be
defined based on their connected Dynkin diagrams, which exhibit the property
that the removal of _any_ node results in a set of connected Dynkin diagrams,
possibly disconnected, all of which are of finite type except for at most one
which may be of affine type. This is easily checked to be true for the Dynkin
diagram at the beginning of section 2 with $\alpha_{-2}$ removed.

The simplest example for such an algebra is the extended affine
$A_{2}$-algebra, i.e. the $\mathbf{(A}_{2}\mathbf{)}_{-1}$-hyperbolic Kac-
Moody algebra for which we define a specific Coxeter element as

$\sigma_{h}:=\sigma_{-1}\sigma_{0}\sigma_{1}\sigma_{2},$ (33)

and calculate its infinite dimensional orbits. Acting on the arbitrary
$\mathbf{(A}_{2}\mathbf{)}_{-2}$ root $\alpha$ in (5) we compute the first
consecutive actions as

$\displaystyle\sigma_{h}({\alpha})$ $\displaystyle=$ $\displaystyle
p\alpha_{-2}+(2l+m-2n+p)\alpha_{-1}+(2l+m-2n+q)\alpha_{0}+(2l-n)\alpha_{1}\qquad\qquad$
$\displaystyle+(l+m-n)\alpha_{2},$ $\displaystyle\sigma_{h}^{2}(\alpha)$
$\displaystyle=$ $\displaystyle
p\alpha_{-2}+(4l-3n+p+2q)\alpha_{-1}+(6l+m-5n+p+2q)\alpha_{0}$
$\displaystyle+(3l+m-3n+2q)\alpha_{1}+(3l-2n+q)\alpha_{2},$

etc. For arbitrary powers we assume a similar expansion as in (19) of the
previous section

$\sigma_{h}^{k}(\alpha):=p\alpha_{-2}+\sum_{\nu=-1}^{2}\sum_{\mu=p,q,l,m,n}b_{\nu}^{\mu}(k)\mu\alpha_{\nu},$
(36)

where $b_{\nu}^{\mu}(k),k\in\mathbb{Z}$. Acting with $\sigma_{h}$ on the
generic expansion in (36) we identify the recurrence relations to be satisfied
by the coefficients $b_{\nu}^{\mu}(k)$

$\displaystyle b_{-1}^{\mu}(k+1)$ $\displaystyle=$ $\displaystyle
b_{-2}^{\mu}(k)+2b_{0}^{\mu}(k)+b_{1}^{\mu}(k)-2b_{2}^{\mu}(k),$ (37)
$\displaystyle b_{0}^{\mu}(k+1)$ $\displaystyle=$ $\displaystyle
b_{-1}^{\mu}(k)+2b_{0}^{\mu}(k)+b_{1}^{\mu}(k)-2b_{2}^{\mu}(k),$ (38)
$\displaystyle b_{1}^{\mu}(k+1)$ $\displaystyle=$ $\displaystyle
2b_{0}^{\mu}(k)-b_{2}^{\mu}(k),$ (39) $\displaystyle b_{2}^{\mu}(k+1)$
$\displaystyle=$ $\displaystyle b_{0}^{\mu}(k)+b_{1}^{\mu}(k)-b_{2}^{\mu}(k).$
(40)

Again we may cast the recurrence relations into matrix form as

$b_{\nu}^{\mu}(k+1)=\left(\hat{M}b^{\mu}(k)\right)_{\nu}=\left(\hat{M}^{k+1}b^{\mu}(0)\right)_{\nu}$
(41)

involving the matrix and the vector

$\hat{M}=\left(\begin{array}[]{ccccc}1&0&0&0&0\\\ 1&0&2&1&-2\\\ 0&1&2&1&-2\\\
0&0&2&0&-1\\\ 0&0&1&1&-1\\\ \end{array}\right),\quad\text{and}\quad
b^{\mu}(k)=[b_{-2}^{\mu}(k),b_{-1}^{\mu}(k),b_{0}^{\mu}(k),b_{1}^{\mu}(k),b_{2}^{\mu}(k)],$
(42)

respectively. Taking $\alpha$ to be of the form (5) the initial conditions are

$b_{-2}^{p}(0)=bc_{-1}^{q}(0)=b_{0}^{l}(0)=b_{1}^{m}(0)=b_{2}^{n}(0)=1,\quad\text{and}\quad
b_{\nu}^{\mu}(0)=0\,\,\text{otherwise}.$ (43)

As in the previous section we may consecutively calculate all the coefficients
$b_{\nu}^{\mu}(k)$ with some given initial condition. We also assume here that
these coefficients satisfy a linear recurrence relation of the form

$b_{\nu}^{\mu}(k+1)=\sum_{i=1}^{N}c_{i}b_{\nu}^{\mu}(k+1-i),$ (44)

with constants $c_{i}\in\mathbb{C}$ and $N\in\mathbb{N}$ that need to be
determined. For $N=5$ we find that indeed the following recurrence relation
holds for all coefficients

$b_{\nu}^{\mu}(k+1)=2b_{\nu}^{\mu}(k)+2b_{\nu}^{\mu}(k-1)-2b_{\nu}^{\mu}(k-2)-2b_{\nu}^{\mu}(k-3)+b_{\nu}^{\mu}(k-4).$
(45)

Following the same procedure as in the previous section we determine the fifth
order characteristic polynomial for (45), by assuming $b_{\nu}^{\mu}(k)\sim
x^{k}$. The characteristic equation then reads

$x^{5}-2x^{4}-2x^{3}+2x^{2}+2x-1=0.$ (46)

We find three real and two complex roots to equation (46)

$\lambda_{1}=1,\quad\lambda_{2/3}=\frac{1}{4}\left[1+\sqrt{21}\pm\sqrt{2\sqrt{21}+6}\right],\quad\lambda_{4/5}=\frac{1}{4}\left[1-\sqrt{21}\pm
i\sqrt{2\sqrt{21}-6}\right].$ (47)

Thus according to the general theory of recurrence relations, see e.g. [28],
the solution to equation (45) can be cast into the form

$b_{\nu}^{\mu}(k)=b_{\mu 1}^{\nu}\lambda_{1}^{k}+b_{\mu
2}^{\nu}\lambda_{2}^{k}+b_{\mu 3}^{\nu}\lambda_{3}^{k}+b_{\mu
4}^{\nu}\lambda_{4}^{k}+b_{\mu 5}^{\nu}\lambda_{5}^{k},$ (48)

where the constant coefficients $b_{\mu i}^{\nu}$, $i=1,\ldots,5$ are
determined from the initial conditions $b_{\nu}^{\mu}(0)$ and the first four
iterations that give $b_{\nu}^{\mu}(1)\ldots,b_{\nu}^{\mu}(4)$ for different
$\nu$ and $\mu$. To stress the reality of these solutions we can equivalently
express the coefficients as

$b_{\nu}^{\mu}(k)=b_{\mu 1}^{\nu}+\hat{b}_{\mu
2}^{\nu}\Lambda_{2}^{k}+\hat{b}_{\mu 3}^{\nu}\Lambda_{3}^{k}+\hat{b}_{\mu
4}^{\nu}\Lambda_{4}^{k}+\hat{b}_{\mu 5}^{\nu}\Lambda_{5}^{k},,$ (49)

where $b_{\mu 2/3}^{\nu}=\hat{b}_{\mu 3}^{\nu}\pm\hat{b}_{\mu 2}^{\nu}$,
$b_{\mu 4/5}^{\nu}=(\hat{b}_{\mu 5}^{\nu}\mp\hat{b}_{\mu 4}^{\nu})/2$ and

$\Lambda_{2/3}^{k}=\frac{1}{\sqrt{21}}(\lambda_{2}^{k}\mp\lambda_{3}^{k}),\qquad\Lambda_{4}^{k}=\frac{1}{2\sqrt{21}}i(\lambda_{5}^{k}-\lambda_{4}^{k})\qquad\Lambda_{5}^{k}=\frac{1}{2\sqrt{21}}(\lambda_{5}^{k}+\lambda_{4}^{k}).$
(50)

Using the initial values as stated above we find the solutions

$\displaystyle b^{p}_{-1}$ $\displaystyle=$
$\displaystyle\Lambda_{3}^{k}-2\Lambda_{5}^{k}$ (51) $\displaystyle
b^{q}_{-1}$ $\displaystyle=$
$\displaystyle\frac{1}{2}[\mu_{-}(\mu_{-}\Lambda_{3}^{k}+2\Lambda_{4}^{k})+\mu_{+}(\Lambda_{2}^{k}+2\mu_{+}\Lambda_{5}^{k})]$
(52) $\displaystyle b^{l}_{-1}$ $\displaystyle=$
$\displaystyle\frac{1}{2}[\varphi_{+}\Lambda_{2}^{k}+3\Lambda_{3}^{k}-2\varphi_{-}\Lambda_{4}^{k}-6\Lambda_{5}^{k}]$
(53) $\displaystyle b^{m}_{-1}$ $\displaystyle=$
$\displaystyle\frac{1}{6}[\sqrt{3}\mu_{-}^{3}\Lambda_{2}^{k}+3\Lambda_{3}^{k}+2\sqrt{3}\mu_{+}^{3}\Lambda_{4}^{k}-6\Lambda_{5}^{k}]$
(54) $\displaystyle b^{n}_{-1}$ $\displaystyle=$
$\displaystyle-\mu_{+}\Lambda_{2}^{k}-\Lambda_{3}^{k}-2\mu_{-}\Lambda_{4}^{k}+2\Lambda_{5}^{k}$
(55) $\displaystyle b^{p}_{0}$ $\displaystyle=$
$\displaystyle-1-\frac{1}{2}\mu_{+}\Lambda_{2}^{k}+\frac{1}{6}\mu_{+}^{4}\Lambda_{3}^{k}-\mu_{-}\Lambda_{4}^{k}-\frac{1}{3}\mu_{-}^{4}\Lambda_{5}^{k}$
(56) $\displaystyle b^{q}_{0}$ $\displaystyle=$
$\displaystyle\mu_{+}\Lambda_{2}^{k}+2\mu_{-}\Lambda_{4}^{k}$ (57)
$\displaystyle b^{l}_{0}$ $\displaystyle=$
$\displaystyle\frac{1}{6}[\mu_{+}^{2}(\sqrt{3}\varphi_{-}\Lambda_{2}^{k}+3\Lambda_{3}^{k})-2\mu_{-}^{2}(\sqrt{3}\varphi_{+}\Lambda_{4}^{k}-3\Lambda_{5}^{k})]\qquad\qquad$
(58) $\displaystyle b^{m}_{0}$ $\displaystyle=$
$\displaystyle\Lambda_{3}^{k}-2\Lambda_{5}^{k}$ (59) $\displaystyle b^{n}_{0}$
$\displaystyle=$
$\displaystyle-\frac{1}{2}\mu_{+}^{3}\Lambda_{2}^{k}-\frac{1}{2}\Lambda_{3}^{k}+\mu_{-}^{3}\Lambda_{4}^{k}+\Lambda_{5}^{k}$
(60)

$\displaystyle b^{p}_{1}$ $\displaystyle=$
$\displaystyle-1+\frac{1}{6}[-\sqrt{21}\mu_{+}\Lambda_{2}^{k}+\mu_{+}^{4}\Lambda_{3}^{k}+2\sqrt{21}\mu_{-}\Lambda_{4}^{k}-2\mu_{-}^{4}\Lambda_{5}^{k}]\qquad\qquad$
(61) $\displaystyle b^{q}_{1}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\varphi_{+}\Lambda_{2}^{k}-\frac{1}{2}\Lambda_{3}^{k}-\varphi_{-}\Lambda_{4}^{k}+\Lambda_{5}^{k}$
(62) $\displaystyle b^{l}_{1}$ $\displaystyle=$
$\displaystyle\mu_{+}\Lambda_{2}^{k}+\Lambda_{3}^{k}+2\mu_{-}\Lambda_{4}^{k}-2\Lambda_{5}^{k}$
(63) $\displaystyle b^{m}_{1}$ $\displaystyle=$
$\displaystyle-\frac{1}{2\sqrt{3}}(\mu_{-}\Lambda_{2}^{k}-2\mu_{+}\Lambda_{4}^{k})+\frac{1}{2\sqrt{21}}(\theta_{-}^{2}\Lambda_{3}^{k}+2\theta_{+}^{2}\Lambda_{5}^{k})$
(64) $\displaystyle b^{n}_{1}$ $\displaystyle=$
$\displaystyle-\frac{1}{6}\tau_{-}\mu_{+}^{3}\Lambda_{2}^{k}+\frac{1}{2}\Lambda_{3}^{k}-\frac{\tau_{+}\mu_{-}^{3}}{3\sqrt{21}}\Lambda_{4}^{k}-\Lambda_{5}^{k}$
(65) $\displaystyle b^{p}_{2}$ $\displaystyle=$
$\displaystyle-1+\frac{1}{6}[-\mu_{+}^{2}(\varphi_{+}\Lambda_{2}^{k}-\sqrt{21}\Lambda_{3}^{k})-2\mu_{-}^{2}(\varphi_{-}\Lambda_{4}^{k}+\sqrt{21}\Lambda_{5}^{k})]$
(66) $\displaystyle b^{q}_{2}$ $\displaystyle=$
$\displaystyle\mu_{+}\Lambda_{2}^{k}-\Lambda_{3}^{k}+2\Lambda_{5}^{k}+\frac{2\mu_{-}}{\sqrt{21}}\Lambda_{4}^{k}$
(67) $\displaystyle b^{l}_{2}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\varphi_{+}\Lambda_{2}^{k}+\frac{1}{2}\Lambda_{3}^{k}-\varphi_{-}\Lambda_{4}^{k}-\Lambda_{5}^{k}$
(68) $\displaystyle b^{m}_{2}$ $\displaystyle=$
$\displaystyle-\frac{\mu_{-}}{3\sqrt{7}}\Lambda_{2}^{k}+\Lambda_{3}^{k}+\frac{2\mu_{+}}{\sqrt{3}}\Lambda_{4}^{k}-2\Lambda_{5}^{k}$
(69) $\displaystyle b^{n}_{2}$ $\displaystyle=$
$\displaystyle-\frac{1}{126}[\theta_{+}^{2}(\mu_{+}^{3}\Lambda_{2}^{k}-3\Lambda_{3}^{k})+2\theta_{-}^{2}(\mu_{-}^{3}\Lambda_{4}^{k}-3\Lambda_{5}^{k})]$
(70)

We abbreviated here some of the lengthy square root constants to achieve a
more compact presentation

$\displaystyle\mu_{\pm}$ $\displaystyle:=$
$\displaystyle\sqrt{\frac{1}{2}\left(\sqrt{21}\pm
3\right)},\quad\mu_{\pm}:=\sqrt{2\sqrt{21}\pm 3},$ (71)
$\displaystyle\theta_{\pm}$ $\displaystyle:=$
$\displaystyle\sqrt{\frac{1}{2}(21\pm\sqrt{21})},\quad\tau_{\pm}:=\sqrt{25\pm
4\sqrt{21}}.$ (72)

Evidently the Coxeter element $\sigma_{h}$ as defined in (33) is of infinite
order.

###  2.4 Infinite dimensional $\mathbf{(A}_{2}\mathbf{)}_{-2}$-Lorentzian Kac-
Moody Coxeter orbits

Adding a further node to the $(A_{2})_{-1}$-diagram at the $\alpha_{-1}$-node
leads to a diagram that no longer satisfies the classification criteria for a
hyperbolic Kac-Moody algebra. This is because when removing the
$\alpha_{-2}$-node one is not left with a set of disconnected diagrams of
finite type except for at most one affine type. The diagram obtained by
removing this node is still of hyperbolic type. Thus the new diagram as
depicted at the beginning of section 2 is of a new type, referred to as
Lorentzian. To characterise these diagrams it was proposed in [18] to relax
the defining criterion for the hyperbolic Kac-Moody algebra in the sense that
the required decomposition must not be obtained for all nodes but only _for at
least one_. Our $(A_{2})_{-2}$-example clearly satisfies this requirement as
it even holds for all nodes except for the $\alpha_{-2}$-node.

We define a Coxeter element for this algebra in the usual manner by the
product of Weyl reflections associated to each simple root in a particular
order

$\sigma_{L}:=\sigma_{-2}\sigma_{-1}\sigma_{0}\sigma_{1}\sigma_{2}.$ (73)

We calculate its orbits by acting with $\sigma_{L}$ consecutively on the root
$\alpha$ in (5), computing

$\displaystyle\sigma_{L}({\alpha})$ $\displaystyle=$
$\displaystyle(2l+m-2n)\alpha_{-2}+(2l+m-2n+p)\alpha_{-1}+(2l+m-2n+q)\alpha_{0}$
$\displaystyle+(2l-n)\alpha_{1}+(l+m-n)\alpha_{2}$
$\displaystyle\sigma_{L}^{2}(\alpha)$ $\displaystyle=$
$\displaystyle(4l-3n+2q)\alpha_{-2}+(6l+m-5n+2q)\alpha_{-1}+(6l+m-5n+p+2q)\alpha_{0},\quad$
$\displaystyle+(3l+m-3n+2q)\alpha_{1}+(3l-2n+q)\alpha_{2}.$

For arbitrary powers we make a similar assumption as in the previous sections

$\sigma_{L}^{k}(\alpha):=\sum_{\nu=-2}^{2}\sum_{\mu=p,q,l,m,n}c_{\nu}^{\mu}(k)\mu\alpha_{\nu},$
(76)

with unknown constants $c_{\nu}^{\mu}(k),k\in\mathbb{Z}$. Acting with
$\sigma_{L}$ on the generic expressions in (76) we easily identify the
recurrence relations to be satisfied by the coefficients $c_{\nu}^{\mu}(k)$

$\displaystyle c_{-2}^{\mu}(k+1)$ $\displaystyle=$ $\displaystyle
2c_{0}^{\mu}(k)+c_{1}^{\mu}(k)-2c_{2}^{\mu}(k),$ (77) $\displaystyle
c_{-1}^{\mu}(k+1)$ $\displaystyle=$ $\displaystyle
c_{-2}^{\mu}(k)+2c_{0}^{\mu}(k)+c_{1}^{\mu}(k)-2c_{2}^{\mu}(k),$ (78)
$\displaystyle c_{0}^{\mu}(k+1)$ $\displaystyle=$ $\displaystyle
c_{-1}^{\mu}(k)+2c_{0}^{\mu}(k)+c_{1}^{\mu}(k)-2c_{2}^{\mu}(k),$ (79)
$\displaystyle c_{1}^{\mu}(k+1)$ $\displaystyle=$ $\displaystyle
2c_{0}^{\mu}(k)-c_{2}^{\mu}(k),$ (80) $\displaystyle c_{2}^{\mu}(k+1)$
$\displaystyle=$ $\displaystyle c_{0}^{\mu}(k)+c_{1}^{\mu}(k)-c_{2}^{\mu}(k).$
(81)

Also in this case a more compact form of the recurrence relations can be
achieved in a matrix formulation

$c_{\nu}^{\mu}(k+1)=\left(\tilde{M}c^{\mu}(k)\right)_{\nu}=\left(\tilde{M}^{k+1}c^{\mu}(0)\right)_{\nu}$
(82)

involving the matrix and the vector

$\tilde{M}=\left(\begin{array}[]{ccccc}0&0&2&1&-2\\\ 1&0&2&1&-2\\\
0&1&2&1&-2\\\ 0&0&2&0&-1\\\ 0&0&1&1&-1\\\
\end{array}\right),\quad\text{and}\quad
c^{\mu}(k)=[c_{-2}^{\mu}(k),c_{-1}^{\mu}(k),c_{0}^{\mu}(k),c_{1}^{\mu}(k),c_{2}^{\mu}(k)],$
(83)

respectively. With $\alpha$ in (5) the initial conditions are

$c_{-2}^{p}(0)=c_{-1}^{q}(0)=c_{0}^{l}(0)=c_{1}^{m}(0)=c_{2}^{n}(0)=1,\quad\text{and}\quad
c_{\nu}^{\mu}(0)=0\,\,\text{otherwise}.$ (84)

We assume once more that these coefficients satisfy a linear recurrence
relation of the form

$c_{\nu}^{\mu}(k+1)=\sum_{i=1}^{N}\tilde{c}_{i}c_{\nu}^{\mu}(k+1-i),$ (85)

with some constants $\tilde{c}_{i}\in\mathbb{R}$ and $N\in\mathbb{N}$ that
need to be determined. For $N=5$ we find that the following recurrence
relation holds for all coefficients

$c_{\nu}^{\mu}(k+1)=c_{\nu}^{\mu}(k)+3c_{\nu}^{\mu}(k-1)+3c_{\nu}^{\mu}(k-2)+c_{\nu}^{\mu}(k-3)-c_{\nu}^{\mu}(k-4).$
(86)

The fifth order characteristic equation for this relation resulting from
$c_{\nu}^{\mu}(k)\sim x^{k}$ is

$x^{5}-x^{4}-3x^{3}-3x^{2}-x+1=0.$ (87)

Similarly as for the hyperbolic case we find three real and two complex roots
to this equation

$\lambda_{1}=-1,\quad\lambda_{2/3}=\frac{1}{2}\left(3\pm\sqrt{5}\right)=\phi^{\pm
2},\quad\lambda_{4/5}=\omega^{\pm 1},$ (88)

where $\omega$ is a third root of unity $\omega:=\exp(2\pi i/3)$ and $\phi$ is
the golden ratio $\phi:=(1+\sqrt{5})/2$ with negative inverse golden ratio
$\bar{\phi}:=(1-\sqrt{5})/2$. Thus the general solution to the recurrence
relation (86) is

$c_{\nu}^{\mu}(k)=c_{1}\lambda_{1}^{k}+c_{2}\lambda_{2}^{k}+c_{3}\lambda_{3}^{k}+c_{4}\lambda_{4}^{k}+c_{5}\lambda_{5}^{k},$
(89)

where the constant coefficients $c_{i}$, $i=1,\ldots,5$ are determined from
the initial conditions $c_{\nu}^{\mu}(0)$ and the first four explicit
iterations $c_{\nu}^{\mu}(1)\ldots,c_{\nu}^{\mu}(4)$ for different $\nu$ and
$\mu$. Performing this calculation we find the solutions

$\displaystyle\tilde{c}_{-2}^{m}(k)$ $\displaystyle=$ $\displaystyle
L_{2k+2}+10\sin\left(\frac{\pi}{6}+\frac{2\pi k}{3}\right)-8(-1)^{k}$ (90)
$\displaystyle\tilde{c}_{-2}^{n}(k)$ $\displaystyle=$ $\displaystyle
L_{2k}-L_{2k+5}+10\sin\left(\frac{2\pi k}{3}-\frac{\pi}{6}\right)+4(-1)^{k},$
(91) $\displaystyle\tilde{c}_{-2}^{p}(k)$ $\displaystyle=$ $\displaystyle
L_{2k+1}+10\sqrt{3}\cos\left(\frac{2\pi k}{3}-\frac{\pi}{6}\right)+4(-1)^{k},$
(92) $\displaystyle\tilde{c}_{-1}^{m}(k)$ $\displaystyle=$ $\displaystyle
2L_{2k}+L_{2k+1}-10\sin\left(\frac{2\pi k}{3}-\frac{\pi}{6}\right),$ (93)
$\displaystyle\tilde{c}_{-1}^{n}(k)$ $\displaystyle=$ $\displaystyle-
L_{2k}-8L_{2k+1}+10\cos\left(\frac{2\pi k}{3}\right),$ (94)
$\displaystyle\tilde{c}_{-1}^{p}(k)$ $\displaystyle=$ $\displaystyle-
L_{2k}+2L_{2k+1}+10\sqrt{3}\sin\left(\frac{2\pi k}{3}\right),$ (95)
$\displaystyle\tilde{c}_{0}^{m}(k)$ $\displaystyle=$ $\displaystyle
4\left(L_{2k}-2\right)=F_{k}^{2},$ (96) $\displaystyle\tilde{c}_{0}^{n}(k)$
$\displaystyle=$ $\displaystyle
4\left(L_{2k}-3L_{2k+1}+1\right)=F_{k}F_{k-1},$ (97)
$\displaystyle\tilde{c}_{0}^{p}(k)$ $\displaystyle=$ $\displaystyle
4\left(L_{2k-1}+1\right)=F_{k}(F_{k}-3F_{k+1}),$ (98)
$\displaystyle\tilde{c}_{1}^{m}(k)$ $\displaystyle=$ $\displaystyle
4L_{2k}-L_{2k+1}+\frac{10}{\sqrt{3}}\cos\left(\frac{2\pi
k}{3}+\frac{\pi}{6}\right)+8(-1)^{k},$ (99)
$\displaystyle\tilde{c}_{1}^{n}(k)$ $\displaystyle=$ $\displaystyle
7L_{2k}-10L_{2k+1}-\frac{10}{\sqrt{3}}\sin\left(\frac{2\pi
k}{3}\right)-4(-1)^{k},$ (100) $\displaystyle\tilde{c}_{1}^{p}(k)$
$\displaystyle=$ $\displaystyle 4L_{2k+1}-5L_{2k}+10\cos\left(\frac{2\pi
k}{3}\right)-4(-1)^{k},$ (101) $\displaystyle\tilde{c}_{2}^{m}(k)$
$\displaystyle=$ $\displaystyle
5L_{2k}-2L_{2k+1}+\frac{10}{\sqrt{3}}\sin\left(\frac{2\pi
k}{3}\right)-8(-1)^{k},$ (102) $\displaystyle\tilde{c}_{2}^{n}(k)$
$\displaystyle=$
$\displaystyle-11L_{2k-1}+\frac{10}{\sqrt{3}}\cos\left(\frac{\pi}{6}-\frac{2\pi
k}{3}\right)+4(-1)^{k},$ (103) $\displaystyle\tilde{c}_{2}^{p}(k)$
$\displaystyle=$ $\displaystyle
5L_{2k+1}-7L_{2k}+10\sin\left(\frac{\pi}{6}+\frac{2\pi
k}{3}\right)+4(-1)^{k},$ (104) $\displaystyle\tilde{c}_{\nu}^{p}(k)$
$\displaystyle=$
$\displaystyle\tilde{c}_{\nu}^{q}(k-1)=\tilde{c}_{\nu}^{l}(k-2),$ (105)

where $c_{\nu}^{\mu}(k)=1/20\tilde{c}_{\nu}^{\mu}(k)$, $F_{n}$ label the
Fibonacci numbers with $n\in\mathbb{N}$ and $L_{n}$ are Lukas numbers
$L_{n}:=\phi^{n}+\bar{\phi}^{n}$. The Coxeter element $\sigma_{L}$ as defined
in (73) is therefore of infinite order.

Our next aim here is to utilise the mathematical structures discussed in this
section in the context of some physical models. In particular we will
formulate Calogero type systems that respect these infinite symmetries. An
important question to address is whether the models based on infinite
dimensional root spaces are integrable? As in the finite case, the existence
and knowledge of the polynomial invariants largely facilitates to answer that
question. One may attempt to construct them from scratch, but there is an
elegant way to obtain them for Weyl groups associated to Dynkin diagrams that
can be bicoloured [13, 14, 15], that is to associate one of two colours to
each node in the Dynkin diagram with no two adjacent nodes of the same colour.
Unfortunately a bicolouration is not possible for our simple example of the
$(A_{2})_{-2}$-diagram. Therefore let us briefly discuss the structure of the
invariants for the $(A_{3})_{-2}$-Lorentzian Kac-Moody algebra.

##  3 $\mathbf{(A}_{3}\mathbf{)}_{-2}$-Lorentzian Kac-Moody algebra

The Cartan matrix $K$ for the $(A_{3})_{-2}$-Lorentzian Kac-Moody algebra and
the corresponding Dynkin diagram are
$\mathbf{(A}_{3}\mathbf{)}_{-2}:$$\small{\alpha_{1}}$$\small{\alpha_{2}}$$\small{\alpha_{3}}$$\circ$$\bullet$$\bullet$$\circ$$\small{\alpha_{0}}$$\bullet$$\small{\alpha_{-1}}$$\circ$$\small{\alpha_{-2}}$$\\!\\!\\!\\!\\!\\!\\!\\!\\!K_{ij}=2\frac{\alpha_{i}\cdot\alpha_{j}}{\alpha_{j}\cdot\alpha_{j}}=\left(\begin{array}[]{cccccc}2&-1&0&0&0&0\\\
-1&2&-1&0&0&0\\\ 0&-1&2&-1&0&-1\\\ 0&0&-1&2&-1&0\\\ 0&0&0&-1&2&-1\\\
0&0&-1&0&-1&2\\\ \end{array}\right)_{ij}.$
Clearly this diagram is of Lorentzian type as for instance the removal of the
$\alpha_{1}$-node leaves us with the finite $A_{5}$-diagram or the removal of
the $\alpha_{2}$-node yields a finite $D_{5}$-diagram. However, the
decomposition rule for the hyperbolic algebras is not satisfied, as the
removal of the $\alpha_{-2}$-node produces a diagram of hyperbolic type.

As indicated in the Dynkin diagram, by the use of $\circ$ and $\bullet$ for
the vertices, the major difference compared to the
$\mathbf{(A}_{2}\mathbf{)}_{-2}$-case is that this version allows for a
bicolouration. This fact can be exploited in a technical manner as will be
seen below. In close analogy to the discussion in the previous section we
choose a 4+4 dimensional standard representation for the
$\mathbf{(A}_{3}\mathbf{)}_{-2}$-roots $\alpha_{i}$, $i=-2,\cdots,3$,

$\displaystyle\alpha_{1}$ $\displaystyle=$
$\displaystyle\left(1,-1,0,0;0,0,0,0\right),\quad\,\alpha_{2}=\left(0,1,-1,0;0,0,0,0\right),\quad\,\,\,\,\,\,\alpha_{3}=\left(0,0,1,-1,0;0,0,0,0\right),$
$\displaystyle\alpha_{0}$ $\displaystyle=$
$\displaystyle\left(-1,0,0,1;1,0,0,0\right),\quad\alpha_{-1}=\left(0,0,0,0;-1,1,0,0\right),\quad\alpha_{-2}=\left(0,0,0,0;1,0,-1,1\right),$

with Lorentzian inner product

$x\cdot
y=x_{1}y_{1}+x_{2}y_{2}+x_{3}y_{3}+x_{4}y_{4}-x_{5}y_{6}-x_{6}y_{5}-x_{7}y_{8}-x_{8}y_{7},$
(106)

for arbitrary 4+4 dimensional vectors $x=(x_{1},\ldots x_{8})$,
$y=(y_{1},\ldots y_{8})$. Demanding the length of a generic root

$\alpha=p\alpha_{-2}+q\alpha_{-1}+l\alpha_{0}+m\alpha_{1}+n\alpha_{2}+r\alpha_{3},\qquad
p,q,l,m,n,r\in\mathbb{Z},$ (107)

to be 2, leads now to the slightly modified Diophantine equation when compared
to the previous sections

$l^{2}-lm-lq-
lr+m^{2}-mn+n^{2}-nr+p^{2}-pq+q^{2}+r^{2}=1\quad\Leftrightarrow\quad\alpha\cdot\alpha=2.$
(108)

The Weyl reflections are defined in the same manner as in equation (7) with
the additional map $\sigma_{3}$ associated to the root $\alpha_{3}$. The
symmetries of the Diophantine equation (108) are now generated by the Weyl
reflections as

$\displaystyle\sigma_{-2}(\alpha):p\rightarrow
q-p,\quad\quad\,\sigma_{-1}(\alpha):q\rightarrow
l+p-q,\quad\,\,\,\,\sigma_{0}(\alpha):l\rightarrow q+m+r-l,\quad$ (109)
$\displaystyle\sigma_{1}(\alpha):m\rightarrow
l+n-m,\quad\sigma_{2}(\alpha):n\rightarrow
m+r-n,\quad\sigma_{3}(\alpha):r\rightarrow l+n-r.$

Using the notations from [15] we can now use the bicolouration to define to
specific elements consisting of commuting Weyl reflections whose mutual
product makes up a Coxeter element

$\sigma_{+}:=\sigma_{-1}\sigma_{1}\sigma_{3},\quad\sigma_{-}:=\sigma_{-2}\sigma_{0}\sigma_{2},\quad\sigma_{L^{\prime}}:=\sigma_{-}\sigma_{+}.\quad$
(110)

We verify now that the Kostant identity [30] established originally for semi-
simple algebras also holds for the $\mathbf{(A}_{3}\mathbf{)}_{-2}$-root space

$\left(\sigma_{-}+\sigma_{+}\right)\alpha_{i}=\sum_{j=-2}^{3}\left(2\delta_{ij}-K_{(3+i)(3+j)}\right)\alpha_{j},\qquad
i=-2,\cdots,3.$ (111)

Using this relation, together with
$\sigma_{L^{\prime}}^{-1}=\sigma_{+}\sigma_{-}$ one readily derives a relation
between the eigensystems of the Cartan matrix and the Coxeter element.
Defining the vectors

$\displaystyle
q_{i}^{+}:=\zeta_{i4}\alpha_{1}+\zeta_{i6}\alpha_{3}+\zeta_{i2}\alpha_{-1},\quad\text{and}\quad
q_{i}^{-}:=\zeta_{i5}\alpha_{2}+\zeta_{i3}\alpha_{0}+\zeta_{i1}\alpha_{-2},$
(112)

where $\zeta=(v_{1},\cdots,v_{6})$ is the matrix with columns corresponding to
the eigenvectors of the Cartan matrix

$\displaystyle Kv_{j}=\lambda_{j}v_{j},$ (113)

we can explicitly define the eigenvectors of the Coxeter element as

$\displaystyle q_{j}=e^{i\theta_{j}/2}q_{j}^{-}+e^{-i\theta_{j}/2}q_{j}^{+}.$
(114)

The eigenvalue equation then reads

$\sigma q_{j}=e^{i2\theta_{j}}q_{j},$ (115)

where the eigenvalues of the Cartan matrix in (113) and those of the Coxeter
element in (115) are related as

$\lambda_{j}=2-2\cos{\theta_{j}}.$ (116)

We notice that there is no equivalent quantity corresponding to the integer
exponents labelling angles $\theta_{j}$ in the finite dimensional case,
however, they still add up pairwise to $\pi$

$\theta_{j}+\theta_{7-j}=\pi,\qquad j=1,\cdots,6.$ (117)

Concretely we have

$\theta_{1}=\operatorname{arcsec}\left(\frac{-2}{\sqrt{3+\sqrt{3}}}\right)=\pi-\theta_{6},\quad\theta_{2}=\operatorname{arcsec}\left(-\sqrt{2+\frac{2}{\sqrt{3}}}\right)\pi-\theta_{5},\quad\theta_{3}=\frac{\pi}{2}=\theta_{4}.\quad$
(118)

It is clear from these values that the order of the Coxeter element can not be
finite, as we expect. Finite order $\sigma^{h}=1$ with $h\in\mathbb{N}$ would
correspond $h\theta_{i}=n\pi$, with $n\in\mathbb{Z}$, for which there is no
solution.

###  3.1 Invariants and integrability

Let us now address the question of whether one can build solvable physical
models that respect an $\mathbf{(A}_{3}\mathbf{)}_{-2}$-symmetry. In many
model, such as those with Calogero type potentials, this requirement relates
to the existence of invariants of the Weyl group. One may either use the
invariants directly to construct a solvable model [8, 9, 10] or identify them
within a given model to establish its integrability [11, 12].

Thus to each vector $x=\sum_{i=1}^{6}x_{i}\alpha_{i-3}\in\mathbb{R}^{8}$ we
associate a polynomials $P(x_{1},\cdots,x_{6})$ that remains invariant under
the action of the Weyl group $\cal{W}$

$P(\omega^{-1}(x))=P(x^{\prime})=P(x)$ (119)

for all $\omega\in\cal{W}$ with $\omega^{-1}(x)\rightarrow x^{\prime}$. The
transformation of the components $x_{i}$ are simply obtained from (109) with
the replacements $p\rightarrow x_{1},q\rightarrow x_{2},l\rightarrow
x_{3},m\rightarrow x_{4},n\rightarrow x_{5},r\rightarrow x_{6}$. It is then
easy to verify that there is no $\mathbf{(A}_{3}\mathbf{)}_{-2}$ polynomial
invariant of order 1. At order 2 we find the standard invariant

$I_{2}(x_{1},\cdots,x_{6})=\sum_{i=1}^{6}x_{i}^{2}+\sum_{i,j=1}^{6}K_{ij}x_{i}x_{j}.$
(120)

To proceed to higher order one can make an Ansatz of a polynomial of a
particular order with arbitrary coefficients and then fix them by
systematically using (119) for every generator of the Weyl group. While this
is possible, especially with the help of symbolic algebra systems software, we
will provide a different more direct option. For this purpose we carry out a
change of variables from $x$ to $w$ defined in an implicit way by the relation

$x=\sum_{i=1}^{6}x_{i}\alpha_{i-3}=\sum_{i=1}^{6}w_{i}q_{i}.$ (121)

From relation (115) it is clear that the coordinates $w_{i}$ transform under
the action of the Coxeter element as $\sigma:w_{j}\rightarrow
e^{2i\theta_{j}}w_{j}$. This implies that in these new coordinates any
invariant of order $n$ must be of the form

$I_{n}(w_{1},\ldots,w_{6})=\sum_{a_{1},\ldots,a_{6}=0}^{n}c(a_{1},\ldots,a_{6})w_{1}^{a_{1}}\ldots
w_{6}^{a_{6}},\quad\,\,\sum_{i=1}^{6}a_{i}=n,\,\,\sum_{i=1}^{6}a_{i}\theta_{i}=m\pi,\,m\in\mathbb{Z}.$
(122)

Since by construction each term in the sum is invariant under the action of
the Coxeter element the constant coefficients $c(a_{1},\ldots,a_{6})$ must be
constrained further by utilising the Weyl reflections associated to the simple
roots. Checking this for every generator of the Weyl group we find

$I_{2}(w_{1},\ldots,w_{6})=i\left[w_{3}^{2}-4\left(3+2\sqrt{3}\right)w_{2}w_{5}-4\left(3-2\sqrt{3}\right)w_{1}w_{6}\right].$
(123)

Alternatively we also obtain (123) with the variable transformation
$x_{i}\rightarrow w_{i}$ of the invariant (120), as specified in (121). It is
now easy to verify that for $n$ odd we can never solve the last two equation
in (122). Moreover, for $n=4$ we verified for the general Ansatz (122) that
the only invariant at that level is $I_{4}=I_{2}^{2}$. This means we can not
build an exactly solvable $\mathbf{(A}_{3}\mathbf{)}_{-2}$ invariant model of
Calogero type. In turn this suggest that the $\mathbf{(A}_{3}\mathbf{)}_{-2}$
symmetric models proposed in the next section are likely to be not integrable.

##  4 Calogero type models associated to Lorentzian Kac–Moody algebras

We return now to our $\mathbf{(A}_{2}\mathbf{)}_{-2}$ example and start the
discussion with the kinetic energy term for which we also need to use the
inner product as defined in equation (4), so that it reads

$H_{\text{kin}}=\frac{1}{2}p\cdot
p=\frac{1}{2}\left(p_{1}^{2}+p_{2}^{2}+p_{3}^{2}-2p_{4}p_{5}-2p_{6}p_{7}\right).$
(124)

Since Weyl reflections are orthogonal transformation, i.e. $p\cdot
p=\sigma_{i}(p)\cdot\sigma_{i}(p)$, the kinetic term is by construction
invariant under the actions of the $(A_{2})_{-2}$-Weyl group. We will specify
below how the Weyl reflections act on the momenta so that one may also verify
this symmetry explicitly.

Next we consider potentials of Calogero type by extending the notions from the
well-studied cases related to finite dimensional semi-simple Lie algebras
$\mathbf{g}$. All potentials considered here are of the general standard
Calogero form

$V(q)=\sum_{\alpha\in\Delta_{\mathbf{g}}}\frac{c_{\alpha}}{\left(\alpha\cdot
q\right)^{2}}=\sum_{\alpha\in\Delta_{\mathbf{g}}}\frac{c_{\alpha}}{\left(\sigma_{i}(\alpha)\cdot
q\right)^{2}}=\sum_{\alpha\in\Delta_{\mathbf{g}}}\frac{c_{\alpha}}{\left(\alpha\cdot\sigma_{i}(q)\right)^{2}},\quad
i=-2,-1,0,1,2.$ (125)

Inteerpreting now $\mathbf{g}$ a Lorentzian algebra, the potentials in (125)
are by construction invariant under the action of the Lorentzian Weyl group as
the sum extends over the entire group. In this case it extents to infinity. As
indicated in (125), we may also realise the transformations on the roots in
the dual coordinate space since the Weyl reflections are orthogonal
transformation, $\left(\sigma_{i}(\alpha)\cdot
q\right)=\left(\sigma_{i}^{2}(\alpha)\cdot\sigma_{i}(q)\right)=\left(\alpha\cdot\sigma_{i}(q)\right)$.
For instance, for our $(A_{2})_{-2}$-Weyl group in the seven dimensional
representation the actions (8) may be realised in the coordinate space as

$\displaystyle\sigma_{-2}(q)$ $\displaystyle:$ $\displaystyle q_{4}\rightarrow
q_{4}+q_{5}+q_{6}-q_{7},\quad q_{6}\rightarrow-q_{5}-q_{6},\quad
q_{7}\rightarrow q_{5}-q_{7},$ (126) $\displaystyle\sigma_{-1}(q)$
$\displaystyle:$ $\displaystyle q_{4}\rightarrow q_{5},\quad q_{5}\rightarrow
q_{4},$ (127) $\displaystyle\sigma_{0}(q)$ $\displaystyle:$ $\displaystyle
q_{1}\rightarrow q_{3}-q_{5},\quad q_{3}\rightarrow q_{1}+q_{5},\quad
q_{4}\rightarrow q_{1}-q_{3}+q_{4}+q_{5},$ (128) $\displaystyle\sigma_{1}(q)$
$\displaystyle:$ $\displaystyle q_{1}\rightarrow q_{2},\quad q_{2}\rightarrow
q_{1},$ (129) $\displaystyle\sigma_{2}(q)$ $\displaystyle:$ $\displaystyle
q_{2}\rightarrow q_{3},\quad q_{3}\rightarrow q_{2}.$ (130)

The affine (17), the hyperbolic (33) and the Lorentzian Coxeter elements (73)
therefore act on the coordinates as

$\displaystyle\sigma_{a}(q)$ $\displaystyle:$ $\displaystyle q_{1}\rightarrow
q_{2},\quad q_{2}\rightarrow q_{1}+q_{5},\quad q_{3}\rightarrow
q_{3}-q_{5},\quad q_{4}\rightarrow q_{1}-q_{3}+q_{4}+q_{5},$ (131)
$\displaystyle\sigma_{h}(q)$ $\displaystyle:$ $\displaystyle q_{1}\rightarrow
q_{2},\quad q_{2}\rightarrow q_{1}+q_{4},\quad q_{3}\rightarrow
q_{3}-q_{4},\quad q_{4}\rightarrow q_{1}-q_{3}+q_{4}-q_{5},\quad
q_{5}\rightarrow q_{4},$ (132) $\displaystyle\sigma_{L}(q)$ $\displaystyle:$
$\displaystyle q_{1}\rightarrow q_{2},\quad q_{2}\rightarrow
q_{1}+q_{4}+q_{5}+q_{6}-q_{7},\quad q_{3}\rightarrow
q_{3}-q_{4}-q_{5}-q_{6}+q_{7},\quad$ $\displaystyle q_{4}\rightarrow
q_{1}-q_{3}+q_{4}-2q_{5}+q_{6}-q_{7},\quad q_{5}\rightarrow
q_{4}+q_{5}+q_{6}-q_{7},\quad q_{6}\rightarrow-q_{5}-q_{6},\quad\quad$
$\displaystyle q_{7}\rightarrow q_{5}-q_{7},$

respectively. The action on the momenta is taken to be the same. Thus using
(126)-(130) with $q\rightarrow p$, we may now verify explicitly the invariance
of the kinetic energy term (124) under the action of the $(A_{2})_{-2}$-Weyl
group.

The two variants of the standard Calogero potential in (1) are now generalised
in a straightforward fashion. Taking $\gamma_{i}$ to be the, say $\ell$,
representatives of each orbit the potential in (125) can be expressed in the
equivalent form

$V_{C}(q)=\sum_{i=1}^{\ell}\sum_{n=-\infty}^{\infty}\frac{g_{in}}{\left[\sigma^{n}(\gamma_{i})\cdot
q\right]^{2}},$ (134)

where the $g_{in}$ are some new real coupling constants related to the
$c_{\alpha}$. It will remain an open question whether $\ell$ can be taken to
be finite in general. The generalisation of the first variant in (1) may be
written more explicitly as

$V_{D}(q)=\\!\\!\\!\\!\\!\\!\\!\sum_{\begin{subarray}{c}p,q,l,m,n=0\\\
\text{Diophantine
equn}\end{subarray}}^{\infty}\\!\\!\\!\\!\\!\frac{g_{pqlmn}}{[(p\alpha_{-2}+q\alpha_{-1}+l\alpha_{0}+m\alpha_{1}+n\alpha_{2})\cdot
q]^{2}}$ (135)

When restricting to particular levels of the Lie algebraic representation as
suggested in [16] for a hyperbolic case, one may achieve to carry out some, or
possibly all of the infinite sums, at that level. However, the potentials
obtained in this manner are not invariant under the infinite Weyl group. In
contrast, aiming to achieve the latter we explore here also the version in
equation (134).

###  4.1 $(A_{2})_{-2}$-Calogero type potentials

We will now investigate the potentials in more detail and discuss under which
circumstances the infinite sums may be carried out explicitly.

####  4.1.1 Invariant potentials of the affine Weyl group

Thus let us now construct some invariant potentials from the version (134),
which has the advantage that it only involves one infinite sum, as in this
case $\ell$ will be shown to be finite. Choosing as an example
$\gamma_{1}=\alpha_{2}$ as a representative of an orbit and taking all
coupling constants to be the same, i.e. $g_{in}=g$, we may compute the
particular term

$V_{1}(q)=\sum_{n=-\infty}^{\infty}\frac{g}{\left[\sigma_{a}^{n}(\alpha_{2})\cdot
q\right]^{2}}.$ (136)

Using our general formula (2.2) for the action of any power of the Coxeter
element action on any $(A_{2})_{-2}$-root we obtain

$V_{1}(q)=\sum_{n=-\infty}^{\infty}\frac{16g}{\left\\{2\left[(-1)^{n}-1\right]q_{1}-2\left[(-1)^{n}+1\right]q_{2}+\left[-6n+(-1)^{n}-1\right]q_{5}+4q_{3}\right\\}^{2}},$
(137)

when acting with $\sigma_{a}$ on $\alpha_{2}$ in the denominator. Splitting
the sum into its even and odd part, i.e. $\sum_{n}=\sum_{2n}+\sum_{2n-1}$,
they are easily evaluated separately when using the general formula
$\sum_{n=-\infty}^{\infty}(A+Bn)^{-2}=\pi^{2}/[\sin^{2}(A\pi/B)B^{2}]$. In
this way we obtain

$V_{1}(q)=\frac{\pi^{2}}{9q_{5}^{2}}\left\\{\frac{g}{\sin^{2}\left[\frac{\pi}{3q_{5}}(q_{2}-q_{3})\right]}+\frac{g}{\sin^{2}\left[\frac{\pi}{3q_{5}}(q_{1}-q_{3}-q_{5})\right]}\right\\}.$
(138)

By construction this term is invariant under the action of the affine Coxeter
element when realised in the coordinate space by (131). This term is, however,
not yet invariant under the entire Weyl group as for this we require more
representatives of the orbits. We may either try with different roots for the
$\gamma_{i}$ representatives, which equivalently corresponds to actions of the
Weyl group on the coordinates as specified in (126)-(130). Let us therefore
consider the potential

$\displaystyle V(q)$ $\displaystyle=$
$\displaystyle\sum_{n=-\infty}^{\infty}\frac{g}{\left[\sigma^{n}(\alpha_{2})\cdot
q\right]^{2}}+\frac{g}{\left[\sigma^{n}(\alpha_{2})\cdot\sigma_{0}(q)\right]^{2}}+\frac{g}{\left[\sigma^{n}(\alpha_{2})\cdot\sigma_{1}(q)\right]^{2}}$
$\displaystyle+\frac{g}{\left[\sigma^{n}(\alpha_{2})\cdot\sigma_{1}\sigma_{0}(q)\right]^{2}}+\frac{g}{\left[\sigma^{n}(\alpha_{2})\cdot\sigma_{0}\sigma_{1}(q)\right]^{2}}+\frac{g}{\left[\sigma^{n}(\alpha_{2})\cdot\sigma_{0}\sigma_{1}\sigma_{0}(q)\right]^{2}}$
$\displaystyle+\frac{g}{\left[\sigma^{n}(\alpha_{2})\cdot\sigma_{2}\sigma_{0}(q)\right]^{2}}+\frac{g}{\left[\sigma^{n}(\alpha_{2})\cdot\sigma_{2}\sigma_{1}(q)\right]^{2}}+\frac{g}{\left[\sigma^{n}(\alpha_{2})\cdot\sigma_{2}\sigma_{0}\sigma_{2}(q)\right]^{2}}.$

At this point the potential appears in an ad hoc manner, but we shall explain
below how is systematically obtained. Crucially, we can evaluate each infinite
sum as outlined above for the first term. In this manner we obtain the
potential

$V(q)=\frac{2\pi^{2}g}{9q_{5}^{2}}\left(V_{12}+V_{13}+V_{23}+V_{125}^{+}+V_{125}^{-}+V_{135}^{+}+V_{135}^{-}+V_{235}^{+}+V_{235}^{-}\right),$
(140)

where we abbreviated

$V_{ij}:=\frac{1}{\sin^{2}\left[\frac{\pi}{3q_{5}}(q_{i}-q_{j})\right]},\quad
V_{ijk}^{\pm}:=\frac{1}{\sin^{2}\left[\frac{\pi}{3q_{5}}(q_{i}-q_{j}\pm
q_{k})\right]},\quad i,j,k=1,2,3,4,5.$ (141)

We verify explicitly that this potential is indeed invariant under the affine
Weyl group. Indicating how each term transforms under the action of the three
generators we have

$\sigma_{0}:$ | $V_{12}\leftrightarrow V_{235}^{+}$ | $V_{13}\leftrightarrow V_{135}^{-}$ | $V_{23}\leftrightarrow V_{125}^{+}$ | $V_{125}^{-}\leftrightarrow V_{235}^{-}$ |  $V_{135}^{+}\circlearrowleft$ ,
---|---|---|---|---|---
$\sigma_{1}:$ | $V_{13}\leftrightarrow V_{23}$ | $V_{125}^{+}\leftrightarrow V_{125}^{-}$ | $V_{135}^{-}\leftrightarrow V_{235}^{-}$ | $V_{135}^{+}\leftrightarrow V_{235}^{+}$ |  $V_{12}\circlearrowleft$ ,
$\sigma_{2}:$ | $V_{12}\leftrightarrow V_{13}$ | $V_{125}^{+}\leftrightarrow V_{135}^{+}$ | $V_{235}^{+}\leftrightarrow V_{235}^{-}$ | $V_{125}^{-}\leftrightarrow V_{135}^{-}$ |  $V_{23}\circlearrowleft$ .
(142)

The action of the Coxeter element is therefore

$\sigma_{a}:$ | $V_{12}\leftrightarrow V_{125}^{+}$ | $V_{13}\leftrightarrow V_{235}^{+}$ | $V_{23}\leftrightarrow V_{135}^{-}$ | $V_{235}^{-}\leftrightarrow V_{135}^{+}$ |  $V_{125}^{-}\circlearrowleft$ .
---|---|---|---|---|---
(143)

Thus the potential (140) is invariant under the entire $(A_{2})_{0}$-affine
Weyl group. It is now also more obvious how the potential (4.1.1) was obtained
in the first place. Starting off with the terms in $V_{1}$, we may utilise the
Weyl symmetries (142) to generate additional terms until the symmetries only
generate terms already included. As it turned out nine terms are sufficient to
achieve that.

Furthermore we notice that we recover the standard $A_{2}$-Calogero model in
the infinite limits

$\lim_{q_{5}\rightarrow\pm\infty}V(q)=2g\left[\frac{1}{(x_{1}-x_{2})^{2}}+\frac{1}{(x_{1}-x_{3})^{2}}+\frac{1}{(x_{2}-x_{3})^{2}}\right].$
(144)

Let us now compare this potential with the one obtained when evaluating the
three infinite sums in (135) constrained by the Diophantine equation (6) with
$p=q=0$. For this purpose we re-write (135) and (134) as

$V_{D}(q)=g\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\sum_{\begin{subarray}{c}l,m,n=0\\\
\text{Diophantine
equn}\end{subarray}}^{\infty}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!v_{lmn},\qquad\text{and}\qquad
V_{C}(q)=g\sum_{n=-\infty}^{\infty}\sum_{i=1}^{9}v_{i}(n),$ (145)

respectively, where the $v_{i}(n)$ are identified in the same order as the
terms appears in (134). We have set all coupling constants to be equal.
Defining the quantities

$v_{ij}=\frac{1}{(q_{i}-q_{j})^{2}},\qquad\text{and}\qquad
v_{ijk}^{\pm}(n)=\frac{1}{(q_{i}-q_{j}\pm nq_{k})^{2}},$ (146)

we list all nonzero terms in (145) with the upper limit in $V_{D}(q)$ taken to
be 5. We find

$v_{001}=v_{1}(0)=v_{23}$, |  $v_{010}=v_{8}(0)=v_{12}$, |  $v_{011}=v_{3}(0)=v_{13}$,
---|---|---
$v_{100}=v_{7}(0)=v^{+}_{135}(1)$, |  $v_{101}=v_{2}(0)=v^{+}_{125}(1)$, |  $v_{110}=v_{9}(1)=v^{+}_{235}(1)$,
$v_{112}=v_{3}(-1)=v^{-}_{235}(1)$, |  $v_{121}=v_{4}(0)=v^{-}_{125}(1)$, |  $v_{122}=v_{9}(0)=v^{-}_{135}(1)$,
$v_{211}=v_{5}(0)=v^{+}_{135}(2)$, |  $v_{212}=v_{5}(-1)=v^{+}_{125}(2)$, |  $v_{221}=v_{6}(0)=v^{+}_{235}(2)$,
$v_{223}=v_{9}(-1)=v^{-}_{235}(2)$, |  $v_{232}=v_{6}(-1)=v^{-}_{125}(2)$, |  $v_{233}=v_{7}(2)=v^{-}_{235}(2)$,
$v_{322}=v_{3}(2)=v^{+}_{135}(3)$, |  $v_{323}=v_{8}(2)=v^{+}_{125}(3)$, |  $v_{332}=v_{1}(2)=v^{+}_{235}(3)$,
$v_{334}=v_{1}(-2)=v^{-}_{235}(3)$, |  $v_{343}=v_{8}(-2)=v^{-}_{125}(3)$, |  $v_{344}=v_{3}(-2)=v^{-}_{135}(3)$,
$v_{433}=v_{9}(2)=v^{+}_{135}(4)$, |  $v_{434}=v_{2}(-2)=v^{+}_{125}(4)$, |  $v_{443}=v_{9}(3)=v^{+}_{235}(4)$,
$v_{445}=v_{3}(-3)=v^{-}_{235}(4)$, |  $v_{454}=v_{5}(3)=v^{-}_{125}(4)$, |  $v_{455}=v_{1}(-3)=v^{-}_{135}(4)$,
$v_{544}=v_{1}(3)=v^{+}_{135}(5)$, |  $v_{545}=v_{5}(-3)=v^{+}_{125}(5)$, |  $v_{554}=v_{3}(3)=v^{+}_{235}(5)$.
(147)

One may have suspected that $V_{C}(q)$ is only a subset of the general
expression $V_{D}(q)$. However, we observe that the potential in the form of
summing over Coxeter orbits, $V_{C}(q)$ captures all the terms from the
version expressed in terms of three infinite sums subject to the Diophantine
equation $V_{D}(q)$. We have verified this beyond the order reported here.
This constitutes of course no rigorous proof of the equality of the two
versions, but very strong evidence.

####  4.1.2 Invariant potentials of the hyperbolic Weyl group

Next we analyse the two versions of the general potential by just keeping
$p=0$ in (135) and by taking in (134) $\sigma=\sigma_{h}$ to be the Coxeter
element related to the hyperbolic Weyl group. Consequently, the resulting
potentials will be invariant under the action of the entire hyperbolic Weyl
group. For this purpose we re-write (135) and (134) as

$V_{D}(q)=g\\!\\!\\!\\!\\!\\!\\!\\!\\!\sum_{\begin{subarray}{c}q,l,m,n=0\\\
\text{Diophantine
equn}\end{subarray}}^{\infty}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!v_{qlmn},\qquad\text{and}\qquad
V_{C}(q)=g\sum_{k=-\infty}^{\infty}\gamma_{i}^{k},$ (148)

respectively. Once again we set all the coupling constants to be equal. We use
the abbreviations

$\gamma_{i}^{k}:=\frac{1}{[\sigma_{h}^{k}(\gamma_{i})\cdot
q]^{2}},\quad\text{and}\quad
w_{r,s,t,u,v}:=\frac{1}{(rq_{1}+sq_{2}+tq_{3}+uq_{4}+vq_{5})^{2}}.$ (149)

In version (134) we need to generate sufficiently many representatives
$\gamma_{i}$ of the respective orbits. We list now all the terms appearing in
the potential $V_{D}(q)$ up to the order $5$ and identify an equivalent way to
express them as terms appearing in the version $V_{C}(q)$. When $q=0$ all the
terms in (147) also appear, but as we now employ a different type of Coxeter
element they are differently realised in formula $V_{C}(q)$. We find

$v_{0001}=v_{23}=\gamma_{1}$, |  $v_{0010}=v_{12}=\gamma_{2}$, |  $v_{0011}=v_{13}=\gamma_{3}$,
---|---|---
$v_{0100}=v^{+}_{135}(1)=\gamma_{4}$, |  $v_{0101}=v^{+}_{125}(1)=\gamma_{2}^{-1}$, |  $v_{0110}=v^{+}_{235}(1)=\gamma_{2}^{2}$,
$v_{0112}=v^{-}_{235}(1)=\gamma_{5}$, |  $v_{0121}=v^{-}_{125}(1)=\gamma_{6}$, |  $v_{0122}=v^{-}_{135}(1)=\gamma_{1}^{-1}$,
$v_{0211}=v^{+}_{135}(2)=\gamma_{7}$, |  $v_{0212}=v^{+}_{125}(2)=\gamma_{8}$, |  $v_{0221}=v^{+}_{235}(2)=\gamma_{9}$,
$v_{0223}=v^{-}_{235}(2)=\gamma_{3}^{-1}$, |  $v_{0232}=v^{-}_{125}(2)=\gamma_{10}$, |  $v_{0233}=v^{-}_{235}(2)=\gamma_{4}^{-3}$,
$v_{0322}=v^{+}_{135}(3)=\gamma_{11}$, |  $v_{0323}=v^{+}_{125}(3)=\gamma_{12}$, |  $v_{0332}=v^{+}_{235}(3)=\gamma_{13}$,
$v_{0334}=v^{-}_{235}(3)=\gamma_{1}^{-3}$, |  $v_{0343}=v^{-}_{125}(3)=\gamma_{14}$, |  $v_{0344}=v^{-}_{135}(3)=\gamma_{7}^{-2}$,
$v_{0433}=v^{+}_{135}(4)=\gamma_{15}$, |  $v_{0434}=v^{+}_{125}(4)=\gamma_{16}$, |  $v_{0443}=v^{+}_{235}(4)=\gamma_{17}$,
$v_{0445}=v^{-}_{235}(4)=\gamma_{18}$, |  $v_{0454}=v^{-}_{125}(4)=\gamma_{19}$, |  $v_{0455}=v^{-}_{135}(4)=\gamma_{20}$,
$v_{0544}=v^{+}_{135}(5)=\gamma_{21}$, |  $v_{0545}=v^{+}_{125}(5)=\gamma_{22}$, |  $v_{0554}=v^{+}_{235}(5)=\gamma_{23}$.
(150)

Extending the sum in $p$ beyond $0$, we find in addition

$v_{1000}=w_{0,0,0,1,-1}=\gamma_{4}^{-1}$, |  $v_{1100}=w_{1,0,-1,1,0}=\gamma_{5}^{1}$, |  $v_{1101}=w_{1,-1,0,1,0}=\gamma_{2}^{1}$,
---|---|---
$v_{1110}=w_{0,1,-1,1,0}=\gamma_{3}^{1}$, |  $v_{1112}=w_{0,1,-1,-1,0}=\gamma_{4}^{-2}$, |  $v_{1121}=w_{1,-1,0,-1,0}=\gamma_{8}^{1}$,
$v_{1122}=w_{1,0,-1,-1,0}=\gamma_{2}^{-2}$, |  $v_{1312}=w_{2,-1,-1,1,2}=\gamma_{18}^{2}$, |  $v_{1321}=w_{1,1,-2,1,2}=\gamma_{6}^{2}$,
$v_{1324}=w_{1,-2,1,1,2}=\gamma_{6}^{-1}$, |  $v_{1342}=w_{1,-2,1,-1,-2}=\gamma_{24}$, |  $v_{1345}=w_{1,1,-2,-1,-2}=\gamma_{5}^{-1}$,
$v_{1354}=w_{2,-1,-1,-1,-2}=\gamma_{25}$, |  $v_{1422}=w_{2,0,-2,1,3}=\gamma_{20}^{2}$ , |  $v_{1424}=w_{2,-2,0,1,3}=\gamma_{26}$,
(151) $v_{1442}=w_{0,2,-2,1,3}=\gamma_{27}$, |  $v_{2211}=w_{1,0,-1,2,0}=\gamma_{1}^{1}$, |  $v_{2212}=w_{1,-1,0,2,0}=\gamma_{6}^{1}$ ,
---|---|---
$v_{2221}=w_{0,1,-1,2,0}=\gamma_{4}^{1}$, |  $v_{2223}=w_{0,1,-1,-2,0}=\gamma_{7}^{-1}$, |  $v_{2232}=w_{1,-1,0,-2,0}=\gamma_{12}^{1}$,
$v_{2233}=w_{1,0,-1,-2,0}=\gamma_{18}^{1}$, |  $v_{2312}=w_{2,-1,-1,2,1}=\gamma_{8}^{2}$, |  $v_{2321}=w_{1,1,-2,2,1}=\gamma_{5}^{2}$,
$v_{2324}=w_{1,-2,1,2,1}=\gamma_{9}^{-1}$, |  $v_{2342}=w_{1,-2,1,-2,-1}=\gamma_{26}^{1}$, |  $v_{2345}=w_{1,1,-2,-2,-1}=\gamma_{8}^{-1}$,
$v_{2354}=w_{2,-1,-1,-2,-1}=\gamma_{28}$, |  $v_{2523}=w_{3,-1,-2,2,3}=\gamma_{29}$, |  $v_{2524}=w_{3,-2,-1,2,3}=\gamma_{30}$ ,
$v_{2532}=w_{2,1,-3,2,3}=\gamma_{10}^{2}$ , |  $v_{2542}=w_{1,2,-3,2,3}=\gamma_{25}^{2}$, |  $v_{3322}=w_{1,0,-1,3,0}=\gamma_{2}^{3}$,
$v_{3323}=w_{1,-1,0,3,0}=\gamma_{10}^{1}$, |  $v_{3332}=w_{0,1,-1,3,0}=\gamma_{7}^{1}$, |  $v_{3334}=w_{0,1,-1,-3,0}=\gamma_{20}^{1}$,
$v_{3343}=w_{1,-1,0,-3,0}=\gamma_{16}^{1}$, |  $v_{3344}=w_{1,0,-1,-3,0}=\gamma_{29}^{-1}$, |  $v_{3422}=w_{2,0,-2,3,1}=\gamma_{3}^{2}$,
$v_{3424}=w_{2,-2,0,3,1}=\gamma_{25}^{1}$, |  $v_{3442}=w_{0,2,-2,3,1}=\gamma_{18}^{3}$, |  $v_{3523}=w_{3,-1,-2,3,2}=\gamma_{12}^{2}$,
$v_{3524}=w_{3,-2,-1,3,2}=\gamma_{28}^{1}$, |  $v_{3532}=w_{2,1,-3,3,2}=\gamma_{1}^{2}$, |  $v_{3542}=w_{1,2,-3,3,2}=\gamma_{8}^{3}$,
$v_{4433}=w_{1,0,-1,4,0}=\gamma_{9}^{1}$, |  $v_{4434}=w_{1,-1,0,4,0}=\gamma_{14}^{1}$, |  $v_{4443}=w_{0,1,-1,4,0}=\gamma_{11}^{1}$,
$v_{4445}=w_{0,1,-1,-4,0}=\gamma_{31}$ , |  $v_{4454}=w_{1,-1,0,-4,0}=\gamma_{22}^{1}$, |  $v_{4455}=w_{1,0,-1,-4,0}=\gamma_{32}$,
$v_{5544}=w_{1,0,-1,5,0}=\gamma_{13}^{1}$, |  $v_{5545}=w_{1,-1,0,5,0}=\gamma_{19}^{1}$, |  $v_{5554}=w_{0,1,-1,5,0}=\gamma_{15}^{1}$.
(152)

We notice here that each term in the multiple infinite sums of the “raw
version” $V_{D}(q)$ has a counterpart in the version $V_{C}(q)$ that is based
on orbits of the Coxeter element. We also observe that we require many more
orbits, e.g. 35 up to level 5, as in the affine case to cover the entire root
space. In this case it remains unresolved whether a systematic way exist to
generate these orbits and whether a finite number of orbits will be sufficient
to generate all terms of the potential. Note here that generating all terms in
the potential is not equivalent to generating the entire Weyl group, as in the
former we may exploit additional symmetries coming from the nature of the
potential.

####  4.1.3 Invariant potentials of the Lorentzian Weyl group

Finally we compute a potential that is invariant under the full Lorentzian
Weyl group. For this we generalise (148) in an obvious manner summing now also
over $p$. We require now the abbreviations

$\gamma_{i}^{k}:=\frac{1}{[\sigma_{L}^{k}(\gamma_{i})\cdot
q]^{2}},\quad\text{and}\quad
w_{r,s,t,u,v,x,y}:=\frac{1}{(rq_{1}+sq_{2}+tq_{3}+uq_{4}+vq_{5}+xq_{6}+yq_{7})^{2}}.$
(153)

We list all the terms in $V_{D}(q)$ up to the order $3$ and identify a
corresponding term in the potential $V_{C}(q)$

$\\!\\!v_{00001}=w_{0,1,-1,0,0,0,0}=\gamma_{1}$, |  $\\!\\!\\!\\!v_{00010}=w_{1,-1,0,0,0,0,0}=\gamma_{2}$, |  $\\!\\!\\!\\!v_{00011}=w_{1,0,-1,0,0,0,0}=\gamma_{3}$,
---|---|---
$\\!\\!v_{00100}=w_{1,0,-1,0,1,0,0}=\gamma_{4}$, |  $\\!\\!\\!\\!v_{00101}=w_{1,-1,0,0,1,0,0}=\gamma_{2}^{-1}$, |  $\\!\\!\\!\\!v_{00110}=w_{0,1,-1,0,1,0,0}=\gamma_{5}$,
$\\!\\!v_{00112}=w_{0,1,-1,0,-1,0,0}=\gamma_{6}$, |  $\\!\\!\\!\\!v_{00121}=w_{1,-1,0,0,-1,0,0}=\gamma_{7}$, |  $\\!\\!\\!\\!v_{00122}=w_{1,0,-1,0,-1,0,0}=\gamma_{1}^{-1}$,
$\\!\\!v_{00211}=w_{1,0,-1,0,2,0,0}=\gamma_{8}$, |  $\\!\\!\\!\\!v_{00212}=w_{1,-1,0,0,2,0,0}=\gamma_{9}$, |  $\\!\\!\\!\\!v_{00221}=w_{0,1,-1,0,2,0,0}=\gamma_{10}$,
$\\!\\!v_{00223}=w_{0,1,-1,0,-2,0,0}=\gamma_{3}^{-1}$, |  $\\!\\!\\!\\!v_{00232}=w_{1,-1,0,0,-2,0,0}=\gamma_{11}$, |  $\\!\\!\\!\\!v_{00233}=w_{1,0,-1,0,-2,0,0}=\gamma_{4}^{-4}$,
$\\!\\!v_{00322}=w_{1,0,-1,0,3,0,0}=\gamma_{12}$, |  $\\!\\!\\!\\!v_{00323}=w_{1,-1,0,0,3,0,0}=\gamma_{13}$, |  $\\!\\!\\!\\!v_{00332}=w_{0,1,-1,0,3,0,0}=\gamma_{14}$,
$\\!\\!v_{01000}=w_{0,0,0,1,-1,0,0}=\gamma_{4}^{-1}$, |  $\\!\\!\\!\\!v_{01100}=w_{1,0,-1,1,0,0,0}=\gamma_{15}$, |  $\\!\\!\\!\\!v_{01101}=w_{1,-1,0,1,0,0,0}=\gamma_{5}^{-1}$,
$\\!\\!v_{01110}=w_{0,1,-1,1,0,0,0}=\gamma_{2}^{2}$, |  $\\!\\!\\!\\!v_{01112}=w_{0,1,-1,-1,0,0,0}=\gamma_{15}^{-2}$, |  $\\!\\!\\!\\!v_{01121}=w_{1,-1,0,-1,0,0,0}=\gamma_{16}$,
$\\!\\!v_{01122}=w_{1,0,-1,-1,0,0,0}=\gamma_{2}^{-2}$, |  $\\!\\!\\!\\!v_{01312}=w_{2,-1,-1,1,2,0,0}=\gamma_{17}$, |  $\\!\\!\\!\\!v_{01321}=w_{1,1,-2,1,2,0,0}=\gamma_{18}$,
$\\!\\!v_{02211}=w_{1,0,-1,2,0,0,0}=\gamma_{19}$, |  $\\!\\!\\!\\!v_{02212}=w_{1,-1,0,2,0,0,0}=\gamma_{20}$, |  $\\!\\!\\!\\!v_{02221}=w_{0,1,-1,2,0,0,0}=\gamma_{21}$,
$\\!\\!v_{02223}=w_{0,1,-1,-2,0,0,0}=\gamma_{8}^{-1}$, |  $\\!\\!\\!\\!v_{02232}=w_{1,-1,0,-2,0,0,0}=\gamma_{22}$, |  $\\!\\!\\!\\!v_{02233}=w_{1,0,-1,-2,0,0,0}=\gamma_{23}$,
(154) $\\!\\!v_{02312}=w_{2,-1,-1,2,1,0,0}=\gamma_{24}$, |  $\\!\\!\\!\\!v_{02321}=w_{1,1,-2,2,1,0,0}=\gamma_{25}$, |  $\\!\\!\\!\\!v_{03322}=w_{1,0,-1,3,0,0,0}=\gamma_{26}$,
---|---|---
$\\!\\!v_{03323}=w_{1,-1,0,3,0,0,0}=\gamma_{27}$, |  $\\!\\!\\!\\!v_{03332}=w_{0,1,-1,3,0,0,0}=\gamma_{28}$, |  $\\!\\!\\!\\!v_{10000}=w_{0,0,0,0,1,1,-1}=\gamma_{4}^{-2}$,
$\\!\\!v_{10111}=w_{0,0,0,0,2,1,-1}=\gamma_{29}$, |  $\\!\\!\\!\\!v_{10222}=w_{0,0,0,0,3,1,-1}=\gamma_{30}$, |  $\\!\\!\\!\\!v_{10333}=w_{0,0,0,0,4,1,-1}=\gamma_{31}$,
$\\!\\!v_{11000}=w_{0,0,0,1,0,1,-1}=\gamma_{15}^{-1}$, |  $\\!\\!\\!\\!v_{11100}=w_{1,0,-1,1,1,1,-1}=\gamma_{6}^{1}$, |  $\\!\\!\\!\\!v_{11101}=w_{1,-1,0,1,1,1,-1}=\gamma_{2}^{1}$,
$\\!\\!v_{11110}=w_{0,1,-1,1,1,1,-1}=\gamma_{3}^{1}$, |  $\\!\\!\\!\\!v_{11112}=w_{0,-1,1,1,1,1,-1}=\gamma_{4}^{-3}$, |  $\\!\\!\\!\\!v_{11121}=w_{-1,1,0,1,1,1,-1}=\gamma_{9}^{1}$,
$\\!\\!v_{11122}=w_{-1,0,1,1,1,1,-1}=\gamma_{5}^{-2}$, |  $\\!\\!\\!\\!v_{11312}=w_{2,-1,-1,1,3,1,-1}=\gamma_{23}^{1}$, |  $\\!\\!\\!\\!v_{11321}=w_{1,1,-2,1,3,1,-1}=\gamma_{20}^{1}$,
$\\!\\!v_{12111}=w_{0,0,0,2,0,1,-1}=\gamma_{29}^{1}$, |  $\\!\\!\\!\\!v_{12311}=w_{2,0,-2,2,2,1,-1}=\gamma_{29}^{2}$, |  $\\!\\!\\!\\!v_{12313}=w_{2,-2,0,2,2,1,-1}=\gamma_{32}$,
$\\!\\!v_{12331}=w_{0,2,-2,2,2,1,-1}=\gamma_{32}^{1}$, |  $\\!\\!\\!\\!v_{13222}=w_{0,0,0,3,0,1,-1}=\gamma_{33}$, |  $\\!\\!\\!\\!v_{13312}=w_{2,-1,-1,3,1,1,-1}=\gamma_{34}$,
$\\!\\!v_{13321}=w_{1,1,-2,3,1,1,-1}=\gamma_{7}^{2}$, |  $\\!\\!\\!\\!v_{21222}=w_{0,0,0,1,3,2,-2}=\gamma_{35}$, |  $\\!\\!\\!\\!v_{21322}=w_{1,0,-1,1,4,2,-2}=\gamma_{36}$,
$\\!\\!v_{21323}=w_{1,-1,0,1,4,2,-2}=\gamma_{37}$, |  $\\!\\!\\!\\!v_{21332}=w_{0,1,-1,1,4,2,-2}=\gamma_{38}$, |  $\\!\\!\\!\\!v_{22211}=w_{1,0,-1,2,2,2,-2}=\gamma_{1}^{1}$,
$\\!\\!v_{22212}=w_{1,-1,0,2,2,2,-2}=\gamma_{7}^{1}$, |  $\\!\\!\\!\\!v_{22221}=w_{0,1,-1,2,2,2,-2}=\gamma_{4}^{1}$, |  $\\!\\!\\!\\!v_{22223}=w_{0,-1,1,2,2,2,-2}=\gamma_{19}^{-1}$,
$\\!\\!v_{22232}=w_{-1,1,0,2,2,2,-2}=\gamma_{13}^{1}$, |  $\\!\\!\\!\\!v_{22233}=w_{-1,0,1,2,2,2,-2}=\gamma_{34}^{-1}$, |  $\\!\\!\\!\\!v_{22312}=w_{2,-1,-1,2,3,2,-2}=\gamma_{16}^{1}$,
$\\!\\!v_{22321}=w_{1,1,-2,2,3,2,-2}=\gamma_{15}^{1}$, |  $\\!\\!\\!\\!v_{23222}=w_{0,0,0,3,1,2,-2}=\gamma_{30}^{1}$, |  $\\!\\!\\!\\!v_{23312}=w_{2,-1,-1,3,2,2,-2}=\gamma_{9}^{2}$,
$\\!\\!v_{23321}=w_{1,1,-2,3,2,2,-2}=\gamma_{6}^{2}$, |  $\\!\\!\\!\\!v_{32333}=w_{0,0,0,2,4,3,-3}=\gamma_{39}$, |  $\\!\\!\\!\\!v_{33322}=w_{1,0,-1,3,3,3,-3}=\gamma_{5}^{1}$,
$\\!\\!v_{33323}=w_{1,-1,0,3,3,3,-3}=\gamma_{11}^{1}$, |  $\\!\\!\\!\\!v_{33332}=w_{0,1,-1,3,3,3,-3}=\gamma_{8}^{1}$,. |
(155)

Once more we see that each term possess at least one counter term in each
version of the potential. We leave it as open issue to construct all orbits,
to find a closed formula or even computing the sums.

####  4.1.4 Partial sums

Alternatively, to converting the five infinity sums into one infinite sum over
orbits of the Coxeter element, we may also consider partial sums for specific
choices of some of the summation indices compatible with the Diophantine
equation with one infinite sum remaining. We already encountered partial sums
in obtaining potentials of the form (138). While this term was not invariant
under the entire Weyl group, it was by construction invariant under the action
of the Coxeter element. We list here some further possibilities:

For instance, the choice $p\rightarrow l,q\rightarrow 1+l,m\rightarrow
l,n\rightarrow l$ satisfies the Diophantine equation (6) and leaves one
infinite sum in the general potential (135) . Taking also all coupling
constants to be the same $g$ and using the 3+4 dimensional representation for
the roots $\alpha_{i}$ we are left with

$V(q)=\sum_{l=0}^{\infty}\frac{g}{A+Bl+\frac{B^{2}}{4A}l^{2}}=\frac{4A}{B^{2}}\Psi\left(\frac{2A}{B}\right),$
(156)

where $\Psi(x):=(\ln\Gamma[x])^{{}^{\prime\prime}}$ is a polygamma function
and $A=q_{4}^{2}-2q_{5}q_{4}+q_{5}^{2}$,
$B=2q_{4}^{2}+2q_{6}q_{4}-2q_{7}q_{4}-2q_{5}^{2}-2q_{5}q_{6}+2q_{5}q_{7}$. The
sum in (156) has been carried out explicitly. Concretely we obtain the
potential

$V(q)=\frac{g}{\left(q_{4}+q_{5}+q_{6}-q_{7}\right)^{2}}\Psi\left(\frac{q_{4}-q_{5}}{q_{4}+q_{5}+q_{6}-q_{7}}\right).$
(157)

For the choice $p\rightarrow l,q\rightarrow l,m\rightarrow 1+l,n\rightarrow
1+l$ we obtain in the same manner

$V(q)=\frac{g}{\left(q_{4}+q_{5}+q_{6}-q_{7}\right)^{2}}\Psi\left(\frac{q_{2}-q_{1}}{q_{4}+q_{5}+q_{6}-q_{7}}\right).$
(158)

For the choice $p\rightarrow k,q\rightarrow 2k,l\rightarrow 2k+1,m\rightarrow
k,n\rightarrow k$ then subsequent infinite sum over $k$ yields

$\displaystyle V_{1}(q)$ $\displaystyle=$
$\displaystyle\frac{\left(q_{1}-q_{3}+q_{5}\right)^{2}}{\left[\left(q_{1}-q_{3}+q_{5}\right)\left(q_{1}-q_{3}+2q_{4}+q_{5}+q_{6}\right)+\left(-q_{1}+q_{3}-7q_{5}\right)q_{7}\right]^{2}}$
$\displaystyle\times\Psi\left[\frac{\left(q_{1}-q_{3}+q_{5}\right){}^{2}}{\left(q_{1}-q_{3}+q_{5}\right)\left(q_{1}-q_{3}+2q_{4}+q_{5}+q_{6}\right)+\left(-q_{1}+q_{3}-7q_{5}\right)q_{7}}\right].$

Another interesting choice is
$p\rightarrow-1-k,q\rightarrow-2-2k,m\rightarrow-1-k,n\rightarrow-1-k,l\rightarrow-2k-1$,
which would mean we extend the sums in (135) to the negative range. In the
case we obtain the potential

$\displaystyle V_{2}(q)$ $\displaystyle=$
$\displaystyle\frac{\left(q_{1}-q_{3}+q_{5}\right)^{2}}{\left[\left(q_{1}-q_{3}+q_{5}\right)\left(q_{1}-q_{3}+2q_{4}+q_{5}+q_{6}\right)+\left(-q_{1}+q_{3}-7q_{5}\right)q_{7}\right]^{2}}$
$\displaystyle\times\Psi\left[1-\frac{\left(q_{1}-q_{3}+q_{5}\right){}^{2}}{\left(q_{1}-q_{3}+q_{5}\right)\left(q_{1}-q_{3}+2q_{4}+q_{5}+q_{6}\right)+\left(-q_{1}+q_{3}-7q_{5}\right)q_{7}}\right].$

Using that fact that $\Psi(x)+\Psi(1-x)=\pi^{2}/\sin(\pi x)^{2}$ the sum of
the two potentials yields

$\displaystyle V_{1}(q)+V_{2}(q)$ $\displaystyle=$
$\displaystyle\frac{\pi^{2}\left(q_{1}-q_{3}+q_{5}\right)^{2}}{\left[\left(q_{1}-q_{3}+q_{5}\right)\left(q_{1}-q_{3}+2q_{4}+q_{5}+q_{6}\right)+\left(-q_{1}+q_{3}-7q_{5}\right)q_{7}\right]^{2}}$
$\displaystyle\times\sin\left[\pi\frac{\left(q_{1}-q_{3}+q_{5}\right){}^{2}}{\left(q_{1}-q_{3}+q_{5}\right)\left(q_{1}-q_{3}+2q_{4}+q_{5}+q_{6}\right)+\left(-q_{1}+q_{3}-7q_{5}\right)q_{7}}\right]^{-2}.$

Evidently, these potentials are not invariant under the action of the full
Lorentzian Weyl group, but they involve no infinite sum and admit a partial
symmetry.

##  5 Conclusions

We discussed general properties of infinite Weyl groups of affine, hyperbolic
and Lorentzian type. As concrete examples we presented the $(A_{2})_{0}$,
$(A_{2})_{-1}$ and $(A_{2})_{-2}$ cases, for which we paid particular
attention to the properties of the respective Coxeter elements. In each case
we derived closed formulae for the infinite orbits of these elements.

For the $(A_{3})_{-2}$ case we exploited the fact that the associated Dynkin
diagram can be bicoloured. We proved that the Kostant identity (111) also
holds for Lorentzian algebras. We also showed that while there are no integer
exponents for these algebras, the analogue quantities of them still add up
pairwise to $\pi$. These features allowed for a systematic analysis of the
respective invariants. We showed that independent invariants at higher degree
do not exist, which in turn indicates that the standard approach to utilize
these invariants in the construction of integrable systems can not be pursued.

Finally, we used the structures found to formulate models of Calogero type
with an in-built infinite symmetry. In their raw form the potential of these
models contain as many infinite sums as simple roots of their associated
algebras simply constrained by a Diophantine equation. However, using the
formulae for the Coxeter orbits we argued that these sums may be reduced. In
the case of the invariant affine potential we showed that the remaining single
infinite sum may even be computed explicitly. For the models based on the
hyperbolic or Lorentzian algebras it remains an open issue how to
systematically generate these representatives, whether there are finitely many
representatives and whether the sum over the Coxeter orbits can be computed
explicitly. We demonstrate, however, that one may perform subsets of the
infinite sums obtaining in this manner potentials that are invariant under the
action of the Coxeter element. We also stress once more that generating all
orbits needed in the potential is different, and more restricted, than
generating the entire root space from all orbits. As we have seen for the case
of the affine Weyl group when the orbits are generated within the potentials
we may use their particular symmetries, such as parity invariance and the fact
that we may shift the summation indices. These properties made it possible to
construct invariant potentials from a finite number of terms.

Besides the numerous fundamental mathematical issues left open, there are many
further interesting questions left for future investigations. The
generalisation beyond rational potentials to models of Calogero-Moser-
Sutherland type appears to be straightforward. It would also be interesting to
consider extended algebraic systems beyond the two examples of $A_{2}$ and
$A_{3}$ presented here. Of course, ultimately one would also like to
understand the quantum versions of the models proposed here.

Acknowledgments: FC is supported by Fondecyt grant 1211356. AF thanks the
Instituto de Ciencias Físicas y Matemáticas of the Universidad Austral de
Chile, where part of this work was completed for kind hospitality and
Francisco Correa for financial support. OQ thanks the finantial support of
Universidad Austral de Chile and Fondecyt grant 1211356 (FC) as well as the
Department of Mathematics of City, University of London for kind hospitality
in the visit that started this work.

## References

  * [1] F. Calogero,  Ground state of one-dimensional N body system,  J. Math. Phys. 10, 2197–2200 (1969).
  * [2] F. Calogero,  Solution of a three-body problem in one-dimension,  J. Math. Phys. 10, 2191–2196 (1969).
  * [3] F. Calogero,  Solution of the one-dimensional N body problems with quadratic and/or inversely quadratic pair potentials,  J. Math. Phys. 12, 419–436 (1971).
  * [4] J. Moser,  Three integrable Hamiltonian systems connected with isospectral deformations,  Adv. Math. 16, 197–220 (1975).
  * [5] B. Sutherland,  Exact results for a quantum many body problem in one- dimension,  Phys. Rev. A4, 2019–2021 (1971).
  * [6] B. Sutherland,  Exact results for a quantum many body problem in one- dimension. 2,  Phys. Rev. A5, 1372–1376 (1972).
  * [7] A. Fring and M. Smith,  Antilinear deformations of Coxeter groups, an application to Calogero models,  J. Phys. A 43, 325201 (2010).
  * [8] O. Haschke and W. Rühl,  Exactly solvable dynamical systems in the neighborhood of the Calogero model,  Int. J. Mod. Phys. A14, 387–408 (1999).
  * [9] O. Haschke and W. Rühl,  Is it possible to construct exactly solvable models?,  Lect. Notes Phys. 539, 118–140 (2000).
  * [10] A. Fring and C. Korff,  Exactly solvable potentials of Calogero type for q-deformed Coxeter groups,  J. Phys. A37, 10931–10950 (2004).
  * [11] C. F. Dunkl,  Differential-difference operators associated to reflection groups,  Transact. Amer. Math. Soc. 311(1), 167–183 (1989).
  * [12] F. Carrillo-Morales, F. Correa, and O. Lechtenfeld,  Integrability, intertwiners and non-linear algebras in Calogero models,  JHEP 2021(5), 1–20 (2021).
  * [13] R. W. Carter,  Simple groups of Lie type, volume 22,  John Wiley & Sons, 1989.
  * [14] P. Dorey,  Root systems and purely elastic S-matrices,  Nucl. Phys. B 358, 654–676 (1991).
  * [15] A. Fring, H. C. Liao, and D. Olive,  The mass spectrum and coupling in affine Toda theories,  Phys. Lett. B 266, 82–86 (1991).
  * [16] O. Lechtenfeld and D. Zagier,  A hyperbolic Kac-Moody Calogero model,  arXiv preprint arXiv:2203.06519 (2022).
  * [17] A. Fring and S. Whittington,  Lorentzian Toda field theories,  Rev. in Math. Phys. 33(06), 2150017 (2021).
  * [18] M. R. Gaberdiel, D. I. Olive, and P. C. West,  A class of Lorentzian Kac–Moody algebras,  Nucl. Phys. B 645, 403–437 (2002).
  * [19] A. Fring and S. Whittington,  n-Extended Lorentzian Kac–Moody algebras,  Lett. Math. Phys , 1–22 (2020).
  * [20] S. Whittington,  Extensions of Integrable Quantum Field Theories Based on Lorentzian Kac-Moody Algebras,  PhD thesis, City, University of London, 2022.
  * [21] P. West,  Hidden superconformal symmetry in M-theory,  JHEP 2000, 007 (2000).
  * [22] P. West,  E11 and M-theory,  Class. Quant. Grav. 18, 4443 (2001).
  * [23] T. Damour, M. Henneaux, and H. Nicolai,  E10 and a small tension expansion of M theory,  Phys. Rev. Lett. 89(22), 221601 (2002).
  * [24] G. Bossard, A. Kleinschmidt, and E. Sezgin,  A master exceptional field theory,  JHEP 2021(6), 1–148 (2021).
  * [25] N. Bourbaki,  Elements of Mathematics, Lie Groups and Lie Algebras, Chapters 4-6,  Springer, Berlin (2002).
  * [26] J. E. Humphreys,  Introduction to Lie algebras and representation theory, volume 9,  Springer Science & Business Media, 2012.
  * [27] V. G. Kac,  Infinite dimensional Lie algebras,  CUP, Cambridge (1990).
  * [28] S. S. Epp,  Discrete mathematics with applications,  Cengage, Boston, MA, 2020.
  * [29] L. Carbone, S. Chung, L. Cobbs, R. McRae, D. Nandi, Y. Naqvi, and D. Penta,  Classification of hyperbolic Dynkin diagrams, root lengths and Weyl group orbits,  J. Phys. A: Math. Theor. 43(15), 155209 (2010).
  * [30] B. Kostant,  The principal three-dimensional subgroup and the Betti numbers of a complex simple Lie group,  Amer. J. Math. 81, 973–1032 (1959).