# A classification of automorphic Lie algebras on complex tori

Vincent Knibbeler, Sara Lombardo, and Casper Oelen

###### Abstract

We classify the automorphic Lie algebras of equivariant maps from a complex
torus to $\mathfrak{sl}_{2}(\mathbb{C})$. For each case we compute a basis in
a normal form. The automorphic Lie algebras correspond precisely to two
disjoint families of Lie algebras parametrised by the modular curve of
$\mathrm{PSL}_{2}({\mathbb{Z}})$, apart from four cases, which are all
isomorphic to Onsager’s algebra.

##  1 Introduction

Automorphic Lie algebras are a class of infinite-dimensional Lie algebras over
the complex field $\mathbb{C}$ that emerged in the context of mathematical
physics, and more precisely in the context of integrable systems [18, 19, 20].
They originated in the study of algebraic reduction of Lax pairs by Lombardo
and Mikhailov [18, 19], related to the notion of _reduction groups_ , proposed
by Mikhailov in [22] and [23]. The first appearance of a notion of automorphic
Lie algebras is to be found in the PhD thesis by Lombardo, and subsequently in
Lombardo and Mikhailov [20], where a systematic study of these algebras within
the theory of integrability began. It then developed independently from this
approach into the search and the description of invariant algebras, with
contributions by Lombardo and Sanders, and later also Knibbeler [21, 12, 15,
16]. The subject of automorphic Lie algebras inspired recent research into
group theory, by the authors [13]. Independently, equivariant map algebras
were introduced and studied by representation theorists Neher, Savage and
Senesi [25] in order to unify many Lie algebras that appear in mathematical
physics. These included automorphic Lie algebras, although they were unaware
of this at the time.

While a precise definition will be given in Section 2, for now, the reader
should think of automorphic Lie algebras as Lie algebras of meromorphic maps
(usually with prescribed poles) from a compact Riemann surface $X$, originally
the Riemann sphere, into a finite dimensional Lie algebra $\mathfrak{g}$,
which are equivariant with respect to a finite group $\Gamma$ acting on $X$
and on $\mathfrak{g}$, both by automorphisms. It is therefore not surprising
that the study of these objects requires notions from algebra, geometry and
analysis. The group $\Gamma$ plays the role of the reduction group in the
context of integrable systems. In this paper, we refer to $\Gamma$ as the
_symmetry group_ of an automorphic Lie algebra.

Considerable work on automorphic Lie algebras based on the Riemann sphere has
been carried out in the past decades. More recently other Riemann surfaces
have been investigated in [5] where the foundations of a representation theory
for automorphic Lie algebras have been developed. Automorphic Lie algebras in
the context of modular forms have been investigated in [17]. In [5], the
authors find a local description of automorphic Lie algebras in the vicinity
of a point on the compact Riemann surface. That is, the quotient of an
automorphic Lie algebra obtained by truncating a local parameter is
determined. However, it remains an open problem to have a global description
of such Lie algebras on surfaces of positive genus, as we have for the Riemann
sphere. First steps in this direction can be found in [26].

This paper develops those concepts further and presents a first classification
in the case where $X$ is a genus 1 Riemann surface and when
${\mathfrak{g}}=\mathfrak{sl}_{2}({\mathbb{C}})$; we will refer to
${\mathfrak{g}}$ as the _base_ Lie algebra. We will exclusively work over the
complex numbers, for which reason we use the notation
$\mathfrak{sl}_{2}:=\mathfrak{sl}_{2}({\mathbb{C}})$.

To give a flavour of automorphic Lie algebras to a newcomer, let us start with
a classic example. Consider a simple complex, finite dimensional Lie algebra
$\mathfrak{g}$ and let $\sigma$ be an automorphism of order $N$ of
$\mathfrak{g}$. Consider the space of Laurent polynomial maps
$f:\mathbb{C}^{*}\rightarrow\mathfrak{g}$ which satisfy $f(\epsilon z)=\sigma
f(z),$ where $\epsilon^{N}=1.$ Endow this space with the pointwise Lie
bracket. This is an example of an automorphic Lie algebra with base Lie
algebra $\mathfrak{g}$, and where the Riemann surface $X$ is the Riemann
sphere $\mathbb{C}_{\infty}$. The space of meromorphic functions on
$\mathbb{C}_{\infty}$ which are holomorphic outside $\\{0,\infty\\}$ is given
by the space of Laurent polynomials $\mathbb{C}[z,z^{-1}]$. The group here is
$C_{N}$ which acts on $\mathbb{C}_{\infty}$ as $z\mapsto\epsilon z$ and on
$\mathfrak{g}$ by the automorphism $\sigma$, such that the equivariance
condition reads $f(\epsilon z)=\sigma f(z)$. It is called a _twisted loop
algebra_ and is probably best known for the role it plays in the construction
of affine Kac-Moody algebras [9]. Automorphic Lie algebras are natural
generalisations of this type of algebra, in the sense illustrated below.

As was recognised only relatively recently [30], another important example of
an automorphic Lie algebra can be traced back to the chemistry Nobel Prize
winner, Lars Onsager, in his pioneering paper [27] on the exact solution of
the planar Ising model. This is not a loop algebra and is now known as the
_Onsager algebra_. It can be defined as the Lie algebra $\mathfrak{O}$ with
complex basis $A_{k}$ and $G_{m}$ where $k\in{\mathbb{Z}}$ and
$m\in\mathbb{N}.$ The brackets are given by

$\displaystyle[A_{k},A_{l}]$ $\displaystyle=4G_{k-l},$
$\displaystyle[A_{k},G_{m}]$ $\displaystyle=2(A_{k-m}-A_{k+m}),$
$\displaystyle[G_{m},G_{n}]$ $\displaystyle=0,$

with $G_{-m}=-G_{m}$ $(m>0)$ and $G_{0}=0$. Onsager’s work on the 2D Ising
model was later simplified using different techniques, cf. [10], in which
there was no appearance of the Onsager algebra. This caused the Onsager
algebra to be less studied until the early 1980s, when it began to appear in
different contexts and in a different form. In a paper by Grady, [4] relations
(the Dolan-Grady relations) between two nonlinear operators were found which
guarantee the existence of infinitely many commuting charges. The Dolan-Grady
relations were later found by Perk in [28] to be connected to the Onsager
algebra. For a concise overview, we refer to El-Chaâr [6].

In [17, Theorem 2.5] it is proven that the Onsager algebra $\mathfrak{O}$ is
isomorphic to the Lie algebra $\mathfrak{A}={\mathbb{C}}\langle
h,e,f\rangle\otimes_{{\mathbb{C}}}{\mathbb{C}}[x]$ with the Lie structure that
is linear over polynomials in the indeterminate $x$ and

$[h,e]=2e,\quad[h,f]=-2f,\quad[e,f]=h\otimes x(x-1).$

This is the form we will encounter when proving that certain automorphic Lie
algebras are isomorphic to the Onsager algebra.

The main result of this work is a classification of automorphic Lie algebras
on complex tori and the construction of certain normal forms. We will present
our classification theorem here and leave the construction of the normal
forms, proving our classification theorem, for the main body of the paper.

Besides the Onsager algebra described above, we find two families of Lie
algebras parametrised by the (open) modular curve. We denote these families by
$\mathfrak{C}_{\tau}$ and $\mathfrak{S}_{\tau}$ and present them in terms of
the traditional elliptic invariants $g_{2}$ and $g_{3}$ defined as

$g_{2}(\tau)=60\sum_{\begin{subarray}{c}a,b\in\mathbb{Z}\\\
(a,b)\neq(0,0)\end{subarray}}(a+b\tau)^{-4},\quad
g_{3}(\tau)=140\sum_{\begin{subarray}{c}a,b\in\mathbb{Z}\\\
(a,b)\neq(0,0)\end{subarray}}(a+b\tau)^{-6}$

where $\tau$ is an element of the upper half plane

$\mathbb{H}=\\{z\in{\mathbb{C}}:\mathrm{Im}(z)>0\\}.$

The Lie algebra $\mathfrak{C}_{\tau}$ is the current algebra

$\mathfrak{C}_{\tau}=\mathfrak{sl}_{2}\otimes_{{\mathbb{C}}}{\mathbb{C}}[x,y]/(y^{2}-4x^{3}+g_{2}(\tau)x+g_{3}(\tau)),$

where, as usual, the Lie bracket is defined by extending the bracket of
$\mathfrak{sl}_{2}$ linearly over the polynomials in $x$ and $y$. The second
family is defined by

$\displaystyle\mathfrak{S}_{\tau}={\mathbb{C}}\langle
E,F,H\rangle\otimes_{{\mathbb{C}}}{\mathbb{C}}[x],$ (1)

where we identify $X$ with $X\otimes 1$ for $X=E,F,H$, with the Lie structure
that is linear over polynomials in $x$ and satisfies

$[H,E]=2E,\quad[H,F]=-2F,\quad[E,F]=H\otimes(4x^{3}-g_{2}(\tau)x-g_{3}(\tau)).$

The elliptic invariants $g_{2}$ and $g_{3}$ are usually defined for any
lattice $\Lambda$ in $\mathbb{C}$ and we could define families of Lie algebras
$\mathfrak{C}_{\Lambda}$ and $\mathfrak{S}_{\Lambda}$ accordingly, but this
does not produce any new Lie algebras: homothetic lattices produce isomorphic
Lie algebras. Indeed, using the fact that
$g_{2}(\alpha\Lambda)=\alpha^{-4}g_{2}(\Lambda)$ and
$g_{3}(\alpha\Lambda)=\alpha^{-6}g_{3}(\Lambda)$ for any nonzero complex
number $\alpha$, we can check that the linear extension of

$A\otimes p(x,y)\mapsto A\otimes p(\alpha^{2}x,\alpha^{3}y)$

descends to an isomorphism from $\mathfrak{C}_{\Lambda}$ to
$\mathfrak{C}_{\alpha\Lambda}$. Likewise we can define an isomorphism from
$\mathfrak{S}_{\Lambda}$ to $\mathfrak{S}_{\alpha\Lambda}$ with the
assignments

$E\otimes p(x)\mapsto E\otimes\alpha^{3}p(\alpha^{2}x),\quad F\otimes
p(x)\mapsto F\otimes\alpha^{3}p(\alpha^{2}x),\quad H\otimes p(x)\mapsto
H\otimes p(\alpha^{2}x).$

Hence we can assume that the lattice $\Lambda$ is scaled to the canonical form
$\mathbb{Z}\oplus\mathbb{Z}\tau$ for $\tau\in\mathbb{H}$ if we are only
concerned with isomorphism classes of Lie algebras.

If $\tau$ is replaced by $\tau^{\prime}=\frac{a\tau+b}{c\tau+d}$ for
$a,b,c,d\in\mathbb{Z}$, $ad-bc=1$ we arrive at isomorphic Lie algebras again.
Indeed,
$\mathbb{Z}\oplus\mathbb{Z}\tau^{\prime}=(c+d\tau)^{-1}\left(\mathbb{Z}\oplus\mathbb{Z}\tau\right)$,
and we have just shown that homothetic lattices produce isomorphic Lie
algebras. Thus, we only need to consider equivalence classes $[\tau]$ in the
modular curve $\mathrm{SL}_{2}(\mathbb{Z})\backslash\mathbb{H}$ if we are
interested in $\mathfrak{C}_{\tau}$ and $\mathfrak{S}_{\tau}$ up to
isomorphism, and we will occasionally write $\mathfrak{C}_{[\tau]}$ and
$\mathfrak{S}_{[\tau]}$ accordingly.

We know, in fact, that $\mathfrak{C}_{\tau}\cong\mathfrak{C}_{\tau^{\prime}}$
if and only if $[\tau]=[\tau^{\prime}]$ due to Fialowski and Schlichenmaier.
In [7, Proposition 4.7] they show that if ${\mathfrak{g}}$ is a semisimple
finite-dimensional Lie algebra, $\mathcal{A}$ and $\mathcal{B}$ two
associative, commutative algebras (with units) and if the current algebras
${\mathfrak{g}}\otimes_{{\mathbb{C}}}\mathcal{A}$ and
${\mathfrak{g}}\otimes_{{\mathbb{C}}}\mathcal{B}$ are isomorphic as Lie
algebras, then $\mathcal{A}$ and $\mathcal{B}$ are isomorphic as associative
algebras. Write
$\mathcal{A}_{\tau}={\mathbb{C}}[x,y]/(y^{2}-4x^{3}+g_{2}(\tau)x+g_{3}(\tau))$.
From algebraic geometry, cf. [8, Corollary 3.7], we learn that
$\mathcal{A}_{\tau}\cong\mathcal{A}_{\tau^{\prime}}$ implies
$[\tau]=[\tau^{\prime}]$ and hence
$\mathfrak{C}_{\tau}\cong\mathfrak{C}_{\tau^{\prime}}$ implies
$[\tau]=[\tau^{\prime}]$. It is not known to the authors whether
$\mathfrak{S}_{[\tau]}\cong\mathfrak{S}_{[\tau^{\prime}]}$ implies
$[\tau]=[\tau^{\prime}]$.

Now that we know the Lie algebras $\mathfrak{C}_{\tau},\,\mathfrak{O}$ and
$\mathfrak{S}_{\tau}$ we can present our classification result.

######  Theorem 1.1.

Let $\Gamma$ be a finite group acting faithfully on a complex torus $T$ with
biholomorphic maps and also faithfully on $\mathfrak{sl}_{2}$ with Lie algebra
isomorphisms. Let $\mathbb{T}$ be the torus minus the orbit
$\Gamma\cdot\\{0\\}$, and let $\mathbb{T}\rightarrow\mathbb{T}/\Gamma$ be the
canonical projection onto the quotient Riemann surface. The isomorphism class
of the automorphic Lie algebra consisting of $\Gamma$-equivariant holomorphic
maps $\mathbb{T}\to\mathfrak{sl}_{2}$, meromorphic at the punctures
$\Gamma\cdot\\{0\\}$, is determined by the number of branch points of
$\mathbb{T}\rightarrow\mathbb{T}/\Gamma$ as in Table 1. No other number of
branch points occur.

The class $[\tau]$ in this table is the element of
$\mathrm{SL}_{2}(\mathbb{Z})\backslash\mathbb{H}$ corresponding to the torus
$T/t(\Gamma)$, where $t(\Gamma)$ is the subgroup of $\Gamma$ generated by the
elements that have no fixed point in $T$.

$\\#$ branch points | Lie algebra
---|---
0 | $\mathfrak{C}_{[\tau]}$
2 | $\mathfrak{O}_{\phantom{\tau}}$
3 | $\mathfrak{S}_{[\tau]}$
Table 1: Lie algebra associated to the number of branch points of the quotient
map $\mathbb{T}\rightarrow\mathbb{T}/\Gamma$.

Here we use the well known one-to-one relation between complex tori up to
isomorphism and elements of $\mathrm{SL}_{2}(\mathbb{Z})\backslash\mathbb{H}$.

The dimension of the abelianisation $\mathfrak{A}/[\mathfrak{A},\mathfrak{A}]$
can be computed directly from the definition for
$\mathfrak{A}=\mathfrak{C}_{\tau},\,\mathfrak{O},\,\mathfrak{S}_{\tau}$, and
is then seen to be equal to the number of branch points of
$\mathbb{T}\rightarrow\mathbb{T}/\Gamma$. In particular, the Lie algebras
$\mathfrak{C}_{\tau},\mathfrak{O}$, and $\mathfrak{S}_{\tilde{\tau}}$, where
$\tau,\tilde{\tau}\in\mathbb{H}$, are pairwise nonisomorphic, and if two
automorphic Lie algebras in our classification are isomorphic, then the number
of branch points of the corresponding quotient maps are equal.

##  2 Definitions, notations and first examples

In this section we will define automorphic Lie algebras, introduce notations
and present two examples: one where the Riemann surface is the Riemann sphere
and one where it is a complex torus.

Let ${\mathfrak{g}}$ be a complex finite dimensional Lie algebra and let $X$
be a compact Riemann surface. Suppose $\Gamma$ is a finite group acting on
${\mathfrak{g}}$ by Lie algebra isomorphisms and on $X$ by biholomorphisms. We
denote the group of Lie algebra isomorphisms by $\mathrm{Aut}({\mathfrak{g}})$
and the group of biholomorphisms of $X$ by $\mathrm{Aut}(X)$. The group
actions can be described by homomorphisms

$\sigma:\Gamma\rightarrow\mathrm{Aut}(X),\quad\rho:\Gamma\rightarrow\mathrm{Aut}({\mathfrak{g}}).$

Consider a finite $\Gamma$-invariant set $\mathcal{S}\subset X$ and denote by
$\mathbb{X}$ the complement $X\setminus\mathcal{S}$. Thus $\mathbb{X}$ is a
punctured compact Riemann surface. We use the notation
$\mathcal{O}_{\mathbb{X}}$ for the ${\mathbb{C}}$-algebra of meromorphic
functions on $X$ holomorphic on $\mathbb{X}$. Denote by $\tilde{\sigma}$ the
induced homomorphism $\Gamma\rightarrow\mathrm{Aut}(\mathcal{O}_{\mathbb{X}})$
defined by $\tilde{\sigma}(\gamma)f=f\circ\sigma(\gamma)^{-1}$ for
$\gamma\in\Gamma$. A _current algebra_ is formed by taking the tensor product
over ${\mathbb{C}}$ of ${\mathfrak{g}}$ and $\mathcal{O}_{\mathbb{X}}$. This
has a natural structure of a Lie algebra by declaring

$\left[\sum_{i}A_{i}\otimes f_{i},\sum_{j}B_{j}\otimes
g_{j}\right]:=\sum_{i,j}[A_{i},B_{j}]\otimes f_{i}g_{j},$

where $A_{i},B_{j}\in{\mathfrak{g}}$, $f_{i},g_{j}\in\mathcal{O}_{\mathbb{X}}$
and the bracket on the right-hand side is the bracket in ${\mathfrak{g}}$. We
let $\Gamma$ act naturally on
${\mathfrak{g}}\otimes_{{\mathbb{C}}}\mathcal{O}_{\mathbb{X}}$ by the diagonal
action

$\gamma\cdot(A\otimes f)=\rho(\gamma)A\otimes\tilde{\sigma}(\gamma)f.$ (2)

We will also write $\rho\otimes\tilde{\sigma}(\gamma)$ for the homomorphism
corresponding to the action defined in (2).

An automorphic Lie algebra is the Lie subalgebra of the current algebra
${\mathfrak{g}}\otimes_{{\mathbb{C}}}\mathcal{O}_{\mathbb{X}}$ of those
elements invariant under the specified action.

######  Definition 2.1.

The automorphic Lie algebra
$\mathfrak{A}=\mathfrak{A}({\mathfrak{g}},\mathbb{X},\Gamma,\rho,\sigma)$ is
defined as the algebra of fixed points of the action of $\Gamma$ on
${\mathfrak{g}}\otimes_{{\mathbb{C}}}\mathcal{O}_{\mathbb{X}}$. That is,

$\mathfrak{A}({\mathfrak{g}},\mathbb{X},\Gamma,\rho,\sigma)=({\mathfrak{g}}\otimes_{{\mathbb{C}}}\mathcal{O}_{\mathbb{X}})^{\rho\otimes\tilde{\sigma}(\Gamma)}=\\{a\in{\mathfrak{g}}\otimes_{{\mathbb{C}}}\mathcal{O}_{\mathbb{X}}:\gamma\cdot
a=a\,\,\text{for}\,\,\text{any}\,\,\gamma\in\Gamma\\}.$

From now onwards, we will use the acronym _aLia_ for “automorphic Lie
algebra”. We will sometimes simply write
$\mathfrak{A}=({\mathfrak{g}}\otimes_{{\mathbb{C}}}\mathcal{O}_{\mathbb{X}})^{\Gamma}$,
suppressing the additional data, if they are clear from the context.

As mentioned in the introduction, the Onsager algebra is a Lie algebra that
appears prominently in the context of aLias. Roan discovered in 1991 [30] that
$\mathfrak{O}$ is in fact an aLia _avant la lettre_. We will give an example
of an aLia, where $X$ is the Riemann sphere, which is isomorphic to the
Onsager algebra.

######  Example 2.2 (The simplest concrete form of the Onsager algebra).

Let $X={\mathbb{C}}_{\infty}$ be the Riemann sphere and let
${\mathfrak{g}}=\mathfrak{sl}_{2}$. Take $\mathcal{S}=\\{0,\infty\\}$. The
punctured surface $\mathbb{X}$ is given by
${X}\setminus\mathcal{S}\cong{\mathbb{C}}^{*}$ and the associated algebra of
functions is $\mathcal{O}_{\mathbb{X}}=\mathbb{C}[z,z^{-1}]$, the Laurent
polynomials. Let $\Gamma$ be the group with two elements, generated by
$\gamma$, and define the homomorphism
$\sigma:\Gamma\rightarrow\mathrm{Aut}(X)$ by $\sigma(\gamma)z=z^{-1}$. The
function

$J=\frac{z+2+z^{-1}}{4}$

is invariant under this action. It is in fact a Hauptmodul, in the sense that
any invariant meromorphic function on the Riemann sphere is a rational
function in $J$. This can be shown using the theory of divisors. If we then
restrict the poles to $\\{0,\infty\\}$ we obtain
$\mathbb{C}[z,z^{-1}]^{\Gamma}=\mathbb{C}[J]$. We denote this space by
$\mathbb{C}[z,z^{-1}]^{+}$ as it is the $+1$ eigenspace of $\gamma$. The $-1$
eigenspace $\mathbb{C}[z,z^{-1}]^{-}$ is given by $\mathbb{C}[J](z-z^{-1})$.
Indeed, if $f\in\mathbb{C}[z,z^{-1}]^{-}$ then it necessarily vanishes at $1$
and $-1$, hence $f/(z-z^{-1})$ is an element of $\mathbb{C}[z,z^{-1}]^{+}$.
Define the homomorphism
$\rho:\Gamma\rightarrow\mathrm{Aut}(\mathfrak{sl}_{2})$ by

$\rho(\gamma)\begin{pmatrix}a&b\\\ c&-a\end{pmatrix}=\begin{pmatrix}a&-b\\\
-c&-a\end{pmatrix}.$

Let

$e=\begin{pmatrix}0&1\\\ 0&0\end{pmatrix},\quad f=\begin{pmatrix}0&0\\\
1&0\end{pmatrix},\quad h=\begin{pmatrix}1&0\\\ 0&-1\end{pmatrix}.$

This action on $\mathfrak{sl}_{2}$ yields the eigenspace decomposition

$\mathfrak{sl}_{2}^{+}={\mathbb{C}}h,\quad\mathfrak{sl}_{2}^{-}={\mathbb{C}}\langle
e,f\rangle.$

We have

$\displaystyle\mathfrak{A}$
$\displaystyle=(\mathfrak{sl}_{2}\otimes_{{\mathbb{C}}}\mathcal{O}_{\mathbb{X}})^{\rho\otimes\tilde{\sigma}(\Gamma)}$
$\displaystyle=\mathfrak{sl}_{2}^{+}\otimes_{{\mathbb{C}}}\mathcal{O}_{\mathbb{X}}^{+}\oplus\mathfrak{sl}_{2}^{-}\otimes_{{\mathbb{C}}}\mathcal{O}_{\mathbb{X}}^{-}$
$\displaystyle={\mathbb{C}}h\otimes_{{\mathbb{C}}}{\mathbb{C}}[J]\oplus{\mathbb{C}}\langle
e,f\rangle\otimes_{{\mathbb{C}}}{\mathbb{C}}[J](z-z^{-1}).$

Define

$H=h\otimes 1,\quad E=e\otimes(z-z^{-1})/4,\quad F=f\otimes(z-z^{-1})/4$

so that we can write $\mathfrak{A}={\mathbb{C}}\langle
E,F,H\rangle\otimes_{{\mathbb{C}}}{\mathbb{C}}[J]$ with brackets

$[H,E]=2E,\quad[H,F]=-2F,\quad[E,F]=H\otimes J(J-1).$

We have chosen the Hauptmodul $J$ with values $0$ and $1$ at the branch points
($-1$ and $1$), but we could have chosen any other two (distinct) values. They
appear as zeros in the last Lie bracket.
This Lie algebra made a lot of appearances in the literature. Roan studied the
Onsager algebra and found this aLia (in another basis) as a concrete form of
the Onsager algebra which enabled him to classify its finite dimensional
irreducible representations [30]. The work of Lombardo and Sanders [21],
Knibbeler, Lombardo and Sanders [14] and Bury and Mikhailov [1] showed that
any aLia of $\Gamma$-equivariant meromorphic maps
${\mathbb{C}_{\infty}}\to\mathfrak{sl}_{2}$ with precisely two $\Gamma$-orbits
of ramification points in its holomorphic domain and precisely one
$\Gamma$-orbit outside its holomorphic domain, is isomorphic to
$\mathfrak{A}$. Knibbeler, Lombardo and Veselov realised this Lie algebra in
terms of modular functions, and provided an explicit isomorphism
$\mathfrak{O}\to\mathfrak{A}$ by

$A_{0}\mapsto H,\quad A_{1}\mapsto H\otimes(2J-1)-2E+2F$

cf. [17, Theorem 2.5].

We will see in Section 6 that the Onsager algebra can also be realised as
certain aLias on a complex torus with enough symmetry, that is, tori on which
the groups $C_{4}$ and $C_{6}$ act. As opposed to the case of the Riemann
sphere, the symmetry group $C_{2}$ will not appear in this context. Instead,
the symmetry groups $C_{3},C_{4},C_{6}$ and $A_{4}$ are the only ones that
give rise to aLias isomorphic to the Onsager algebra.

For our first example of an aLia on a complex torus, we will use the
_Weierstrass $\wp$-function_. This function is defined as follows. Let
${\Lambda}$ be a lattice and consider the complex torus
$T={\mathbb{C}}/{\Lambda}$. The function
$\wp_{{\Lambda}}:T\rightarrow{\mathbb{C}}$ defined by

$\displaystyle\wp_{\Lambda}(z)=\frac{1}{z^{2}}+\sum_{0\neq\omega\in\Lambda}\left(\frac{1}{(z-\omega)^{2}}-\frac{1}{\omega^{2}}\right),$
(3)

is called the Weierstrass $\wp$-function associated with the lattice
${\Lambda}$. It is a meromorphic function on $T$ with poles precisely at the
lattice points $\omega\in{\Lambda}$. The reference to a lattice $\Lambda$ will
be suppressed when there is no confusion. It is clear that $\wp(-z)=\wp(z)$
and $\wp^{\prime}(-z)=-\wp^{\prime}(z)$, where $\wp^{\prime}$ denotes the
derivative of $\wp$. Furthermore, $\wp$ satisfies the equation
$(\wp^{\prime})^{2}=4\wp^{3}-g_{2}\wp-g_{3}$ where
$g_{2},g_{3}\in{\mathbb{C}}$ are the elliptic invariants. The space of even
meromorphic functions on $T$ which are holomorphic on $T\setminus\\{0\\}$ is
given by the polynomials in $\wp$ with complex coefficients,
${\mathbb{C}}[\wp]$, cf. [2, Proposition V.3.1]. Using this and the relation
$(\wp^{\prime})^{2}=4\wp^{3}-g_{2}\wp-g_{3}$, it is not difficult to see that
any odd meromorphic function on $T$, holomorphic on $T\setminus\\{0\\}$, is an
element of ${\mathbb{C}}[\wp]\wp^{\prime}$. Since any function on $T$ is a sum
of an even and odd function, we have established the following fact.

######  Lemma 2.3.

The algebra of meromorphic function on complex torus $T$, holomorphic on
$T\setminus\\{0\\}$, is given by
$\mathcal{O}_{T\setminus\\{0\\}}={\mathbb{C}}[\wp,\wp^{\prime}]$.

We now have all the ingredients to present our first, simple example of an
aLia on a complex torus with symmetry group $C_{2}$. We will write
$\mathbb{T}$ for $T\setminus\Gamma\cdot\\{0\\}$.

######  Example 2.4.

Fix a complex torus $T={\mathbb{C}}/{\Lambda}$ with
$\Lambda={\mathbb{Z}}\oplus{\mathbb{Z}}\tau$. Let
${\mathfrak{g}}=\mathfrak{sl}_{2}$ and $\Gamma=C_{2}$, generated by $\gamma$,
where $\sigma:\Gamma\rightarrow\mathrm{Aut}(T)$ is the homomorphism defined by
$\sigma(\gamma)z=-z.$ The punctured torus $\mathbb{T}$ is given by
$T\setminus\\{0\\}$. Write $\wp=\wp_{{\Lambda}}$. Keeping in mind that
$\mathcal{O}_{\mathbb{T}}={\mathbb{C}}[\wp,\wp^{\prime}]$ and
$(\wp^{\prime})^{2}=4\wp^{3}-g_{2}\wp-g_{3}$, it is easily seen that the
algebra $\mathcal{O}_{\mathbb{T}}$ is given by
$\mathcal{O}_{\mathbb{T}}=\mathcal{O}_{\mathbb{T}}^{+}\oplus\mathcal{O}_{\mathbb{T}}^{-}$,
where

$\mathcal{O}_{\mathbb{T}}^{+}={\mathbb{C}}[\wp],\quad\mathcal{O}_{\mathbb{T}}^{-}={\mathbb{C}}[\wp]\wp^{\prime}$

denote the space of even and odd functions, respectively. Define the
homomorphism $\rho:\Gamma\rightarrow\mathrm{Aut}(\mathfrak{sl}_{2})$ by

$\rho(\gamma)\begin{pmatrix}a&b\\\ c&-a\end{pmatrix}=\begin{pmatrix}a&-b\\\
-c&-a\end{pmatrix}.$

Let

$e=\begin{pmatrix}0&1\\\ 0&0\end{pmatrix},\quad f=\begin{pmatrix}0&0\\\
1&0\end{pmatrix},\quad h=\begin{pmatrix}1&0\\\ 0&-1\end{pmatrix}.$

This action on $\mathfrak{sl}_{2}$ decomposes as
$\mathfrak{sl}_{2}^{+}\oplus\mathfrak{sl}_{2}^{-}$, where

$\mathfrak{sl}_{2}^{+}={\mathbb{C}}h,\quad\mathfrak{sl}_{2}^{-}={\mathbb{C}}\langle
e,f\rangle$

are the eigenspaces with eigenvalue $1$ and $-1$, respectively. We have

$\mathfrak{A}=(\mathfrak{sl}_{2}\otimes_{{\mathbb{C}}}\mathcal{O}_{\mathbb{T}})^{\rho\otimes\tilde{\sigma}(C_{2})}=\mathfrak{sl}_{2}^{+}\otimes_{{\mathbb{C}}}\mathcal{O}_{\mathbb{X}}^{+}\oplus\mathfrak{sl}_{2}^{-}\otimes_{{\mathbb{C}}}\mathcal{O}_{\mathbb{X}}^{-}={\mathbb{C}}\langle
h,e\otimes\wp^{\prime},f\otimes\wp^{\prime}\rangle\otimes_{{\mathbb{C}}}{\mathbb{C}}[\wp].$

Define

$E=e\otimes\wp^{\prime},\quad F=f\otimes\wp^{\prime},\quad H=h\otimes 1.$

Then
$\mathfrak{A}=(\mathfrak{sl}_{2}\otimes_{{\mathbb{C}}}\mathcal{O}_{\mathbb{X}})^{\rho\otimes\tilde{\sigma}(C_{2})}={\mathbb{C}}\langle
E,F,H\rangle\otimes_{{\mathbb{C}}}{\mathbb{C}}[\wp]\cong\mathfrak{S}_{\tau}$
with brackets

$[H,E]=2E,\quad[H,F]=-2F,\quad[E,F]=H\otimes(4\wp^{3}-g_{2}(\tau)\wp-
g_{3}(\tau)).$

Observe that
$(\mathfrak{sl}_{2}\otimes_{{\mathbb{C}}}\mathcal{O}_{\mathbb{T}})^{\rho\otimes\tilde{\sigma}(C_{2})}\not\cong\mathfrak{sl}_{2}\otimes_{{\mathbb{C}}}\mathcal{O}_{\mathbb{T}}^{\tilde{\sigma}(C_{2})}$,
since the right-hand side is perfect, whereas the left-hand side is not.

In the above example,
$(\mathfrak{sl}_{2}\otimes_{{\mathbb{C}}}\mathcal{O}_{\mathbb{T}})^{\rho\otimes\tilde{\sigma}(C_{2})}\not\cong\mathfrak{sl}_{2}\otimes_{{\mathbb{C}}}\mathcal{O}_{\mathbb{T}}^{\tilde{\sigma}(C_{2})}$.
This is a consequence of the fact that with this choice of homomorphism
$\sigma$, the quotient $T/\sigma(C_{2})$ has genus 0 or equivalently, the
canonical projection $\pi:{\mathbb{T}}\rightarrow{\mathbb{T}}/\sigma(C_{2})$
has ramification points. It is a general fact, cf. [5], that for a punctured
compact Riemann surface $\mathbb{X}$ and a complex finite dimensional Lie
algebra ${\mathfrak{g}}$, we have
$({\mathfrak{g}}\otimes_{{\mathbb{C}}}\mathcal{O}_{\mathbb{X}})^{\Gamma}\not\cong{\mathfrak{g}}\otimes_{{\mathbb{C}}}\mathcal{O}_{\mathbb{X}}^{\Gamma}$
whenever $\mathbb{X}\rightarrow\mathbb{X}/\Gamma$ contains a ramification
point $x_{0}$, when $\Gamma_{x_{0}}$ acts nontrivially on ${\mathfrak{g}}$. We
will later see how choosing a different homomorphism of
$C_{2}\rightarrow\mathrm{Aut}(T)$ does yield an isomorphism.

##  3 A classification scheme

The aim of this section is to explain what we will precisely classify. Our
objects of interest are aLias of the form

$\mathfrak{A}(\mathfrak{sl}_{2},{\mathbb{T}},\Gamma,\rho,\sigma)=(\mathfrak{sl}_{2}\otimes_{{\mathbb{C}}}\mathcal{O}_{\mathbb{T}})^{\rho\otimes\tilde{\sigma}(\Gamma)},$

where $\Gamma$ is a finite group,
$\rho:\Gamma\rightarrow\mathrm{Aut}(\mathfrak{sl}_{2})$ and
$\sigma:\Gamma\rightarrow\mathrm{Aut}(T)$ homomorphisms,
${\mathbb{T}}=T\setminus\mathcal{S}$, where $T$ is a complex torus and
$\mathcal{S}$ an orbit of $\Gamma$ in $T$. We would like to determine the
isomorphism classes of these aLias, which is the content of Theorem 1.1. We
may assume that $\mathcal{S}$ is the orbit of $0$ because $\mathrm{Aut}(T)$
acts transitively on $T$ (using Lemma 3.3 below), and we may assume that
$\rho$ and $\sigma$ are faithful, almost without losing generality, explained
by Lemma 3.1 and 3.2. These restrictions constitute our classification scheme.

We will first address some general aspects of invariant spaces with respect to
some group action. Let $X$ be a compact Riemann surface, ${\mathfrak{g}}$ a
complex finite dimensional Lie algebra and let
$\rho:\Gamma\rightarrow\mathrm{Aut}(\mathfrak{g})$ and
$\sigma:\Gamma\rightarrow\mathrm{Aut}(X)$ be homomorphisms. Given a finite
group $\Gamma$ and a $\Gamma$-module $M$, a standard technique for obtaining
the space of invariants of the action of $\Gamma$ on $M$, denoted by
$M^{\Gamma}$, is that of averaging over the group $\Gamma$. We will explain
this method. Define the _averaging operator_ (also known as the _Reynolds
operator_) $\langle\cdot\rangle_{\Gamma}:M\rightarrow M$ by

$\langle m\rangle_{\Gamma}=\frac{1}{|\Gamma|}\sum_{\gamma\in\Gamma}\gamma\cdot
m.$

Clearly $\langle\cdot\rangle_{\Gamma}$ is linear if the action of $\Gamma$ on
$M$ is linear. One sees that its image is $M^{\Gamma}$ and $\langle\langle
m\rangle_{\Gamma}\rangle_{\Gamma}=\langle m\rangle_{\Gamma}$, thus
$\langle\cdot\rangle_{\Gamma}$ is a projection onto $M^{\Gamma}$. Suppose $K$
is a normal subgroup of $\Gamma$, which we denote by $K\lhd\Gamma$. Then there
is an obvious action of the quotient group $\Gamma/K$ on $M^{K}$ and we can
check directly that
$\langle\langle\cdot\rangle_{K}\rangle_{\Gamma/K}=\langle\cdot\rangle_{\Gamma}$.
In particular, an aLia with symmetry group $\Gamma$ can be computed making use
of the fact that it has a normal subgroup $K$. Phrased in terms of our
setting:

######  Lemma 3.1.

Let $K\lhd\Gamma$. Then
$({\mathfrak{g}}\otimes_{{\mathbb{C}}}\mathcal{O}_{\mathbb{X}})^{\Gamma}=(({\mathfrak{g}}\otimes_{{\mathbb{C}}}\mathcal{O}_{\mathbb{X}})^{K})^{\Gamma/K}.$

Let $\tilde{\rho}=\rho\otimes\tilde{\sigma}$ and
$\tilde{K}=\ker\rho\cdot\ker\sigma$. First of all, observe that we may replace
$\Gamma$ by $\Gamma/\ker(\rho\otimes\tilde{\sigma})$, making use of the fact
that $\ker(\rho\otimes\tilde{\sigma})$ is a normal subgroup of $\Gamma$ and
invoking Lemma 3.1. Trivially,
$({\mathfrak{g}}\otimes_{{\mathbb{C}}}\mathcal{O}_{\mathbb{X}})^{\ker(\rho\otimes\tilde{\sigma})}={\mathfrak{g}}\otimes_{{\mathbb{C}}}\mathcal{O}_{\mathbb{X}}$.
We can therefore assume that $\ker\rho\cap\ker\tilde{\sigma}=1$ without losing
generality.

######  Lemma 3.2.

$({\mathfrak{g}}\otimes_{{\mathbb{C}}}\mathcal{O}_{\mathbb{X}})^{\rho\otimes\tilde{\sigma}(\Gamma)}=({\mathfrak{g}}^{\rho(\ker\sigma)}\otimes_{{\mathbb{C}}}\mathcal{O}_{\mathbb{X}}^{\tilde{\sigma}(\ker\rho)})^{\tilde{\rho}(\Gamma)/\tilde{\rho}(\tilde{K})}.$

###### Proof.

Clearly $\tilde{\rho}(\tilde{K})\lhd\tilde{\rho}(\Gamma)$ since
$\tilde{K}\lhd\Gamma$ and $\tilde{\rho}$ is a homomorphism. Suppose
$A\in{\mathfrak{g}}$ and $f\in\mathcal{O}_{\mathbb{X}}$. Now, if
$\tilde{\rho}(\gamma_{1}\gamma_{2})\in\tilde{\rho}(\tilde{K})$ with
$\gamma_{1}\in\ker(\rho)$ and $\gamma_{2}\in\ker(\sigma)$, then

$\tilde{\rho}(\gamma_{1}\gamma_{2})(A\otimes
f)=\rho(\gamma_{1}\gamma_{2})\otimes\tilde{\sigma}(\gamma_{1}\gamma_{2})(A\otimes
f)=\rho(\gamma_{2})A\otimes\tilde{\sigma}(\gamma_{1})f,$

where the homomorphism $\rho\otimes\tilde{\sigma}$ is defined in (2). Hence
$({\mathfrak{g}}\otimes_{{\mathbb{C}}}\mathcal{O}_{\mathbb{X}})^{\tilde{\rho}(\tilde{K})}={\mathfrak{g}}^{\rho(\ker\sigma)}\otimes_{{\mathbb{C}}}\mathcal{O}_{\mathbb{X}}^{\tilde{\sigma}(\ker\rho)}.$
By employing Lemma 3.1, this proves the claim. ∎

Assume now the case of $X=T$ and denote again by $\mathbb{T}$ the punctured
complex torus $T\setminus\sigma(\Gamma)\cdot\\{0\\}$ (see Section 2). We know
that
$\mathcal{O}_{\mathbb{T}}^{\tilde{\sigma}(\ker\rho)}\cong\mathcal{O}_{\mathbb{T}/\sigma(\ker\rho)}.$
Now, $\mathbb{T}/\sigma(\ker\rho)$ is a punctured compact Riemann surface of
genus 0 or 1. In the former case, we refer to previous literature [21, 15, 16,
1]. In the latter case, we may assume that
$\sigma(\ker\rho)=\tilde{\sigma}(\ker\rho)=1$. Together with
$\ker\sigma\cap\ker\rho=1$ this is equivalent to $\ker\rho=1$. Now consider
the homomorphism $\sigma$. If $\ker\sigma$ is nontrivial, the Lie algebra
$\mathfrak{sl}_{2}^{\rho(\ker\sigma)}$ has dimension less than 3 and therefore
is abelian, and we will rule out this class.

The following simple lemma plays an important role in our classification. It
appears in a more general setting in the context of equivariant map algebras
in [25]. Recall that $\mathbb{X}$ stands for the punctured compact Riemann
surface $X\setminus\mathcal{S}.$

######  Lemma 3.3.

Suppose $\rho:\Gamma\rightarrow\mathrm{Aut}(\mathfrak{g})$ and
$\sigma:\Gamma\rightarrow\mathrm{Aut}(X)$ are homomorphisms and
$\tilde{\rho}=\rho\otimes\tilde{\sigma}:\Gamma\rightarrow\mathrm{Aut}({\mathfrak{g}}\otimes_{{\mathbb{C}}}\mathcal{O}_{\mathbb{X}})$.
Let $\mathfrak{A}$ be the aLia defined by these actions:

$\mathfrak{A}=\\{a\in{\mathfrak{g}}\otimes_{{\mathbb{C}}}\mathcal{O}_{\mathbb{X}}:\tilde{\rho}(\gamma)a=a\,\,\text{for}\,\,\text{any}\,\,\gamma\in\Gamma\\},$

where $a=A\otimes f$, with $A\in{\mathfrak{g}}$ and
$f\in\mathcal{O}_{\mathbb{X}}$. Define a second action of $\Gamma$ on
${\mathfrak{g}}$ and $X$ by
$\rho^{\prime}(\gamma)=\tau_{1}\rho(\gamma)\tau_{1}^{-1}$ and
$\sigma^{\prime}(\gamma)=\tau_{2}\sigma(\gamma)\tau_{2}^{-1}$, where
$\tau_{1}\in\mathrm{Aut}({\mathfrak{g}})$ and $\tau_{2}\in\mathrm{Aut}(X)$
such that $\tau_{2}(\mathcal{S})=\mathcal{S}$. Let $\tilde{\rho}^{\prime}$ be
given by $\tilde{\rho}^{\prime}=\rho^{\prime}\otimes\tilde{\sigma}^{\prime}$
and define

$\mathfrak{A}^{\prime}=\\{a\in{\mathfrak{g}}\otimes_{{\mathbb{C}}}\mathcal{O}_{\mathbb{X}}:\tilde{\rho}^{\prime}(\gamma)a=a\,\,\text{for}\,\,\text{any}\,\,\gamma\in\Gamma\\}.$

Then $\mathfrak{A}\cong\mathfrak{A}^{\prime}$ as Lie algebras.

###### Proof.

Define
$\varphi:{\mathfrak{g}}\otimes_{{\mathbb{C}}}\mathcal{O}_{\mathbb{X}}\rightarrow{\mathfrak{g}}\otimes_{{\mathbb{C}}}\mathcal{O}_{\mathbb{X}}$
on simple tensors by $\varphi(A\otimes
f)=\tau_{1}(A)\otimes\tilde{\tau}_{2}f$, where $A\in{\mathfrak{g}}$,
$f\in\mathcal{O}_{\mathbb{X}}$ and $\tilde{\tau}_{2}f(z)=f(\tau_{2}^{-1}z)$
for $z\in\mathbb{X}$, and extend $\varphi$ ${\mathbb{C}}$-linearly to the
whole space. It is a straightforward verification that $\varphi$ is Lie
algebra isomorphism and that
$\varphi\circ\tilde{\rho}(\gamma)=\tilde{\rho}^{\prime}(\gamma)\circ\varphi$
for all $\gamma\in\Gamma$, that is, it intertwines $\tilde{\rho}$ and
$\tilde{\rho}^{\prime}$. This implies that
$a\in{\mathfrak{g}}\otimes_{{\mathbb{C}}}\mathcal{O}_{\mathbb{X}}$ is
invariant with respect to the action defined by $\tilde{\rho}$ if and only if
$\varphi(a)$ is invariant with respect to the action defined by
$\tilde{\rho}^{\prime}$. Thus $\varphi$ restricts to an isomorphism between
$\mathfrak{A}$ and $\mathfrak{A}^{\prime}.$ ∎

##  4 Symmetry groups with ${\mathfrak{g}}=\mathfrak{sl}_{2}$

In this section we will investigate which symmetry groups $\Gamma$ will play a
role in the confined context of our classification scheme. This requires an
understanding of the finite subgroups of $\mathrm{Aut}(\mathfrak{sl}_{2})$ and
of $\mathrm{Aut}\\!\left(T\right)$.

######  Lemma 4.1 (Klein).

Finite subgroups of $\mathrm{Aut}(\mathfrak{sl}_{2})$ are classified by the
list

$C_{N},\quad D_{N},\quad A_{4},\quad S_{4},\quad A_{5}.$

If two finite subgroups of $\mathrm{Aut}(\mathfrak{sl}_{2})$ are isomorphic,
then they are conjugate.

###### Proof.

The group $\mathrm{Aut}(\mathfrak{sl}_{2})$ is isomorphic to
$\mathrm{PSL}_{2}({\mathbb{C}})$ by the adjoint representation. A finite
subgroup of $\mathrm{PSL}_{2}({\mathbb{C}})$ leaves a Hermitian inner product
invariant, and is therefore conjugate to a subgroup of $\mathrm{PSU}_{2}$
which in turn is isomorphic to $\mathrm{SO}_{3}$. Therefore, the statement of
the lemma is equivalent to the analogue statement for the Lie group
$\mathrm{SO}_{3}$ instead of $\mathrm{Aut}(\mathfrak{sl}_{2})$. Klein showed
that the subgroups of $\mathrm{SO}_{3}$ are precisely the orientation
preserving isometries of $\mathbb{R}^{3}$ that fix the regular pyramids,
regular polygons, regular tetrahedrons, regular octahedrons and regular
icosahedrons, centred at the origin [11]. This yields the groups listed in the
statement, respectively. We may moreover assume that the vertices of the
polyhedra have norm $1$. If we then take two polyhedra of the same type, there
is an element of $\mathrm{SO}_{3}$ that transforms one to the other. This
group element conjugates the symmetry groups of the two polyhedra, and proves
the last statement of the lemma. ∎

######  Lemma 4.2.

A finite group $G$ embeds in $\mathrm{Aut}(\mathfrak{sl}_{2})$ and
$\mathrm{Aut}\\!\left(T\right)$ if and only if $G$ equals

  1. 1.

$C_{N},\quad N\geq 1$

  2. 2.

$D_{N},\quad N\geq 2$

  3. 3.

$A_{4},\quad(g_{2}(\tau)=0)$

###### Proof.

By Lemma 4.1, we know that the finite subgroups of
$\mathrm{Aut}(\mathfrak{sl}_{2})$ are given by

$C_{N},\quad D_{N},\quad A_{4},\quad S_{4},\quad A_{5}.$

We will show that only the subgroups as given in the statement, are
simultaneously subgroups of $\mathrm{Aut}(T)$ as well.

Let $T$ be a complex torus. It follows from [24, Proposition III.1.11] that
any automorphism $\sigma$ of $T$ is of the form $\sigma(z)=\epsilon z+\alpha$,
where $\epsilon$ is a suitable root of unity (for which $T$ has multiplication
by $\epsilon$) and $\alpha\in T$. This immediately gives a semi-direct product
structure $\mathrm{Aut}(T)=\mathrm{Aut}_{0}(T)\ltimes t(T)$ where
$\mathrm{Aut}_{0}(T)$ is the group of automorphisms fixing zero and $t(T)$ is
the group of translations of $T$. Now, take two automorphisms
$\sigma,\sigma^{\prime}$ defined by $\sigma(z)=\epsilon z+\alpha$ and
$\sigma^{\prime}(z)=\epsilon^{\prime}z+\alpha^{\prime}$. Then
$[\sigma,\sigma^{\prime}](z)=z-\alpha^{\prime}-\epsilon^{\prime}\alpha+\epsilon\alpha^{\prime}+\alpha$,
so that $[\sigma,\sigma^{\prime}]$ is indeed a translation. Hence
$[\mathrm{Aut}(T),\mathrm{Aut}(T)]\subset t(T)$.

Now, let $r\in t(T)$ be any translation of $T$, say $r(z)=z+\alpha$. We will
show that there are $s\in\mathrm{Aut}_{0}(T)$ and
$r^{\prime}\in\mathrm{Aut}(T)$ such that $r=[s,r^{\prime}]$. Suppose $s(z)=-z$
(any torus has multiplication by $-1$) and let
$r^{\prime}(z)=z-\frac{\alpha}{2}$. Then $sr^{\prime}(z)=-z+\frac{\alpha}{2}$.
Hence
$[s,r^{\prime}](z)=-(-z-\frac{\alpha}{2}-\frac{\alpha}{2})=z+\alpha=r(z)$.
This proves the claim that $[\mathrm{Aut}(T),\mathrm{Aut}(T)]=t(T)$.

For the proof that $C_{N}$, $D_{N}$ are subgroups of $\mathrm{Aut}(T)$, as
well as $A_{4}$ for a suitable torus $T$, we refer to Lemma 4.4 below.

Let us now argue that $S_{4}$ and $A_{5}$ are not subgroups of
$\mathrm{Aut}(T)$ for any complex torus $T$. To see that $S_{4}$ is not a
subgroup of $\mathrm{Aut}(T)$, note that $[S_{4},S_{4}]=A_{4}$, which is
nonabelian, whereas for $\Gamma\subset\mathrm{Aut}(T)$, the commutator
subgroup $[\Gamma,\Gamma]\subset t(T)$ is abelian.

Finally, suppose for a contradiction that $A_{5}\subset\mathrm{Aut}(T)$ for
some complex torus $T$. Then $A_{5}\cong C_{\ell}\ltimes H$ for some normal
subgroup $H\neq 1$, since $\mathrm{Aut}(T)=\mathrm{Aut}_{0}(T)\ltimes t(T)$.
However, $A_{5}$ is simple and thus it cannot have a proper normal subgroup.
This shows that the list above is all there is in the intersection of finite
subgroups of $\mathrm{Aut}(\mathfrak{sl}_{2})$ and $\mathrm{Aut}(T)$, for all
complex tori $T$. ∎

######  Remark 4.3.

Observe that we can write the groups from Lemma 4.2 in terms of semidirect
products as follows:

$1\ltimes C_{N},\quad C_{2}\ltimes C_{N},\quad C_{3}\ltimes(C_{2}\times
C_{2}),\quad C_{\ell}\ltimes 1,$

where $\ell\in\\{1,2,3,4,6\\}$ and $N\in\mathbb{N}.$ When we write $G\ltimes
K\subset\mathrm{Aut}_{0}(T)\ltimes t(T)$, we shall tacitly assume that
$G\subset\mathrm{Aut}_{0}(T)$ and $K\subset t(T)$.

######  Lemma 4.4.

The subgroups of $\mathrm{Aut}\\!\left(T\right)$ which are isomorphic to one
of the finite groups of Lemma 4.2 are classified by the following list, up to
conjugation.

  1. 1.

$C_{N}=\langle r:r^{N}=1\rangle$,

    1. (a)

$C_{\ell}\subset\mathrm{Aut}_{0}(T),\quad r(z)=e^{2\pi
i/\ell}z\quad(\ell\in\\{2,3,4,6)\\}).$

    2. (b)

$C_{N}\subset t(T),\quad r(z)=z+\alpha\quad(\alpha\,\,\text{is
a}\,\,N\text{-torsion point in }T).$

  2. 2.

$D_{N}=\langle s,r:s^{2}=r^{N}=1,\,(sr)^{2}=1\rangle$,

    1. (a)

$C_{2}\times C_{2}\subset t(T),\quad s(z)=z+\tau/2,\quad r(z)=z+1/2.$

    2. (b)

$C_{2}\ltimes C_{N}\subset\mathrm{Aut}_{0}(T)\ltimes t(T),\quad s(z)=-z,\quad
r(z)=z+\alpha$
$(\alpha\,\,\text{is a}\,\,N\text{-torsion point in }T)$.

  3. 3.

$A_{4}=\langle
s,r_{1},r_{2}:s^{3}=r_{1}^{2}=r_{2}^{2}=1,\,sr_{1}s^{-1}=r_{1}r_{2}=r_{2}r_{1},\,sr_{2}s^{-1}=r_{1}\rangle$,
$\tau=e^{2\pi i/3}$,

$s(z)=e^{2\pi i/3}z,\quad r_{1}(z)=z+1/2.$

Here we recall that the case 1a with $\ell=3,4$ or $6$ and the case 3 only
occurs for special tori, cf. [24, Proposition III.1.12], and the other cases
occur in any torus.

###### Proof.

We leave it to the reader to verify that the subgroups in the statements
satisfy the group relations and thus are indeed of the mentioned isomorphism
class.

We start with the cyclic groups. If an element $r$ of $\mathrm{Aut}(T)$ fixes
an element $p$ of $T$, then $t(p)rt(p)^{-1}\in\mathrm{Aut}_{0}(T)$ (where
$t(p)(z)=z+p$) and $t(p)\langle r\rangle t(p)^{-1}$ is as described in 1a. If
on the other hand $r$ has no fixed points, then it is as described in 1b.

Now for the dihedral groups, consider first the abelian case
$D_{2}=C_{2}\times C_{2}$. This group occurs precisely once in $t(T)$, as in
2a, since both groups have precisely 3 elements of order $2$.

Suppose now that $D_{2}\subset\mathrm{Aut}(T)$ has an element $s$ which is not
contained in $t(T)$. Using a conjugation as we did to classify the cyclic
groups, we may assume that $s\in\mathrm{Aut}_{0}(T)$, so that $s(z)=-z$. There
must also be a nontrivial element of $D_{2}$, say $r$, contained in $t(T)$.
Indeed, the elements of order $2$ in $\mathrm{Aut}(T)\setminus t(T)$ are the
maps $z\mapsto-z+b$. A product of two such maps is in $t(T)$. Thus the group
$D_{2}$ is as described in 2b with $N=2$.

For the remaining dihedral groups $D_{N}\subset\mathrm{Aut}(T)$ ($N\geq 3$) we
notice that $[D_{N},D_{N}]=\langle
r^{2}\rangle\subset[\mathrm{Aut}(T),\mathrm{Aut}(T)]\subset t(T)$ implies that
$r\in t(T)$. At least one of the order $2$ elements $D_{N}\setminus\langle
r\rangle$ must be outside of $t(T)$ because $D_{N}$ is nonabelian. Taking a
conjugate of the group, we may again assume that such an element, say $s$,
fixes $0$. Thus we arrive at the remaining groups described in 2b.

Finally, for the group $A_{4}$ we argue as follows. The derived subgroup
$[A_{4},A_{4}]=\langle
r_{1},r_{2}\rangle\subset[\mathrm{Aut}(T),\mathrm{Aut}(T)]\subset t(T)$ is
uniquely determined by the only $3$ elements in $t(T)$ of order $2$. The
element $s$ must be outside of $t(T)$ for $A_{4}$ is nonabelian. Taking a
conjugate of the group, we may assume that $s$ fixes zero, which leaves two
options: $s(z)=e^{2\pi i/3}z$ and $s(z)=e^{4\pi i/3}z$. Both options generate
the same group, since there is an automorphism of $A_{4}$ sending $s$ to
$s^{2}$. ∎

######  Remark 4.5.

The lemma above gives a classification of subgroups
$\Gamma\subset\mathrm{Aut}(T)$. One may be interested in a classification of
embeddings (injective homomorphisms) $\Gamma\to\mathrm{Aut}(T)$ instead. The
difference is in the consideration of $\mathrm{Aut}(\Gamma)$: two embeddings
$\Gamma\to\mathrm{Aut}(T)$ have the same image if and only if one is the
composition of the other with an automorphism of $\Gamma$.

######  Lemma 4.6.

Let $\Gamma\subset\mathrm{Aut}_{0}(T)\ltimes t(T)$ be a finite subgroup and
${\mathbb{T}}=T\setminus\Gamma\cdot\\{0\\}$. Then for the canonical projection
$\pi:{\mathbb{T}}\rightarrow{\mathbb{T}}/\Gamma$, we have

$\\#(\text{branch points of
}\pi:{\mathbb{T}}\rightarrow{\mathbb{T}}/\Gamma)=\begin{cases}0\quad\text{if}\,\,\Gamma\subset
t(T),\\\ 2\quad\text{if}\,\,\Gamma=C_{\ell}\ltimes
1\,\,\text{or}\,\,\Gamma=C_{3}\ltimes(C_{2}\times C_{2}),\\\
3\quad\text{if}\,\,\Gamma=C_{2}\ltimes C_{N},\end{cases}$

where $\ell\in\\{3,4,6\\}$.

###### Proof.

A point $p\in T$ is a ramification point of $\pi$ if the multiplicity of $\pi$
at $p$, denoted by $\mathrm{mult}_{p}(\pi)$, is at least 2. By [24, Theorem
III.3.4], $\mathrm{mult}_{p}(\pi)$ equals $|\Gamma_{p}|$, where
$\Gamma_{p}=\\{\gamma\in\Gamma:\gamma\cdot p=p\\}$. It is clear there are no
branch points if $\Gamma\subset t(T)$ since a translation has no fixed points.
Let now $\Gamma=C_{3}\subset\mathrm{Aut}_{0}(T)$ where
$T={\mathbb{C}}/{\mathbb{Z}}\oplus{\mathbb{Z}}\omega_{6}$. It is
straightforward to check that $|\Gamma_{p}|>1$ if and only if
$p\in\\{0,1/2,\omega_{6}/2,(1+\omega_{6})/2\\}$. The only points that are in
the same $\Gamma$-orbit, are $1/2$ and $\omega_{6}/2$. Hence the total number
of branch points of $\pi$ after deleting $\Gamma\cdot\\{0\\}$ equals 2. The
other cases follow by similar arguments. ∎

##  5 Functional aspects of aLia on complex tori

We will now discuss the functional aspects of aLias on complex tori. Since we
are considering meromorphic $\mathfrak{sl}_{2}$-valued maps on a complex
torus, a starting point is to understand meromorphic functions on a complex
torus. There are multiple (equivalent) approaches to this, e.g. via _Jacobi
theta-functions_ or _Weierstrass functions_. We shall further develop the
approach with the Weierstrass $\wp$-function, as we have started to do before
Example 2.4.

Below we will formulate some elementary properties of the Weierstrass
$\wp$-function. Define the values of the half lattice points of a given
lattice ${\Lambda}_{\tau}={\mathbb{Z}}\oplus{\mathbb{Z}}\tau$, under
$\wp_{{\Lambda}_{\tau}}$:

$\displaystyle e_{1}=\wp_{{\Lambda}_{\tau}}(1/2),\quad
e_{2}=\wp_{{\Lambda}_{\tau}}(\tau/2),\quad
e_{3}=\wp_{{\Lambda}_{\tau}}((1+\tau)/2).$ (4)

Observe that $\wp_{{\Lambda}_{\tau}}^{\prime}$ vanishes at the half lattice
points, since
$\wp_{{\Lambda}_{\tau}}^{\prime}(-z)=-\wp_{{\Lambda}_{\tau}}^{\prime}(z)$.

As we have remarked before Example 2.4, there is the relation between the
square of the derivative $\wp^{\prime}$ and $\wp$ itself, where we now write
$\wp$ without specifying the lattice:

$\displaystyle(\wp^{\prime})^{2}=4\wp^{3}-g_{2}\wp-g_{3}.$ (5)

Because $\wp^{\prime}$ vanishes at the half lattice points, the right-hand
side of (5) factors as $4(\wp-e_{1})(\wp-e_{2})(\wp-e_{3})$. From this, one
gets the following relations between the $e_{i}$:

$e_{1}+e_{2}+e_{3}=0,\quad e_{1}e_{2}+e_{1}e_{3}+e_{2}e_{3}=-g_{2}/4,\quad
e_{1}e_{2}e_{3}=g_{3}/4.$

The next basic lemma records how Weierstrass $\wp$-functions associated to
homothetic lattices are related.

######  Lemma 5.1.

For $\alpha\in\mathbb{C}^{*}$, we have
$\wp_{\alpha\Lambda}(z)=\alpha^{-2}\wp_{\Lambda}\left(\alpha^{-1}z\right).$

We write out explicitly two special cases of Lemma 5.1 that will play an
important role in the course of the paper. These cases correspond to the
square lattice ${\Lambda}_{i}={\mathbb{Z}}\oplus{\mathbb{Z}}i$ and the
hexagonal lattice
${\Lambda}_{\omega_{6}}={\mathbb{Z}}\oplus{\mathbb{Z}}\omega_{6}$, which are
the only lattices (up to homothety) that satisfy the property
$\alpha{\Lambda}={\Lambda}$ for some $\alpha\neq\pm 1$.

$\displaystyle\wp_{{\Lambda}_{i}}(i^{-1}z)$
$\displaystyle=-\wp_{{\Lambda}_{i}}(z),$
$\displaystyle\wp_{{\Lambda}_{i}}^{\prime}(i^{-1}z)$
$\displaystyle=-i\wp_{{\Lambda}_{i}}^{\prime}(z),$ (6)
$\displaystyle\wp_{{\Lambda}_{\omega_{6}}}(\omega_{6}^{-1}z)$
$\displaystyle=\omega_{6}^{2}\wp_{{\Lambda}_{\omega_{6}}}(z),$
$\displaystyle\wp_{{\Lambda}_{\omega_{6}}}^{\prime}(\omega_{6}^{-1}z)$
$\displaystyle=-\wp_{{\Lambda}_{\omega_{6}}}^{\prime}(z).$ (7)

The next lemma describes the isotypical components of the action
$C_{\ell}\subset\mathrm{Aut}_{0}(T)$ on $\mathcal{O}_{\mathbb{T}}$, where
$\ell\in\\{2,3,4,6\\}$ and $T$ is a suitable complex torus. These
decompositions will be used in Theorem 6.10, where we construct a basis for
aLias with symmetry group $C_{\ell}$. We denote by $\chi_{j}$,
$j=0,\ldots,\ell-1$ the characters of the group $C_{\ell}$, which are defined
as $\chi_{j}(r^{k})=\omega_{\ell}^{jk}$, where $r$ generates $C_{\ell}$.

######  Lemma 5.2.

Let $\sigma_{\ell}:C_{\ell}\rightarrow\mathrm{Aut}_{0}(T)$ be the homomorphism
$\sigma_{\ell}(r)(z)=\omega_{\ell}z$ and let $\wp:=\wp_{{\Lambda}}$. Suppose
that $\mathcal{S}=\\{0\\}$ and $\mathbb{T}=T\setminus\mathcal{S}$. Then for
$\Gamma=C_{2}$ we have

$\mathcal{O}_{\mathbb{T}}^{\chi_{0}}={\mathbb{C}}[\wp],\quad\mathcal{O}_{\mathbb{T}}^{\chi_{1}}={\mathbb{C}}[\wp]\wp^{\prime}.$

For $\Gamma=C_{3}$ we have

$\mathcal{O}_{\mathbb{T}}^{\chi_{0}}={\mathbb{C}}[\wp^{\prime}],\quad\mathcal{O}_{\mathbb{T}}^{\chi_{1}}={\mathbb{C}}[\wp^{\prime}]\wp^{2},\quad\mathcal{O}_{\mathbb{T}}^{\chi_{2}}={\mathbb{C}}[\wp^{\prime}]\wp.$

For $\Gamma=C_{4}$ we have

$\displaystyle\mathcal{O}_{\mathbb{T}}^{\chi_{0}}$
$\displaystyle={\mathbb{C}}[\wp^{2}],$
$\displaystyle\mathcal{O}_{\mathbb{T}}^{\chi_{1}}$
$\displaystyle={\mathbb{C}}[\wp^{2}]\wp\wp^{\prime},$
$\displaystyle\mathcal{O}_{\mathbb{T}}^{\chi_{2}}$
$\displaystyle={\mathbb{C}}[\wp^{2}]\wp,$
$\displaystyle\mathcal{O}_{\mathbb{T}}^{\chi_{3}}$
$\displaystyle={\mathbb{C}}[\wp^{2}]\wp^{\prime}.$

For $\Gamma=C_{6}$ we have

$\displaystyle\mathcal{O}_{\mathbb{T}}^{\chi_{0}}$
$\displaystyle={\mathbb{C}}[\wp^{3}],$
$\displaystyle\mathcal{O}_{\mathbb{T}}^{\chi_{1}}$
$\displaystyle={\mathbb{C}}[\wp^{3}]\wp^{2}\wp^{\prime},$
$\displaystyle\mathcal{O}_{\mathbb{T}}^{\chi_{2}}$
$\displaystyle={\mathbb{C}}[\wp^{3}]\wp,$
$\displaystyle\mathcal{O}_{\mathbb{T}}^{\chi_{3}}$
$\displaystyle={\mathbb{C}}[\wp^{3}]\wp^{\prime},$
$\displaystyle\mathcal{O}_{\mathbb{T}}^{\chi_{4}}$
$\displaystyle={\mathbb{C}}[\wp^{3}]\wp^{2},$
$\displaystyle\mathcal{O}_{\mathbb{T}}^{\chi_{5}}$
$\displaystyle={\mathbb{C}}[\wp^{3}]\wp\wp^{\prime}.$

###### Proof.

We will only prove the claims for $\Gamma=C_{3}$; the rest follows in the same
way. We know by Lemma 2.3 that
$\mathcal{O}_{\mathbb{T}}=\mathbb{C}[\wp,\wp^{\prime}]$. Using (7) we see that
${\mathbb{C}}[\wp^{\prime}]\subset\mathcal{O}_{\mathbb{T}}^{\chi_{0}}$,
${\mathbb{C}}[\wp^{\prime}]\wp^{2}\subset\mathcal{O}_{\mathbb{T}}^{\chi_{1}}$
and ${\mathbb{C}}[\wp^{\prime}]\wp\subset\mathcal{O}_{\mathbb{T}}^{\chi_{2}}.$
It remains to show that $\mathbb{C}[\wp,\wp^{\prime}]$ is a subset of
${\mathbb{C}}[\wp^{\prime}]\oplus{\mathbb{C}}[\wp^{\prime}]\wp^{2}\oplus{\mathbb{C}}[\wp^{\prime}]\wp$.
If $\wp^{a}\wp^{\prime b}$ is a monomial with $a\geq 3$, then we can
substitute
$\wp^{3}=\frac{1}{4}(\wp^{\prime})^{2}+\frac{g_{2}}{4}\wp+\frac{g_{3}}{4}$,
using (5), to obtain a polynomial in $\wp$ and $\wp^{\prime}$ where all
exponents of $\wp$ are less than $a$. Repeating this substitution finitely
many times, we see that $\wp^{a}\wp^{\prime b}$ is an element of
${\mathbb{C}}[\wp^{\prime}]\oplus{\mathbb{C}}[\wp^{\prime}]\wp^{2}\oplus{\mathbb{C}}[\wp^{\prime}]\wp$.
Hence $\mathbb{C}[\wp,\wp^{\prime}]$ is indeed a subset of
${\mathbb{C}}[\wp^{\prime}]\oplus{\mathbb{C}}[\wp^{\prime}]\wp^{2}\oplus{\mathbb{C}}[\wp^{\prime}]\wp$.
∎

ALias on complex tori with symmetry group $\Gamma$ and base Lie algebra
$\mathfrak{sl}_{2}$, as in our classification, turn out to be generated by
three generators over their algebra of invariants, as we will establish in
Section 6. We would therefore like to know what the algebra of invariants
$\mathcal{O}_{\mathbb{T}}^{\Gamma}$ is, to obtain explicit forms of the Lie
algebras. For aLias with symmetry group $C_{N}\subset t(T)$, this is described
in the following lemma. For $\alpha\in T$ of finite order, let
${\Lambda}_{(\alpha)}={\mathbb{Z}}+{\mathbb{Z}}\alpha+{\mathbb{Z}}\tau$ and
$T_{(\alpha)}={\mathbb{C}}/{\Lambda}_{(\alpha)}$.

######  Lemma 5.3.

Let $\sigma_{\alpha}:C_{N}\rightarrow\mathrm{Aut}(T)$ be given by
$\sigma_{\alpha}(r)(z)=z+\alpha$. Then

$\mathcal{O}_{\mathbb{T}}^{\tilde{\sigma}_{\alpha}(C_{N})}={\mathbb{C}}[\wp_{\Lambda_{(\alpha)}},\wp_{\Lambda_{(\alpha)}}^{\prime}],$

###### Proof.

We know that $\mathcal{O}^{\tilde{\sigma}_{\alpha}(C_{N})}_{\mathbb{T}}$ is
the space of $\Lambda_{(\alpha)}$-periodic meromorphic functions which are
holomorphic on $T\setminus\Lambda_{(\alpha)}$. With this perspective,
$\mathcal{O}_{\mathbb{T}}^{\tilde{\sigma}_{\alpha}(C_{N})}=\mathcal{O}_{\mathbb{T}/\sigma_{\alpha}(C_{N})}$.
Now use Lemma 2.3 with $T$ taken to be the complex torus
$T/\sigma_{\alpha}(C_{N})$. Thus
$\mathcal{O}_{\mathbb{T}}^{\tilde{\sigma}_{\alpha}(C_{N})}=\mathbb{C}[\wp_{\Lambda_{(\alpha)}},\wp_{\Lambda_{(\alpha)}}^{\prime}].$
∎

We will now construct meromorphic functions on complex tori that will play a
fundamental role throughout, especially in the construction of normal forms.
Given a meromorphic function $f$ on $T$, the _divisor_ $(f)$ of $f$ is defined
as the following formal ${\mathbb{Z}}$-linear combination of points $p\in T$,
$(f)=\sum_{p\in T}\mathrm{ord}_{p}(f)(p)$, where $\mathrm{ord}_{p}(f)$ is the
order of a zero or pole of $f$ at $p$ (if $p$ is neither a zero nor a pole,
$\mathrm{ord}_{p}(f)=0$). Let $\\{\chi_{0},\dots,\chi_{N-1}\\}$ denote the set
of characters of the cyclic group $C_{N}$. Embed this group as
$\sigma(C_{N})\subset\mathrm{Aut}(T)$, where $\sigma(r)z=z+\alpha.$ Let
$\tilde{\wp}=\wp-\wp(\alpha)$. Let $\pi_{\chi_{j}}$ denote the projection
$\mathcal{O}_{\mathbb{T}}\rightarrow\mathcal{O}_{\mathbb{T}}^{\chi_{j}}$ given
by

$\pi_{\chi_{j}}=\frac{1}{N}\sum_{r\in
C_{N}}\overline{\chi_{j}(r)}\tilde{\sigma}(r).$

We shall sometimes drop the notation that includes $\sigma$ and only write
$\gamma f$ for a group action of $\Gamma$ on some
$f\in\mathcal{O}_{\mathbb{T}}$. However, when this notation causes potential
confusion, we opt for readability and write a dot. Consider the function
$\wp^{\prime}/\tilde{\wp}$. By (5) and the fact that $\wp^{\prime}$ is odd, we
have $(\wp^{\prime})=-3(0)+(1/2)+(\tau/2)+((1+\tau)/2)$. If we assume $N\geq
3$, then $\alpha$ cannot be a half lattice point and thus $\alpha\neq-\alpha$.
The divisor of $\wp^{\prime}/\tilde{\wp}$ is then given by

$\displaystyle(\wp^{\prime}/\tilde{\wp})=-(0)-(\alpha)-(-\alpha)+(1/2)+(\tau/2)+((1+\tau)/2),$
(8)

and thus we see that $\wp^{\prime}/\tilde{\wp}\in\mathcal{O}_{\mathbb{T}}$. If
$N=2$, then one can verify that $\wp^{\prime}/\tilde{\wp}$ has poles in
$\\{0,\alpha\\}$ and thus again
$\wp^{\prime}/\tilde{\wp}\in\mathcal{O}_{\mathbb{T}}$.

We define $P_{j}:T\rightarrow{\mathbb{C}}$ to be the projection of
$\wp^{\prime}/\tilde{\wp}$ under $\pi_{\chi_{j}}$, multiplied by $N$:

$\displaystyle
P_{j}=N\pi_{\chi_{j}}\left(\frac{\wp^{\prime}}{\tilde{\wp}}\right)=\sum_{k=0}^{N-1}\frac{r^{k}\wp^{\prime}}{\omega_{N}^{kj}r^{k}\tilde{\wp}}.$
(9)

Notice that indeed $r\cdot P_{j}=\chi_{j}(r)P_{j}$. We would like to stress
that $P_{j}$ depends both on a choice of a homomorphism
$\sigma:C_{N}\rightarrow\mathrm{Aut}(T)$ and the character $\chi_{j}$ of
$C_{N}$, while the notation only shows $j$ dependence. To avoid any confusion,
we could write $P_{\sigma,\chi_{j}}$ instead - now risking an overload of
notation however.

The next lemma states how $P_{j}$ transforms under the action of
$s:z\mapsto-z$. It is a simple consequence of the fact that
$\wp^{\prime}/\tilde{\wp}$ is odd and that if $r$ is a translation of the
torus, then $sr^{k}=r^{-k}s$.

######  Lemma 5.4.

Consider the action on $T$ given by $s:z\mapsto-z$. Then $s$ acts on $P_{j}$
as $sP_{j}=-P_{-j}.$

We will now list some basic properties of the functions $P_{j}$. Recall that
if a function $f$ has a Laurent expansion
$f(z)=\sum_{n\in{\mathbb{Z}}}c_{n}(z-z_{0})^{n}$ about $z=z_{0}$, then the
_residue_ of $f$ at $z=z_{0}$ is given by the coefficient $c_{-1}$. Notation:
$\mathrm{Res}_{z=z_{0}}(f)=c_{-1}.$

######  Proposition 5.5.

Let $\alpha$ be an $N$-torsion point with $N\geq 2$, and let
$\mathcal{S}=\\{0,\alpha,2\alpha,\dots,(N-1)\alpha\\}$. Then

  1. 1.

$P_{j}$ is a meromorphic function on $T$ for all $j$, with at most order 1
poles in $\mathcal{S}$.

  2. 2.

If $P_{j}$ has a pole, then it has poles everywhere in $\mathcal{S}$.

  3. 3.

The residue of $P_{j}$ about $z=0$ is given by

$\mathrm{Res}_{z=0}(P_{j})=-2+\omega_{N}^{-j}+\omega_{N}^{j}.$

In particular $P_{j}$ is constant if and only if $j\equiv 0\mod N$. In fact,
$P_{0}=0$.

###### Proof.

Let $v=\frac{\wp^{\prime}}{\tilde{\wp}}$ and first take $N\geq 3$. Assume $r$
is given by $r:z\mapsto z+\alpha$. To prove the first item, note that
$(r^{k}\tilde{\wp})=-2(k\alpha)+((k+1)\alpha)+((k-1)\alpha)$ and hence

$(1/r^{k}\tilde{\wp})=-((k-1)\alpha)-((k+1)\alpha)+2(k\alpha).$

Using the divisor in (8), we see that
$r^{k}v=r^{k}\left(\frac{\wp^{\prime}}{\tilde{\wp}}\right)$ has order 1 poles
at $(k-1)\alpha$, $k\alpha$ and $(k+1)\alpha$, which are distinct when $N\geq
3$. Therefore
$P_{j}=\sum_{k=0}^{N-1}\frac{r^{k}\wp^{\prime}}{\omega_{N}^{kj}r^{k}\tilde{\wp}}$
has at most order 1 poles in
$\mathcal{S}=\\{0,\alpha,2\alpha,\dots,(N-1)\alpha\\}$.

To prove the second statement, we observe that the divisor $(P_{j})$ is
invariant under $r$, since $r$ acts by a nonzero scalar on $P_{j}$. Hence if
$P_{j}$ has a pole, then it has poles of order 1 everywhere in $\mathcal{S}$.

For the third item we consider a number of expansions about $z=0$. The
summands of $P_{j}$ that are responsible for the potential occurrence of pole
at $z=0$ are $r^{-1}v,v$ and $rv$. It follows from the definition of $\wp$ in
(3) that about $z=0$, we have

$\frac{1}{\wp(z)-\wp(\alpha)}=z^{2}+O(z^{4}),\quad\wp^{\prime}(z)=-\frac{2}{z^{3}}+O(z).$

Thus

$\displaystyle v(z)=\frac{\wp^{\prime}}{\tilde{\wp}}(z)=-\frac{2}{z}+O(z).$
(10)

For $N\geq 3$ we have $r\tilde{\wp}(z)=-\wp^{\prime}(\alpha)z+O(z^{2}).$
Observe that $\wp^{\prime}(\alpha)\neq 0$ as soon as $N\geq 3$, because zeros
of $\wp^{\prime}$ appear at half lattice points (i.e. 2-torsion points). We
see
$\frac{1}{r\tilde{\wp}}(z)=-\frac{1}{\wp^{\prime}(\alpha)}\frac{1}{z}+O(1)$
and $r\tilde{\wp}^{\prime}(z)=-\wp^{\prime}(\alpha)+O(z)$ from which we infer

$rv(z)=\frac{1}{z}+O(1).$

Similarly, one argues that $r^{-1}v(z)=\frac{1}{z}+O(1).$ Thus indeed

$\mathrm{Res}_{z=0}(P_{j})=\mathrm{Res}_{z=0}\left(\sum_{k=0}^{N-1}\omega_{N}^{-kj}r^{k}v\right)=-2+\omega_{N}^{-j}+\omega_{N}^{j}.$

If $N=2$, then
$P_{1}=\frac{\wp^{\prime}}{\tilde{\wp}}-r\frac{\wp^{\prime}}{\tilde{\wp}}$ and
it is easy to see that it has order 1 poles in $\\{0,\alpha\\}$. The expansion
in (10) still holds, thus
$\frac{\wp^{\prime}}{\tilde{\wp}}(z)=-\frac{2}{z}+O(z)$ and one can show that
$r\frac{\wp^{\prime}}{\tilde{\wp}}(z)=\frac{2}{z}+O(1)$ about $z=0$.

Now consider $P_{j}$ for $j\equiv 0\mod N$. Observe that it is invariant under
$r$. Thus $P_{0}$ descends to a function $\tilde{P}_{0}$ on the torus
$\tilde{T}=T/\langle r\rangle$. If $P_{0}$ has poles, they are of order 1 and
if this is the case, then $\tilde{P}_{0}$ would have a simple pole on
$\tilde{T}$. This is not possible due to the well known fact that if a compact
Riemann surface has a meromorphic function with a single simple pole, it must
be isomorphic to the Riemann sphere, cf. [24, Proposition II.4.11]. Hence
$P_{0}$ is constant.

Finally, $sP_{0}=-P_{0}$ from which it follows that $P_{0}=0$. This concludes
the proof. ∎

In Section 6, we will introduce intertwining maps which will be used in the
construction of bases of aLias. These maps are constructed via the $P_{j}$’s.
As a consequence, the bases will be formulated in terms of the functions
$P_{j}$. These functions are also important in another way: they generate the
ring $\mathcal{O}_{\mathbb{T}}$ over the complex numbers, as we will prove
now.

######  Proposition 5.6.

Let $r\in\mathrm{Aut}(T)$ be defined by $r(z)=z+\alpha$ with $\alpha$ an
$N$-torsion point, where $N\geq 3$. Define $P_{j}$ with respect to this $r$.
Then

$\mathcal{O}_{\mathbb{T}}={\mathbb{C}}[P_{1},P_{2},\ldots,P_{N-1}],$

where ${\mathbb{T}}=T\setminus\langle r\rangle\cdot\\{0\\}.$ Furthermore, for
any integer $j\not\equiv 0\mod N$, we have

${\mathbb{C}}[\wp_{{\Lambda}_{(\alpha)}}]={\mathbb{C}}[P_{-j}P_{j}].$

Formulated differently, $P_{-j}P_{j}$ is a generator for the algebra of even
meromorphic functions on the torus
$T^{\prime}={\mathbb{C}}/{\Lambda}_{(\alpha)}$, holomorphic on
$T^{\prime}\setminus\\{0\\}$.

###### Proof.

The space of meromorphic functions on a complex torus $T$ is denoted by
$\mathcal{M}(T)$. Let $D$ be a divisor on $T$. The degree $\deg(D)$ of
$D=\sum_{p\in T}n_{p}(p)$ is defined as $\deg(D)=\sum_{p\in T}n_{p}.$
Introduce the space of meromorphic functions on $T$ with poles bounded by $D$,
denoted by $L(D)$:

$L(D)=\\{f\in\mathcal{M}(T):\mathrm{div}(f)\geq-D\\}.$

Note that $L(D)$ is a complex vector space. The Riemann-Roch theorem,
specialised to genus 1 Riemann surfaces [24, Proposition V.3.14], says that if
$D$ is a divisor on a complex torus $T$ and $\deg(D)>0$, then
$\dim_{{\mathbb{C}}}L(D)=\deg(D)$. Let $D=\sum_{j=0}^{N-1}(j\alpha).$ We see
that $\dim_{{\mathbb{C}}}L(D)=\deg(D)=N$. By construction, we have $P_{j}\in
L(D)^{\chi_{j}}$, and hence $\dim_{{\mathbb{C}}}L(D)^{\chi_{j}}=1$ if
$j\not\equiv 0\mod N$. Therefore, $L(D)={\mathbb{C}}\langle
1,P_{1},\ldots,P_{N-1}\rangle.$ We proceed inductively. For each
$k\in\\{1,\ldots,N-1\\}$, going from $L((m-1)D)^{\chi_{k}}$ to
$L(mD)^{\chi_{k}}$, we add the 1-dimensional subspace generated by a product
of $P_{j}$’s such that $0\neq\prod_{j\in\\{j_{1},\ldots,j_{m}\\}}P_{j}\in
L(mD)^{\chi_{k}}$ to $L((m-1)D)^{\chi_{k}}$. In particular, for $k\not\equiv
0\mod N/2$ and letting $Q=(P_{-j}P_{j})^{\lfloor\frac{m-1}{2}\rfloor}$ (where
$\lfloor x\rfloor$ denotes the integer part of $x$), we claim that

$L(mD)^{\chi_{k}}/L((m-1)D)^{\chi_{k}}=\begin{cases}{\mathbb{C}}P_{k}Q+L((m-1)D)^{\chi_{k}},\quad\text{if}\,\,m\,\,\text{odd}\\\
{\mathbb{C}}P_{-k}P_{2k}Q+L((m-1)D)^{\chi_{k}},\quad\text{if}\,\,m\,\,\text{even}.\end{cases}$

Here we used that $N\geq 3$, so that there are enough integers $k$ such that
$P_{2k}\neq 0$. For $k\equiv 0\mod N/2$ and $m$ even, we need to change
$P_{-k}P_{2k}$ to $P_{-k-n}P_{2k+n}\in\mathcal{O}_{\mathbb{T}}^{\chi_{k}}$ for
some suitable $n\in{\mathbb{Z}}$. This shows that
$\mathcal{O}_{\mathbb{T}}=\bigcup_{j\in\mathbb{N}}L(jD)$ is generated as a
${\mathbb{C}}$-algebra by $P_{1},\ldots,P_{N-1}$ whenever $N\geq 3$.
To prove the second claim, notice that $rP_{-j}P_{j}=P_{-j}P_{j}$ and that it
has poles of order 2 in ${\Lambda}_{(\alpha)}$ if and only if $j\not\equiv
0\mod N$. If $z_{0}$ is a zero of $P_{-j}$, then $-z_{0}$ is a zero of $P_{j}$
since $P_{-j}(z_{0})=-P_{j}(-z_{0})$ by Lemma 5.4. Hence the divisor of
$P_{-j}P_{j}$ equals $(P_{-j}P_{j})=-2(0)+(z_{0})+(-z_{0})$. Now recall that
$\wp_{{\Lambda}_{(\alpha)}}$ has an order 2 pole at $z=0$ and is even, hence
$(\wp_{{\Lambda}_{(\alpha)}})=-2(0)+(w_{0})+(-w_{0})$ for some $w_{0}\in
T^{\prime}$. We see that indeed
$P_{-j}P_{j}=c_{1}\wp_{{\Lambda}_{(\alpha)}}+c_{2}$ for some
$c_{1},c_{2}\in{\mathbb{C}}$ and $c_{1}\neq 0$ if and only if $j\not\equiv
0\mod N$. Therefore
${\mathbb{C}}[\wp_{{\Lambda}_{(\alpha)}}]={\mathbb{C}}[P_{-j}P_{j}]$ if and
only if $j\not\equiv 0\mod N$. ∎

The following identities related to the $P_{j}$’s are recorded in the next
lemma. These will play an important role in the construction of intertwiners
in Section 6.

######  Lemma 5.7.

Let $N\in\mathbb{N}$. For every $j\in{\mathbb{Z}}$ and $k\not\equiv 0\mod N$,
there are $\lambda,\mu\in{\mathbb{C}}$ such that

$P_{2j}P_{-j}^{2}-P_{-2j}P_{j}^{2}=\lambda P_{-k}P_{k}+\mu.$

If in addition $N\geq 3$, $k=\pm j$ and $j\not\equiv 0\mod N/2$, then $\mu$ is
nonzero.

###### Proof.

Let $j\in{\mathbb{Z}}$ and write $f=P_{2j}P_{-j}^{2}-P_{-2j}P_{j}^{2}$.
Clearly $\gamma f=f$ for all $\gamma\in\langle s,r\rangle\cong D_{N}$ for any
$N\in\mathbb{N}$. In particular, $f$ is even and hence has at most order 2
poles, for any choice of $j$. By Proposition 5.6, we see that for all
$k\not\equiv 0\mod N$ we have that $f=\lambda P_{-k}P_{k}+\mu$ for some
$\lambda,\mu\in{\mathbb{C}}$.

Assume now that $N\geq 3$, $k=\pm j$ and $j\not\equiv 0\mod N/2$. If $\mu$
would be zero and $z_{0}\in T$ is a zero of $P_{j}$, then
$f(z_{0})=P_{2j}(z_{0})P_{-j}^{2}(z_{0})=0.$ Thus $z_{0}$ would also be a zero
of $P_{-j}$ or $P_{2j}$. However, since the set of zeros of $P_{k}$ is
invariant under $r$, the set of zeros of $P_{j}$ must then coincide with the
zero set of $P_{-j}$ or $P_{2j}$. Since the order and location of the poles of
$P_{j}$, $P_{-j}$ and $P_{2j}$ already coincide, this means that
$(P_{j})=(P_{-j})$ or $(P_{j})=(P_{2j})$, that is, $P_{j}$ is either a
constant multiple of $P_{-j}$ or $P_{2j}$. This is absurd, which therefore
means that $\mu$ cannot be zero. ∎

######  Remark 5.8.

One could find explicit formulas for $\lambda$ and $\mu$ using Proposition 5.5
and more detailed expansions. However, the obtained formulas are unlikely to
benefit the transparency of the normal form we will later obtain. Related
constructions will be carried out for $\Gamma=C_{2}\times C_{2}$ in Section
6.2, and here it does seem appropriate to keep track of the constants, mainly
because there is simply one group instead of a collection of groups, and the
constants have transparent formulas in terms of well known functions.

##  6 Construction of normal forms

In Section 2, we have seen a basic example of an aLia on a complex torus with
symmetry group $C_{2}$, embedded inside $\mathrm{Aut}(T)$ as $z\mapsto-z$, and
developed a number of functional aspects of the theory. We have observed that
this $C_{2}$-aLia has a particularly transparent structure - it is generated
over its algebra of invariants and its Cartan subalgebra has eigenvalues in
${\mathbb{C}}$ (in fact, in this case it is a constant matrix). We will see
that this is a general feature of aLias on complex tori with base Lie algebra
$\mathfrak{sl}_{2}$.

In this section we shall give explicit realisations of aLias and determine
their isomorphism type. It turns out that if the symmetry group acts on a
complex torus without fixed points, then we obtain aLias that are isomorphic
to a current algebra. Formulated differently: they are untwisted. If on the
other end the group acts with fixed points, the corresponding aLias are
twisted. The twisted aLias turn out to be isomorphic to either the Onsager
algebra $\mathfrak{O}$ or $\mathfrak{S}_{\tau}$.

We will bring the aLias into a _normal form_ , which will allow us to decide
the isomorphism type of the aLias in question. The normal form will be defined
in Definition 6.2. As a consequence, we settle the question whether the aLias
are hereditary [16], as defined in Definition 6.1.

Our main tools in constructing bases for aLias, are certain
$SL_{2}({\mathbb{C}})$-valued maps on a complex torus
${\mathbb{T}}=T\setminus\Gamma\cdot\\{0\\}$ whose composition with
$\mathrm{Ad}:SL_{2}({\mathbb{C}})\rightarrow\mathrm{Aut}(\mathfrak{sl}_{2})$
will intertwine the actions of $\Gamma$ on $T$ and $\mathfrak{sl}_{2}$.

######  Definition 6.1 (Hereditary aLias [16]).

An aLia is said to be _hereditary_ if it contains a _constant spectrum Cartan
subalgebra (CSA)_. This is a Cartan subalgebra which is generated by an
element $H$ with constant $\mathrm{ad}(H)$-eigenvalues.

######  Definition 6.2 (Normal form of aLias).

We say that an aLia
$\mathfrak{A}=(\mathfrak{sl}_{2}\otimes_{{\mathbb{C}}}\mathcal{O}_{\mathbb{T}})^{\Gamma}$
is in _normal form_ if

$\mathfrak{A}={\mathbb{C}}\langle
E,F,H\rangle\otimes_{{\mathbb{C}}}\mathcal{O}_{\mathbb{T}}^{\Gamma}$

such that

$[H,E]=2E,\quad[H,F]=-2F,\quad[E,F]=H\otimes p,$

where $p\in\mathcal{O}_{\mathbb{T}}^{\Gamma}$.

We remark that for the normal form as in Definition 6.2, we automatically have
that $H$ has eigenvalues in ${\mathbb{C}}$. The CSA
$\mathfrak{h}={\mathbb{C}}H\otimes\mathcal{O}_{\mathbb{T}}^{\Gamma}$ is a
constant spectrum CSA.

In the upcoming sections, we will construct normal forms for all aLias on
complex tori with base Lie algebra $\mathfrak{sl}_{2}$, in our classification.
This will therefore also establish the property of being hereditary of this
class of aLias.

Let the symmetry group be given by $\Gamma=G\ltimes
K\subset\mathrm{Aut}_{0}(T)\ltimes t(T)$, where we recall the notation $t(T)$
for the normal subgroup of translations of the torus $T$. We start with a
lemma that says that, under certain circumstances, a normal form for the aLia
with symmetry group $K$ automatically determines a normal form for $\Gamma$ as
well.

######  Lemma 6.3.

Let $\Gamma=G\ltimes K\subset\mathrm{Aut}_{0}(T)\ltimes t(T)$. If
$\mathfrak{A}=(\mathfrak{sl}_{2}\otimes_{{\mathbb{C}}}\mathcal{O}_{\mathbb{T}})^{K}$
has a normal form, and $H\in\mathfrak{A}$ is $G$-invariant, such that it
generates a CSA of $\mathfrak{A}$, then
$(\mathfrak{sl}_{2}\otimes_{{\mathbb{C}}}\mathcal{O}_{\mathbb{T}})^{\Gamma}$
has a normal form as well.

###### Proof.

First recall that $G=C_{\ell}\subset\mathrm{Aut}_{0}(T)$ and that $K$ consists
of translations of $T$. By Lemma 5.3, we therefore have
$\mathcal{O}_{\mathbb{T}}^{K}={\mathbb{C}}[\wp,\wp^{\prime}]$ for some $\wp$
associated to a suitable lattice. Observe that the statement is trivially true
for $\ell=1$, hence we take $\ell\in\\{2,3,4,6\\}$.

Assume the normal form of
$(\mathfrak{sl}_{2}\otimes_{{\mathbb{C}}}\mathcal{O}_{\mathbb{T}})^{K}$ to be

$(\mathfrak{sl}_{2}\otimes_{{\mathbb{C}}}\mathcal{O}_{\mathbb{T}})^{K}={\mathbb{C}}\langle
E,F,H\rangle\otimes_{{\mathbb{C}}}\mathcal{O}_{\mathbb{T}}^{K}={\mathbb{C}}\langle
E,F,H\rangle\otimes_{{\mathbb{C}}}{\mathbb{C}}[\wp,\wp^{\prime}]\cong\mathfrak{sl}_{2}\otimes_{{\mathbb{C}}}\mathcal{O}_{\mathbb{T}/K},$

with $\mathcal{O}_{\mathbb{T}}^{K}$-linear brackets $[H,E]=2E$, $[H,F]=-2F$
and $[E,F]=H$. Now, since $K$ is normal in $\Gamma$, $g\in G$ acts on
${\mathbb{C}}\langle
E,F,H\rangle\otimes_{{\mathbb{C}}}\mathcal{O}_{\mathbb{T}}^{K}$ and $\\{g\cdot
E,g\cdot F,g\cdot H\\}$ forms again a standard $\mathfrak{sl}_{2}$-triple,
where the action is defined in (2). By assumption, $g\cdot H=H$. Furthermore,
$[H,g\cdot E]=2g\cdot E$, so $g\cdot E=E\otimes k_{1}$ for some function
$k_{1}\in\mathcal{O}_{\mathbb{T}}^{K}$. Similarly, $g\cdot F=F\otimes k_{2}$
for some $k_{2}\in\mathcal{O}_{\mathbb{T}}^{K}$. We have $[g\cdot E,g\cdot
F]=H$ and this implies that $k_{2}=k_{1}^{-1}$. The only units in
$\mathcal{O}_{\mathbb{T}}^{K}={\mathbb{C}}[\wp,\wp^{\prime}]$ are the nonzero
constants. Therefore $g\cdot E=\omega_{\ell}^{j}E$ and $g\cdot
F=\omega_{\ell}^{-j}F$, for some $j\in\\{1,\ldots,\ell-1\\}$ such that
$\gcd(j,\ell)=1$. Notice that this forces $j\in\\{1,\ell-1\\}$, due to the
limited amount of choices for $\ell$.

We now decompose ${\mathbb{C}}\langle
E,F,H\rangle\otimes_{{\mathbb{C}}}{\mathbb{C}}[\wp,\wp^{\prime}]$ into its
$C_{\ell}$ isotypical components:

$({\mathbb{C}}\langle
E,F,H\rangle\otimes_{{\mathbb{C}}}{\mathbb{C}}[\wp,\wp^{\prime}])^{G}={\mathbb{C}}\langle
E\otimes p_{1},F\otimes p_{2},H\rangle\otimes_{{\mathbb{C}}}{\mathbb{C}}[J],$

where $p_{1}\in{\mathbb{C}}[\wp,\wp^{\prime}]^{\chi_{\ell-j}}$,
$p_{2}\in{\mathbb{C}}[\wp,\wp^{\prime}]^{\chi_{j}}$ and $J$ generates
$\mathcal{O}_{\mathbb{T}}^{\Gamma}$, which follows from Lemma 5.2.

Its Lie structure is given by

$[H,E\otimes p_{1}]=2E\otimes p_{1},\quad[H,F\otimes p_{2}]=-2F\otimes
p_{2},\quad[E\otimes p_{1},F\otimes p_{2}]=H\otimes p_{1}p_{2},$

where $p_{1}p_{2}\in{\mathbb{C}}[J]$. We may assume that $g\cdot
E=\omega_{\ell}E$ and $g\cdot F=\omega_{\ell}^{-1}F$ after applying an
automorphism that interchanges the isotypical components if necessary (e.g.
$\mathrm{Ad}\begin{pmatrix}0&1\\\ -1&0\end{pmatrix}$) and assume therefore
that $p_{1}\in{\mathbb{C}}[\wp,\wp^{\prime}]^{\chi_{\ell-1}}$,
$p_{2}\in{\mathbb{C}}[\wp,\wp^{\prime}]^{\chi_{1}}$.

∎

Observe that Lemma 6.3 implies that in order to understand aLias with
nonabelian symmetry groups $\Gamma=G\ltimes K$, it is sufficient to understand
the aLias with abelian symmetry groups $G$ and $K$ separately, granted that we
have found a $K$-invariant element $H$ in a CSA, which is simultaneously
$G$-invariant.

The proof of Lemma 6.3 gives us a construction to obtain the normal for the
aLia with symmetry group $\Gamma$ as well, if the condition on a generator of
the CSA is being satisfied.

The next proposition gives a criterion for the existence of normal forms of
aLias and extends Lemma 6.3 by providing a normal form of the full group
$\Gamma=G\ltimes K$.

######  Proposition 6.4.

Let $\Gamma=G\ltimes K\subset\mathrm{Aut}_{0}(T)\ltimes t(T)$ and let
$\rho:\Gamma\rightarrow\mathrm{Aut}(\mathfrak{sl}_{2})$ be a monomorphism and
$\tilde{\rho}:\Gamma\rightarrow\mathrm{Aut}(\mathfrak{sl}_{2})$ a homomorphism
whose kernel equals $K$. If there exists an inner automorphism
$\Psi\in\mathrm{Aut}_{\mathcal{O}_{\mathbb{T}}}(\mathfrak{sl}_{2}\otimes_{{\mathbb{C}}}\mathcal{O}_{\mathbb{T}})$
such that for all $z\in{\mathbb{T}}$ and $\gamma\in\Gamma$, we have

$\displaystyle\Psi(\gamma\cdot
z)=\rho(\gamma)\Psi(z)\tilde{\rho}(\gamma)^{-1},$ (11)

then

  1. 1.

$\mathfrak{K}:=(\mathfrak{sl}_{2}\otimes_{{\mathbb{C}}}\mathcal{O}_{\mathbb{T}})^{K}\cong\mathfrak{sl}_{2}\otimes_{{\mathbb{C}}}\mathcal{O}_{\mathbb{T}}^{K},$

  2. 2.

There exists a normal form for
$\mathfrak{A}=(\mathfrak{sl}_{2}\otimes_{{\mathbb{C}}}\mathcal{O}_{\mathbb{T}})^{\Gamma}$
and if the normal form of $\mathfrak{K}$ is given by ${\mathbb{C}}\langle
E,F,H\rangle\otimes_{{\mathbb{C}}}\mathcal{O}_{\mathbb{T}}^{K}$, with its Lie
structure inherited from $\mathfrak{sl}_{2}$, then the normal form for
$\mathfrak{A}$ given by

$(\mathfrak{sl}_{2}\otimes_{{\mathbb{C}}}\mathcal{O}_{\mathbb{T}})^{\Gamma}={\mathbb{C}}\langle
E\otimes p_{1},F\otimes
p_{2},H\rangle\otimes_{{\mathbb{C}}}\mathcal{O}_{\mathbb{T}}^{\Gamma},$

with Lie structure

$[H,E\otimes p_{1}]=2E\otimes p_{1},\quad[H,F\otimes p_{2}]=-2F\otimes
p_{2},\quad[E\otimes p_{1},F\otimes p_{2}]=H\otimes p_{1}p_{2},$

where $p_{1}$ generates $\mathcal{O}_{\mathbb{T}}^{\chi_{\ell-1}}$, where
$|G|=\ell$, $p_{2}$ generates $\mathcal{O}_{\mathbb{T}}^{\chi_{1}}$ and
$p_{1}p_{2}\in\mathcal{O}_{\mathbb{T}}^{\Gamma}$.

###### Proof.

First of all, we would like to stress that we can view
$\Psi\in\mathrm{Aut}_{\mathcal{O}_{\mathbb{T}}}(\mathfrak{sl}_{2}\otimes_{{\mathbb{C}}}\mathcal{O}_{\mathbb{T}})$
as a holomorphic $\mathrm{Aut}(\mathfrak{sl}_{2})$-valued function on
$\mathbb{T}$, in which case we use the notation $\Psi(z)$. We shall use both
interpretations interchangeably and let the argument of $\Psi$ be either an
element of $\mathbb{T}$ or an element of
$\mathfrak{sl}_{2}\otimes_{{\mathbb{C}}}\mathcal{O}_{\mathbb{T}}$.

The first part claims that for $k\in K$, if we have $\Psi(k\cdot
z)=\rho(k)\Psi(z)$, then there is a normal form for $\mathfrak{K}$. Let
$\\{x_{1},x_{2},x_{3}\\}$ be a basis for $\mathfrak{sl}_{2}$. Then for
$j=1,2,3,$ we have that $X_{j}(z):=\Psi(z)x_{j}$ is $K$-invariant:

$k\cdot X_{j}(z)=k\cdot\Psi(z)x_{j}=\rho(k)\Psi(k^{-1}\cdot z)x_{j}=X_{j}(z).$

Since
$\Psi\in\mathrm{Aut}_{\mathcal{O}_{\mathbb{T}}}(\mathfrak{sl}_{2}\otimes_{{\mathbb{C}}}\mathcal{O}_{\mathbb{T}})$
is an inner automorphism, we have

${\mathbb{C}}\langle
x_{1},x_{2},x_{3}\rangle\otimes_{{\mathbb{C}}}\mathcal{O}_{\mathbb{T}}=\Psi({\mathbb{C}}\langle
x_{1},x_{2},x_{3}\rangle\otimes_{{\mathbb{C}}}\mathcal{O}_{\mathbb{T}})={\mathbb{C}}\langle
X_{1},X_{2},X_{3}\rangle\otimes_{{\mathbb{C}}}\mathcal{O}_{\mathbb{T}}.$

Taking the invariants, we obtain a normal form

$\mathfrak{K}={\mathbb{C}}\langle
X_{1},X_{2},X_{3}\rangle\otimes_{{\mathbb{C}}}\mathcal{O}_{\mathbb{T}}^{K}\cong\mathfrak{sl}_{2}\otimes_{{\mathbb{C}}}{\mathbb{C}}[\wp,\wp^{\prime}],$

with the brackets inherited from $\mathfrak{sl}_{2}$ and where
$\wp,\wp^{\prime}$ generate $\mathcal{O}_{\mathbb{T}}^{K}$.

To prove the second item, we will argue as follows. Let
$\gamma=gk\in\Gamma=G\ltimes K$ and $x\in\mathfrak{sl}_{2}$. By (11),

$\gamma\cdot\Psi(z)x=\Psi(z)\tilde{\rho}(\gamma)x.$

Since $G$ is cyclic, it leaves a CSA of $\mathfrak{sl}_{2}$ invariant. Hence
there is a basis $\\{X_{1},X_{2},X_{3}\\}\subset\mathfrak{sl}_{2}$, with
$X_{1}$ a generator in the CSA, such that with respect to this basis,
$\tilde{\rho}(\gamma)=\tilde{\rho}(gk)=\mathrm{diag}(1,\chi_{1}(g),\chi_{\ell-1}(g))$,
where $\chi_{j}$ are the characters of $G$, $g$ generates $G$ and $k\in K$.
Now, $\Psi(X_{1}\otimes 1)$ generates a CSA of $\mathfrak{K}$ and
$g\cdot\Psi(X_{1}\otimes 1)=\Psi(X_{1}\otimes 1)$ for all $g\in G$ by (11). We
can therefore apply Lemma 6.3 to arrive at the conclusion of the existence of
a normal form for $\mathfrak{A}$ and the stated form of it. ∎

Recall that the groups $\Gamma=G\ltimes K\subset\mathrm{Aut}_{0}(T)\ltimes
t(T)$ considered in our classification, are isomorphic to one of the following
groups:

$1\ltimes C_{N},\quad C_{2}\ltimes C_{N},\quad C_{3}\ltimes(C_{2}\times
C_{2}),\quad C_{\ell}\ltimes 1,$

where $\ell\in\\{1,2,3,4,6\\}$ and $N\in\mathbb{N}$. The main object of our
considerations for the rest of the paper is finding a suitable
$\Psi\in\mathrm{Aut}_{\mathcal{O}_{\mathbb{T}}}(\mathfrak{sl}_{2}\otimes_{{\mathbb{C}}}\mathcal{O}_{\mathbb{T}})$
such that (11) holds. Section 6.1 is devoted to the cases $\Gamma=C_{N}$ and
$\Gamma=D_{N}\cong C_{2}\ltimes C_{N}$ while Section 6.2 concerns
$\Gamma=C_{2}\times C_{2}$ and $\Gamma=A_{4}\cong C_{3}\ltimes(C_{2}\times
C_{2})$.

###  6.1 The cases $\Gamma=C_{N}$ and $\Gamma=D_{N}$

We will start with the case $\Gamma=C_{N}$ where $C_{N}$ is embedded into
$\mathrm{Aut}(T)$ as translations by the homomorphism
$\sigma_{\alpha}:C_{N}\rightarrow\mathrm{Aut}(T)$, defined by
$\sigma_{\alpha}(r)z=z+\alpha$ for some $N$-torsion point $\alpha\in T$. The
main tool we use in the construction of aLias with symmetry group $C_{N}$, is
the construction of map $\Phi_{j}:{\mathbb{T}}\rightarrow
SL_{2}({\mathbb{C}})$ ($j\not\equiv 0\mod N/2)$ for which the
$\mathcal{O}_{\mathbb{T}}$-linear automorphism $\mathrm{Ad}(\Phi_{j})$ will
map a basis of $\mathfrak{sl}_{2}$ to a subalgebra of the current algebra
$\mathfrak{sl}_{2}\otimes_{{\mathbb{C}}}\mathcal{O}_{\mathbb{T}}$. The
automorphism $\mathrm{Ad}(\Phi_{j})$ is designed such that it satisfies the
equivariance condition (13) for a given homomorphism
$\rho:C_{N}\rightarrow\mathrm{Aut}(\mathfrak{sl}_{2})$, which will imply that
we obtain a basis of invariant matrices (with respect to the action induced by
the homomorphism $\rho\otimes\tilde{\sigma})$. From here, one quickly obtains
normal forms for aLias with symmetry group $C_{N}$. Having established the
relevant results, one easily obtains aLias with symmetry group $D_{N}$, since
$D_{N}\cong C_{2}\ltimes C_{N}$. Due to Lemma 4.1 and 3.3, we can choose any
faithful action of $C_{N}$ on $\mathfrak{sl}_{2}$ without loss of generality.

A crucial step in our construction of normal forms relies on the existence of
intertwining operators between the actions of a symmetry group $\Gamma$ on
$\mathfrak{sl}_{2}$ and $T$. Let us briefly mention some general facts about
these intertwiners to provide some motivation on the choice of these maps.
Consider monomorphisms (that is, faithful homomorphisms)
$\sigma:\Gamma\rightarrow\mathrm{Aut}(T)$ and $\delta:\Gamma\rightarrow
SL_{2}({\mathbb{C}})$. We would like to construct a matrix-valued map
$\Phi:T\rightarrow SL_{2}({\mathbb{C}})$,

$\displaystyle\Phi(z)=\begin{pmatrix}p_{1}(z)&q_{1}(z)\\\
p_{2}(z)&q_{2}(z)\end{pmatrix},$ (12)

where $p_{j},q_{j}\in\mathcal{O}_{\mathbb{T}}$, such that
$\Phi(\sigma(\gamma)z)=\delta(\gamma)\Phi(z)$ for all $\gamma\in\Gamma$ and
$z\in\mathbb{T}$. Taking the adjoint $\mathrm{Ad}$ of both sides, gives us the
equivariance property with respect to the actions induced by
$\sigma:\Gamma\rightarrow\mathrm{Aut}(T)$ and
$\rho:\Gamma\rightarrow\mathrm{Aut}(\mathfrak{sl}_{2})$, with
$\rho=\mathrm{Ad}\circ\delta$:

$\displaystyle\mathrm{Ad}(\Phi(\sigma(\gamma)z))=\rho(\gamma)\mathrm{Ad}(\Phi(z)).$
(13)

Notice that $\mathrm{Ad}(\Phi)$ is an element of
$\mathrm{Aut}_{\mathcal{O}_{\mathbb{T}}}(\mathfrak{sl}_{2}\otimes_{{\mathbb{C}}}\mathcal{O}_{\mathbb{T}})$
\- the group of $\mathcal{O}_{\mathbb{T}}$-linear automorphisms of
$\mathfrak{sl}_{2}\otimes_{{\mathbb{C}}}\mathcal{O}_{\mathbb{T}}$. The
difficult part in constructing such an equivariant map, is designing it in
such a way that it is invertible over ${\mathbb{C}}$. We will present a
procedure for the group $C_{N}$ below. The case of $C_{2}\times C_{2}$ will be
dealt with in Section 6.2. As we will see later, it will be enough to consider
these two cases in the construction of normal forms.

Take an element $\gamma\in\Gamma$ of order $N$ and let
$\delta:\Gamma\rightarrow SL_{2}({\mathbb{C}})$ and
$\sigma:\Gamma\rightarrow\mathrm{Aut}(T)$ be monomorphisms. We call a vector
$v=(v_{1},v_{2})^{T}$ consisting of functions
$v_{j}\in\mathcal{O}_{\mathbb{T}}$ an _equivariant vector_ with respect to the
actions defined by $\delta$ and $\sigma$, if

$v(\sigma(\gamma)z)=\delta(\gamma)v(z),\quad\text{for
all}\,\,\gamma\in\Gamma,\,\,z\in\mathbb{T}.$

The columns of $\Phi$ consist of equivariant vectors $p:=(p_{1},p_{2})^{T}$
and $q:=(q_{1},q_{2})^{T}$ as in (12). We will now fix a choice of $\delta$,
without loss of generality as we will see later. Let
$\delta=\delta_{j}:\Gamma\rightarrow SL_{2}({\mathbb{C}})$ be the monomorphism
defined by $\delta_{j}(\gamma)=\mathrm{diag}(\omega_{N}^{j},\omega_{N}^{-j})$
for some $j$ coprime to $N$. The condition
$\Phi(\sigma(\gamma)z)=\delta_{j}(\gamma)\Phi(z)$ implies that
$p_{1},q_{1}\in\mathcal{O}_{\mathbb{T}}^{\chi_{N-j}}$, and
$p_{2},q_{2}\in\mathcal{O}_{\mathbb{T}}^{\chi_{j}}$.

We will now present a construction for the matrix $\Phi$ as defined in (12),
that will be used in the construction of normal forms of both $C_{N}$ and
$D_{N}$.

Let $N\geq 3$ and fix some integer $j\not\equiv 0\mod N/2$. Take
$\lambda,\mu\in{\mathbb{C}}$ such that
$P_{2j}P_{-j}^{2}-P_{-2j}P_{j}^{2}=\lambda P_{-j}P_{j}+\mu.$ Lemma 5.7
guarantees that $\mu$ is nonzero. Let the columns of $\Phi$ be given by $p$
and $q$. In a sense, the simplest $C_{N}$-equivariant vector we can choose for
$p$ is $(P_{-j},P_{j})^{T}$. Given this choice for $p$, for $q$ we need to
find $q_{1}\in\mathcal{O}_{\mathbb{T}}^{\chi_{N-j}}$ and
$q_{2}\in\mathcal{O}_{\mathbb{T}}^{\chi_{j}}$ such that
$\det(\Phi)=P_{-j}q_{2}-P_{j}q_{1}=1$. Let $s$ be given by $s(z)=-z$ and
recall that $s\cdot P_{j}=-P_{-j}$ (cf. Lemma 5.4). One finds

$p(s\cdot z)=-\begin{pmatrix}0&1\\\ 1&0\end{pmatrix}p(z).$

Let $S=\begin{pmatrix}0&1\\\ 1&0\end{pmatrix}$. In light of the equivariance
condition (11), we will impose the condition $q(s\cdot z)=Sq(z),$ from which
we get

$\displaystyle\Phi(s\cdot z)=S\Phi(z)\begin{pmatrix}-1&0\\\ 0&1\end{pmatrix}.$
(14)

Adding this condition to the equation $P_{-j}q_{2}-P_{j}q_{1}=1$, yields a
unique solution in the space $L(2D)\subset\mathcal{O}_{\mathbb{T}}$, where $D$
is the divisor $D=\Gamma\cdot(0)=\sum_{j=0}^{N-1}(j\alpha)$. The solution to
this equation in $L(2D)$ is provided by Lemma 5.7. By the proof of Proposition
5.6, we deduce that for each integer $j\not\equiv 0\mod N/2$, we have the
vector space decomposition
$L(2D)^{\chi_{j}}={\mathbb{C}}P_{j}\oplus{\mathbb{C}}P_{-j}P_{2j}$. Then, if
we choose

$q_{1}=\frac{1}{\mu}P_{j}P_{-2j}+\frac{\lambda}{2\mu}P_{-j}\in\mathcal{O}_{\mathbb{T}}^{\chi_{N-j}},\quad
q_{2}=\frac{1}{\mu}P_{-j}P_{2j}-\frac{\lambda}{2\mu}P_{j}\in\mathcal{O}_{\mathbb{T}}^{\chi_{j}},$

we satisfy the requirement by Lemma 5.7, and this solution is unique by
considering the vector space decompositions of $L(2D)^{\chi_{j}}$ and
$L(2D)^{\chi_{N-j}}$. Thus the matrix $\Phi$ with the $p_{j}$ and $q_{j}$
defined above, is the unique matrix with lowest order poles satisfying the
intertwining condition.

We can summarise this as follows: given a complex torus $T$, define a faithful
homomorphism $\sigma:C_{N}\rightarrow t(T)$. Set ${\mathbb{T}}=T\setminus
C_{N}\cdot\\{0\\}$ and consider the functions $P_{j}$ defined with respect to
the choice of $\sigma$. For every integer $j\not\equiv 0\mod N/2$, we
associate to $\delta_{j}$ and $\sigma$ a meromorphic, $C_{N}$-equivariant map
$\Phi_{j}:T\rightarrow SL_{2}({\mathbb{C}})$, holomorphic on ${\mathbb{T}}$,
given by

$\displaystyle\Phi_{j}(z)=\begin{pmatrix}P_{-j}(z)&\frac{1}{\mu}P_{j}(z)P_{-2j}(z)+\frac{\lambda}{2\mu}P_{-j}(z)\\\
P_{j}(z)&\frac{1}{\mu}P_{-j}(z)P_{2j}(z)-\frac{\lambda}{2\mu}P_{j}(z)\end{pmatrix},$
(15)

where $\lambda$ and $\mu$ are constants defined in Lemma 5.7. We will record
our discussion below in two lemmata.

######  Lemma 6.5.

For any integer $j\not\equiv 0\mod N/2$ it holds that $\det(\Phi_{j})=1$.

######  Lemma 6.6.

Let $\sigma:C_{N}\rightarrow\mathrm{Aut}(T)$ be a monomorphism and
$\delta_{j,N}:C_{N}\rightarrow SL_{2}({\mathbb{C}})$ be defined by
$\delta_{j,N}(r)=\mathrm{diag}(\omega_{N}^{j},\omega_{N}^{-j})$. For all
integers $j\not\equiv 0\mod N/2$, $z\in\mathbb{T}$ and $r\in C_{N}$, we have

$\displaystyle\Phi_{j}(\sigma(r)z)=\delta_{j,N}(r)\Phi_{j}(z).$ (16)

Define $\rho_{j,N}=\mathrm{Ad}\circ\delta_{j,N}$. Taking the adjoint of both
sides of $\Phi_{j}(\sigma(r)z)=\delta_{j,N}(r)\Phi_{j}(z)$, we obtain
$\mathrm{Ad}(\Phi_{j}(\sigma(r)z))=\rho_{j,N}(r)\mathrm{Ad}(\Phi_{j}(z))$.
Notice that $\ker\rho_{j,N}\cong C_{2}$ if $N$ is even and $\gcd(j,N)=1$.

We have that for any integer $j\not\equiv 0\mod N/2$,
$\mathrm{Ad}(\Phi_{j})\in\mathrm{Aut}_{\mathcal{O}_{\mathbb{T}}}(\mathfrak{sl}_{2}\otimes_{{\mathbb{C}}}\mathcal{O}_{\mathbb{T}})$
by Lemma 6.5. We can also consider $\mathrm{Ad}(\Phi_{j})$ to be a meromorphic
map from $T$ to $\mathrm{Aut}(\mathfrak{sl}_{2})$,
$z\mapsto\mathrm{Ad}(\Phi_{j}(z))$ holomorphic on ${\mathbb{T}}$. The map
$\mathrm{Ad}(\Phi_{j})$ intertwines the actions defined by $\sigma$ and
$\rho$, or stated differently, it is equivariant with respect the action of
$C_{N}$.

######  Remark 6.7.

The matrix $\Phi(z):=\Phi_{j}(z)$ is constructed via a particular choice of
$\rho$. If $\rho^{\prime}$ is an equivalent representation of $C_{N}$,
$\rho^{\prime}(r)=B\rho(r)B^{-1}$ for all $r\in C_{N}$ for some
$B\in\mathrm{Aut}(\mathfrak{sl}_{2})$, and we define $\Phi^{\prime}=B\Phi
B^{-1}$, then one can verify that
$\rho^{\prime}(r)\mathrm{Ad}(\Phi^{\prime}(z))=\mathrm{Ad}(\Phi^{\prime}(\sigma(r)z))$.

The next two theorems describe the aLias with a cyclic symmetry group. We
split the results according to the genus $g$ of $T/\sigma(C_{N})$. First, we
consider $g(T/\sigma(C_{N}))=1$.

######  Theorem 6.8 (Genus 1 case).

Let $\rho:C_{N}\rightarrow\mathrm{Aut}(\mathfrak{sl}_{2})$ and
$\sigma:C_{N}\rightarrow t(T)$ be monomorphisms. Then

$(\mathfrak{sl}_{2}\otimes_{{\mathbb{C}}}\mathcal{O}_{\mathbb{T}})^{\rho\otimes\tilde{\sigma}(C_{N})}\cong\mathfrak{sl}_{2}\otimes_{{\mathbb{C}}}\mathcal{O}_{\mathbb{T}}^{\tilde{\sigma}(C_{N})}.$

A normal form is given by

$(\mathfrak{sl}_{2}\otimes_{{\mathbb{C}}}\mathcal{O}_{\mathbb{T}})^{\rho\otimes\tilde{\sigma}(C_{N})}={\mathbb{C}}\langle
E,F,H\rangle\otimes_{{\mathbb{C}}}{\mathbb{C}}[\wp_{{\Lambda}},\wp_{{\Lambda}}^{\prime}],$

where ${\Lambda}$ is a suitable lattice, and with brackets

$[H,E]=2E,\quad[H,F]=-2F,\quad[E,F]=H,$

where $E,F$ and $H$ are the images under $\mathrm{Ad}(\Phi_{j})$, for some
integer $j\not\equiv 0\mod N/2$, of $e\otimes 1$, $f\otimes 1$ and $h\otimes
1$, respectively.

###### Proof.

The main ingredients are Proposition 6.4 and Lemma 6.6. Fix a lattice
${\Lambda}={\mathbb{Z}}\oplus{\mathbb{Z}}\tau$ and let
$T={\mathbb{C}}/{\Lambda}$. Let $\alpha^{\prime}\in{\mathbb{C}}$ be such that
the projection of $\alpha^{\prime}$ under
$\pi_{{\Lambda}}:{\mathbb{C}}\rightarrow{\mathbb{C}}/{\Lambda}$,
$\alpha:=\pi(\alpha^{\prime})$ is an $N$-torsion point of $T$. Thus $\alpha$
satisfies $N\alpha=0\mod{\Lambda}$. We may assume that $\sigma$ is given by
$\sigma_{\alpha}(r)z=z+\alpha$.

First, let $N$ be odd. We may also assume that
$\delta(r)=\delta_{j,N}(r):=\mathrm{diag}(\omega_{N}^{j},\omega_{N}^{-j})$ for
some $j$ with $\gcd(j,N)\neq 1$ and that $\rho$ is given by
$\mathrm{Ad}\circ\delta$. The explanation is that if $\delta^{\prime}$ is
another monomorphism, and $\rho^{\prime}=\mathrm{Ad}\circ\delta^{\prime}$,
then we know by Lemma 4.1 that $\rho(C_{N})$ and $\rho^{\prime}(C_{N})$ are
conjugated subgroups. Lemma 3.3 then tells us that the aLias defined by $\rho$
and $\rho^{\prime}$ are isomorphic.

The algebra of invariants
$\mathcal{O}_{\mathbb{T}}^{\tilde{\sigma}_{\alpha}(C_{N})}$ is given by
${\mathbb{C}}[\wp_{{\Lambda}_{(\alpha)}},\wp_{{\Lambda}_{(\alpha)}}^{\prime}]$
by Lemma 5.3. Now, $\mathrm{Ad}(\Phi_{j})$ satisfies (11) because of Lemma 6.5
and Lemma 6.6, for which reason we can apply Proposition 6.4. We therefore
have, as in the proof of Proposition 6.4, that the following elements form a
basis of
$\mathfrak{A}=(\mathfrak{sl}_{2}\otimes_{{\mathbb{C}}}\mathcal{O}_{\mathbb{T}})^{C_{N}}$
over
${\mathbb{C}}[\wp_{\Lambda_{(\alpha)}},\wp_{\Lambda_{(\alpha)}}^{\prime}]$:

$E_{j}=\mathrm{Ad}(\Phi_{j})e\otimes 1,\quad
F_{j}=\mathrm{Ad}(\Phi_{j})f\otimes 1,\quad
H_{j}=\mathrm{Ad}(\Phi_{j})h\otimes 1,$

where $\Phi_{j}$ is given in (15). Remark 6.7 can be used to find a normal
form if instead of $\delta$, we had chosen $\delta^{\prime}$ as defined in the
beginning of the proof.

Assume now $N$ is even. We may assume that $\rho$ is given by

$\rho(r)\begin{pmatrix}a&b\\\
c&-a\end{pmatrix}=\begin{pmatrix}a&\omega_{N}b\\\
\omega_{N}^{-1}c&-a\end{pmatrix}.$

Observe that the image of $\rho$ and
$\hat{\rho}=\mathrm{Ad}\circ\delta_{j,2N}$ coincide. Notice that
$\ker\hat{\rho}=\langle r^{N}\rangle\cong C_{2}$, where $r\in C_{2N}$. Define

$\hat{{\Lambda}}=\begin{cases}{\mathbb{Z}}2\oplus{\mathbb{Z}}\tau&\,\,\text{if}\,\,N\alpha/2\in\\{1/2\mod{\Lambda},(1+\tau)/2\mod{\Lambda}\\},\\\
{\mathbb{Z}}\oplus{\mathbb{Z}}2\tau&\,\,\text{if}\,\,N\alpha/2=\tau/2\mod{\Lambda}.\end{cases}$

Let $\hat{T}={\mathbb{C}}/\hat{{\Lambda}}$ and define $\hat{{\mathbb{T}}}$
accordingly. Define $\sigma_{1}:C_{2N}\rightarrow\mathrm{Aut}(\hat{T})$ by
$\sigma_{1}(r)z=z+\pi_{\hat{{\Lambda}}}(\alpha^{\prime})$ and notice that
$\pi_{\hat{{\Lambda}}}(N\alpha^{\prime})=N\alpha$ is a half lattice point of
$\hat{{\Lambda}}$.

Recall that in the proof of Lemma 5.3 we used that for any $\alpha\in T$,
$\mathcal{O}^{\tilde{\sigma}_{\alpha}(C_{N})}_{\mathbb{T}}$ is the space of
$\Lambda_{(\alpha)}$-periodic meromorphic functions which are holomorphic on
$T\setminus\Lambda_{(\alpha)}$. This allowed us to say that
$\mathcal{O}_{\mathbb{T}}^{\Gamma}=\mathcal{O}_{{\mathbb{T}}/\Gamma}$. We
adopt this perspective again. Suppose $N\alpha/2=1/2\mod{\Lambda}$. Then
$\hat{T}/\sigma_{1}(C_{2N})={\mathbb{C}}/({\mathbb{Z}}2+{\mathbb{Z}}\tau+{\mathbb{Z}}\alpha^{\prime})$.
Notice that $N\alpha^{\prime}=1+2n_{1}+2n_{2}\tau$ for some
$n_{1},n_{2}\in{\mathbb{Z}}$. We see that
$1\in{\mathbb{Z}}2+{\mathbb{Z}}\tau+{\mathbb{Z}}\alpha^{\prime}$ and hence

$\hat{T}/\sigma_{1}(C_{2N})={\mathbb{C}}/({\mathbb{Z}}2+{\mathbb{Z}}\tau+{\mathbb{Z}}\alpha^{\prime})={\mathbb{C}}/({\mathbb{Z}}+{\mathbb{Z}}\tau+{\mathbb{Z}}\alpha^{\prime}).$

Also observe that
$T/\sigma(C_{N})={\mathbb{C}}/({\mathbb{Z}}+{\mathbb{Z}}\tau+{\mathbb{Z}}\alpha^{\prime})$.
Therefore $\hat{T}/\sigma_{1}(C_{2N})=T/\sigma(C_{N})$. Similarly we have
$\hat{{\mathbb{T}}}/\langle r^{N}\rangle={\mathbb{T}}$. The same claim holds
in the cases where $N\alpha/2\in\\{\tau/2,(1+\tau)/2\\}$.

Define $\Phi:=\Phi_{j}$ associated to $\delta_{j,2N}$ and $\sigma_{1}$, for
some integer $j\not\equiv N/2$. It is clear that
$\mathrm{Ad}(\Phi)\in\mathrm{Aut}_{\mathcal{O}_{\hat{{\mathbb{T}}}}}(\mathfrak{sl}_{2}\otimes_{{\mathbb{C}}}\mathcal{O}_{\hat{{\mathbb{T}}}})$.
By similar reasoning as in Proposition 6.4, we have that $\mathrm{Ad}(\Phi)$
establishes an isomorphism

$\mathfrak{B}:=(\mathfrak{sl}_{2}\otimes_{{\mathbb{C}}}\mathcal{O}_{\hat{{\mathbb{T}}}})^{\hat{\rho}\otimes\tilde{\sigma}_{1}(C_{2N})}\cong\mathfrak{sl}_{2}\otimes_{{\mathbb{C}}}\mathcal{O}_{\hat{{\mathbb{T}}}}^{\tilde{\sigma}_{1}(C_{2N})}.$

We know $\hat{T}/\sigma_{1}(C_{2N})=T/\sigma(C_{N})$ and therefore
$\mathcal{O}_{\hat{{\mathbb{T}}}}^{\tilde{\sigma}_{1}(C_{2N})}=\mathcal{O}_{{\mathbb{T}}}^{\tilde{\sigma}(C_{N})}.$
Hence
$\mathfrak{B}\cong\mathfrak{sl}_{2}\otimes_{{\mathbb{C}}}\mathcal{O}_{{\mathbb{T}}}^{\tilde{\sigma}(C_{N})}.$
We claim that $\mathfrak{B}$ is precisely
$(\mathfrak{sl}_{2}\otimes_{{\mathbb{C}}}\mathcal{O}_{\mathbb{T}})^{\rho\otimes\tilde{\sigma}(C_{N})}.$
By Lemma 3.2 we have that
$\mathfrak{B}=(\mathfrak{sl}_{2}\otimes_{{\mathbb{C}}}\mathcal{O}_{\hat{\mathbb{T}}/\langle
r^{N}\rangle})^{\hat{\Gamma}}$, where $\hat{\Gamma}=\langle r\rangle/\langle
r^{N}\rangle\cong C_{N}$. Now, since $\hat{{\mathbb{T}}}/\langle
r^{N}\rangle={\mathbb{T}}$ and the action of $\hat{\Gamma}$ is the same as the
one induced from $\sigma$, we conclude that
$\mathfrak{B}=(\mathfrak{sl}_{2}\otimes_{{\mathbb{C}}}\mathcal{O}_{\mathbb{T}})^{\rho\otimes\tilde{\sigma}(C_{N})}.$

We know $\mathrm{Ad}(\Phi(r^{N}\cdot z))=\mathrm{Ad}(\Phi(z))$, since
$r^{N}\in\ker(\hat{\rho})$. Therefore, any invariant element
$X:=\mathrm{Ad}(\Phi)x$, with $x\in\mathfrak{sl}_{2}$, consists of entries
which are invariant under $\langle r^{N}\rangle\cong C_{2}$. The remainder of
the proof for $N$ is even, follows as it did for $N$ odd. A normal form is
given by

$E=\mathrm{Ad}(\Phi)e\otimes 1,\quad F=\mathrm{Ad}(\Phi)f\otimes 1,\quad
H=\mathrm{Ad}(\Phi)h\otimes 1.$

This proves the claim. ∎

Explicitly, the generators $H_{j}:=\mathrm{Ad}(\Phi_{j})h\otimes 1$,
$E_{j}:=\mathrm{Ad}(\Phi_{j})e\otimes 1$ and
$F_{j}:=\mathrm{Ad}(\Phi_{j})f\otimes 1$ ($j\not\equiv 0\mod N/2$) are given
by

$\displaystyle
H_{j}=\frac{1}{\mu}\begin{pmatrix}P_{-j}^{2}P_{2j}+P_{-2j}P_{j}^{2}&-2P_{-j}P_{j}P_{-2j}-\lambda
P_{-j}^{2}\\\ 2P_{-j}P_{j}P_{2j}-\lambda
P_{j}^{2}&-P_{-j}^{2}P_{2j}-P_{-2j}P_{j}^{2}\end{pmatrix},\quad
E_{j}=\begin{pmatrix}-P_{-j}P_{j}&P_{-j}^{2}\\\
-P_{j}^{2}&P_{-j}P_{j}\end{pmatrix},$ (17)

and

$\displaystyle
F_{j}=\frac{1}{4\mu^{2}}\begin{pmatrix}4P_{-j}P_{j}P_{-2j}P_{2j}+\lambda^{2}P_{-j}P_{j}+2\lambda\mu&-4P_{j}^{2}P_{-2j}^{2}-4\lambda
P_{-j}P_{j}P_{-2j}-\lambda^{2}P_{-j}^{2}\\\ 4P_{-j}^{2}P_{2j}^{2}-4\lambda
P_{-j}P_{j}P_{2j}+\lambda^{2}P_{j}^{2}&-4P_{j}P_{-j}P_{2j}P_{-2j}-\lambda^{2}P_{-j}P_{j}-2\lambda\mu\end{pmatrix}.$
(18)

In the next corollary, we shall omit the $j$-dependence of $E,F,H$ and assume
this implicitly.

######  Corollary 6.9.

Let $\rho:D_{N}\rightarrow\mathrm{Aut}(\mathfrak{sl}_{2})$ and
$\sigma:D_{N}\rightarrow\mathrm{Aut}(T)$ be monomorphisms. Then

$(\mathfrak{sl}_{2}\otimes_{{\mathbb{C}}}\mathcal{O}_{\mathbb{T}})^{\rho\otimes\tilde{\sigma}(D_{N})}\cong\mathfrak{S}_{\tau},$

for some $\tau\in\mathbb{H}$. A normal form is given by

$(\mathfrak{sl}_{2}\otimes_{{\mathbb{C}}}\mathcal{O}_{\mathbb{T}})^{\rho\otimes\tilde{\sigma}(D_{N})}={\mathbb{C}}\langle\tilde{E},\tilde{F},\tilde{H}\rangle\otimes_{{\mathbb{C}}}{\mathbb{C}}[\wp_{\Lambda}],$

where ${\Lambda}$ is a suitable lattice and
$\tilde{E}=E\otimes\wp_{\Lambda}^{\prime}$,
$\tilde{F}=F\otimes\wp_{\Lambda}^{\prime}$ and $\tilde{H}=H$, where $E,F,H$
are defined in the proof of Theorem 6.8. The Lie structure is given by

$\displaystyle[\tilde{H},\tilde{E}]=2\tilde{E},\quad[\tilde{H},\tilde{F}]=-2\tilde{F},\quad[\tilde{E},\tilde{F}]=\tilde{H}\otimes(4\wp_{\Lambda}^{3}-g_{2}\wp_{\Lambda}-g_{3}).$
(19)

###### Proof.

We may assume that $\sigma$ is given by $\sigma(s)z=-z$ and
$\sigma(r)z=z+\alpha$ for some $N$-torsion point $\alpha\in T$. Let
$S=\begin{pmatrix}0&1\\\ 1&0\end{pmatrix}$. We will use Proposition 6.4.
Recall $H=\mathrm{Ad}(\Phi)h,$ where $\Phi=\Phi_{j}$ for some $j\not\equiv
0\mod N/2$. To satisfy condition (11), we only need to verify condition (14):

$\Phi(s\cdot z)=S\Phi(z)\begin{pmatrix}-1&0\\\ 0&1\end{pmatrix},$

where $s$ generates $C_{2}\subset\mathrm{Aut}_{0}(T)$. This holds as a result
of the construction of $\Phi$ in the beginning of this section. Thus $H$ is a
$D_{N}$-invariant as well. Hence we can apply Lemma 6.3 and use Example 2.4 to
obtain the stated normal form. ∎

Notice that above corollary also covers the aLia
$(\mathfrak{sl}_{2}\otimes_{{\mathbb{C}}}\mathcal{O}_{\mathbb{T}})^{C_{2}}$,
where $g(T/C_{2})=0$, since $D_{1}=C_{2}\ltimes 1\cong C_{2}.$

So far, we have considered the groups $C_{N}$ and $D_{N}\cong C_{2}\ltimes
C_{N}$ for which $C_{N}$ is embedded as translations in $\mathrm{Aut}(T)$.
There are a number of cyclic groups of small order which also allow an
embedding as a rotation in $\mathrm{Aut}(T)$ for a torus $T$ with more
symmetry than the generic case, cf. Proposition 4.4. The groups $C_{\ell}$ for
$\ell\in\\{3,4,6\\}$ can be embedded inside $\mathrm{Aut}(T)$ as rotations,
for tori either isomorphic to $T_{i}$ or $T_{\omega_{6}}$. The following
theorem describes the aLias with these symmetry groups.

######  Theorem 6.10 (Genus 0 case).

Let $\rho:C_{\ell}\rightarrow\mathrm{Aut}(\mathfrak{sl}_{2})$ and
$\sigma:C_{\ell}\rightarrow\mathrm{Aut}(T)$ be monomorphisms and assume that
$g(T/\sigma(C_{\ell}))=0$. Then

$(\mathfrak{sl}_{2}\otimes_{{\mathbb{C}}}\mathcal{O}_{\mathbb{T}})^{\rho\otimes\tilde{\sigma}(C_{\ell})}\cong\mathfrak{O},$

if and only if $\ell\in\\{3,4,6\\}.$

###### Proof.

By Lemma 3.3, we only have to prove the statement for a particular choice of
monomorphisms. Let $\ell=3$ and define the homomorphism
$\rho:C_{3}\rightarrow\mathrm{Aut}(\mathfrak{sl}_{2})$ by

$\rho(s)\begin{pmatrix}a&b\\\
c&-a\end{pmatrix}=\begin{pmatrix}a&\omega_{3}b\\\
\omega_{3}^{-1}c&-a\end{pmatrix}.$

Let $s\in C_{3}$ act on $T\cong T_{\omega_{6}}$ as $s\cdot z=\omega_{3}z$.
Recall that $s\cdot\wp(z)=\wp(\omega_{3}^{-1}z)=\omega_{3}^{2}\wp(z)$ and
$s\cdot\wp^{\prime}=\wp^{\prime}$ by Equations (6) and (7). By Proposition 5.2
we have

$\mathcal{O}_{\mathbb{T}}^{\tilde{\sigma}(C_{3})}=\mathcal{O}_{\mathbb{T}}^{\chi_{0}}={\mathbb{C}}[\wp^{\prime}],\quad\mathcal{O}_{\mathbb{T}}^{\chi_{1}}={\mathbb{C}}[\wp^{\prime}]\wp^{2},\quad\mathcal{O}_{\mathbb{T}}^{\chi_{2}}={\mathbb{C}}[\wp^{\prime}]\wp.$

Define the matrices

$H=\begin{pmatrix}1&0\\\ 0&-1\end{pmatrix},\quad E=\begin{pmatrix}0&1\\\
0&0\end{pmatrix}\otimes\wp,\quad F=\begin{pmatrix}0&0\\\
1&0\end{pmatrix}\otimes\wp^{2},$

which are easily seen to be invariant. By decomposing the action of $C_{3}$ on
both factors of the tensor product, we get
$(\mathfrak{sl}_{2}\otimes_{{\mathbb{C}}}\mathcal{O}_{\mathbb{T}})^{\rho\otimes\tilde{\sigma}(C_{3})}=\bigoplus_{j=0}^{2}\mathfrak{sl}_{2}^{\chi_{j}}\otimes_{{\mathbb{C}}}\mathcal{O}_{\mathbb{T}}^{\overline{\chi_{j}}}$,
where $\overline{\chi_{j}}(g)=\overline{\chi_{j}(g)}$, and we obtain

$\mathfrak{A}:=(\mathfrak{sl}_{2}\otimes_{{\mathbb{C}}}\mathcal{O}_{\mathbb{T}})^{\rho\otimes\tilde{\sigma}(C_{3})}={\mathbb{C}}\langle
E,F,H\rangle\otimes_{{\mathbb{C}}}\mathcal{O}_{\mathbb{T}}^{\tilde{\sigma}(C_{3})}={\mathbb{C}}\langle
E,F,H\rangle\otimes_{{\mathbb{C}}}{\mathbb{C}}[\wp^{\prime}]$

with brackets

$[H,E]=2E,\quad[H,F]=-2F,\quad[E,F]=H\otimes\wp^{3}.$

We will show that after performing a sequence of scalings and translations,
$[E,F]$ can be written as $[E,F]=H\otimes\wp^{\prime}(\wp^{\prime}-1)$, after
which we will recognise Onsager’s algebra $\mathfrak{O}$ in it.

By Lemma 5.2, a generator $J$ of
$\mathcal{O}_{\mathbb{T}}^{\tilde{\sigma}(C_{3})}$ can be chosen to be
$\wp^{\prime}$. The last bracket can be rewritten as

$\displaystyle[E,F]=H\otimes\frac{1}{4}((\wp^{\prime})^{2}+g_{3})=H\otimes\frac{1}{4}(J^{2}+g_{3}).$

Notice that $g_{3}\neq 0$ since $g_{2}=0$ and they cannot be both zero. Let
$E^{\prime}=-\frac{4}{g_{3}}E$. Then
$[E^{\prime},F]=H\otimes(-\frac{1}{g_{3}}J^{2}-1).$ Replace
$-\frac{1}{g_{3}}J^{2}$ by $J^{2}$ to obtain
$[E^{\prime},F]=H\otimes(J^{2}-1)$. Replace $J-1$ by $J$ to get
$[E^{\prime},F]=H\otimes J(J+2)$ and transform further to obtain
$[E^{\prime},F]=H\otimes J(J-1)$ to see that $\mathfrak{A}$ is isomorphic to
the Onsager algebra, cf. [17, Theorem 2.5].

Another possibility for embedding $C_{3}$, is given by
$\sigma^{\prime}(r)(z)=\omega_{3}^{-1}z.$ The $\chi_{1}$ and $\chi_{2}$
isotypical components interchange when we consider this action. Define the
elements

$\tilde{H}=-\begin{pmatrix}1&0\\\
0&-1\end{pmatrix},\quad\tilde{E}=-\begin{pmatrix}0&1\\\
0&0\end{pmatrix}\otimes\wp^{2},\quad\tilde{F}=-\begin{pmatrix}0&0\\\
1&0\end{pmatrix}\otimes\wp.$

Observe how $\mathrm{Ad}\begin{pmatrix}0&1\\\ -1&0\end{pmatrix}$ maps $H,E,F$
to $\tilde{H},\tilde{F},\tilde{E}$, respectively. By Lemma 3.3, we see that

$\displaystyle(\mathfrak{sl}_{2}\otimes_{{\mathbb{C}}}\mathcal{O}_{\mathbb{T}})^{\rho\otimes\tilde{\sigma}^{\prime}(C_{3})}\cong(\mathfrak{sl}_{2}\otimes_{{\mathbb{C}}}\mathcal{O}_{\mathbb{T}})^{\rho\otimes\tilde{\sigma}(C_{3})}.$

The cases of $\ell=4$ and $\ell=6$ follow in an analogous manner. By Example
2.4,
$(\mathfrak{sl}_{2}\otimes_{{\mathbb{C}}}\mathcal{O}_{\mathbb{T}})^{C_{2}}\not\cong\mathfrak{O}$.
This proves the claim. ∎

The proof of Theorem 6.10 shows that a normal form of
$(\mathfrak{sl}_{2}\otimes_{{\mathbb{C}}}\mathcal{O}_{\mathbb{T}})^{\rho\otimes\tilde{\sigma}(C_{3})}$
is given by ${\mathbb{C}}\langle
E_{3},F_{3},H\rangle\otimes_{{\mathbb{C}}}{\mathbb{C}}[\wp^{\prime}]$, where
$H=h$, $E_{3}=e\otimes\wp$, $F_{3}=f\otimes\wp^{2},$ and
$\wp=\wp_{{\Lambda}_{\omega_{6}}}$. In a similar fashion, one can obtain
normal forms for remaining aLias with $\Gamma=C_{4}$ and $\Gamma=C_{6}$ using
Proposition 5.2. For $\ell\in\\{4,6\\}$, choose the homomorphism
$\rho_{\ell}:C_{\ell}\rightarrow\mathrm{Aut}(T_{\omega_{\ell}})$ defined by

$\rho_{\ell}(s)\begin{pmatrix}a&b\\\
c&-a\end{pmatrix}=\begin{pmatrix}a&\omega_{\ell}b\\\
\omega_{\ell}^{-1}c&-a\end{pmatrix}$

and $\sigma_{\ell}:C_{\ell}\rightarrow\mathrm{Aut}(T_{\omega_{\ell}})$ defined
by $\sigma(s)z=\omega_{\ell}z$. Then

$\displaystyle(\mathfrak{sl}_{2}\otimes_{{\mathbb{C}}}\mathcal{O}_{\mathbb{T}})^{\rho_{4}\otimes\tilde{\sigma}_{4}(C_{4})}={\mathbb{C}}\langle
E_{4},F_{4},H\rangle\otimes_{{\mathbb{C}}}{\mathbb{C}}[\wp^{2}]\cong\mathfrak{O},$
(20)

where $E_{4}=e\otimes\wp\wp^{\prime}$, $F_{4}=f\otimes\wp^{\prime}$, and
$\wp=\wp_{{\Lambda}_{i}}$.

Finally, for $C_{6}$ we have the following normal form:

$\displaystyle(\mathfrak{sl}_{2}\otimes_{{\mathbb{C}}}\mathcal{O}_{\mathbb{T}})^{\rho_{6}\otimes\tilde{\sigma}_{6}(C_{6})}={\mathbb{C}}\langle
E_{6},F_{6},H\rangle\otimes_{{\mathbb{C}}}{\mathbb{C}}[\wp^{3}]\cong\mathfrak{O},$
(21)

where $E_{6}=e\otimes\wp\wp^{\prime}$, $F_{6}=f\otimes\wp^{2}\wp^{\prime}$,
and $\wp=\wp_{{\Lambda}_{\omega_{6}}}$.

###  6.2 The cases $\Gamma=C_{2}\times C_{2}$ and $\Gamma=A_{4}$

We will now consider the final two aLias in our classification, namely those
with symmetry group $C_{2}\times C_{2}$ and a homomorphism $\sigma:C_{2}\times
C_{2}\rightarrow\mathrm{Aut}(T)$, such that $g(T/\sigma(C_{2}\times
C_{2}))=1$, where $T={\mathbb{C}}/{\mathbb{Z}}\oplus{\mathbb{Z}}\tau$, and
those with symmetry group $A_{4}$. We may assume that the homomorphism
$\sigma$ is given by $\sigma(r_{1})z=z+\frac{1}{2}$ and
$\sigma(r_{2})z=z+\frac{\tau}{2}$, where $r_{1},r_{2}$ generate $C_{2}\times
C_{2}$, cf. Lemma 4.4.

Let $\\{\alpha_{00},\alpha_{01},\alpha_{10},\alpha_{11}\\}$ denote the set of
characters of $C_{2}\times C_{2}$, defined by
$\alpha_{ij}(r_{1},r_{2})=\chi_{i}(r_{1})\chi_{j}(r_{2})$, where $\chi_{0}$,
$\chi_{1}$ are the characters of $C_{2}$. Let
$\mathbb{T}=T\setminus\mathcal{S}$, with $\mathcal{S}=\sigma(C_{2}\times
C_{2})\cdot\\{0\\}=\\{0,1/2,\tau/2,(1+\tau)/2\\}$ and write
$\wp=\wp_{{\Lambda}}$. The divisor of $1/\wp^{\prime}$ is given by

$\displaystyle(1/\wp^{\prime})=-(1/2)-(\tau/2)-((1+\tau)/2))+3(0),$ (22)

and therefore $1/\wp^{\prime}\in\mathcal{O}_{\mathbb{T}}$. Let
$\pi_{\alpha_{ij}}=\frac{1}{4}\sum_{r\in C_{2}\times
C_{2}}\overline{\alpha_{ij}(r)}r$ be the projection of
$\mathcal{O}_{\mathbb{T}}$ onto the isotypical component
$\mathcal{O}_{\mathbb{T}}^{\alpha_{ij}}$. Define
$p_{ij}:=4\pi_{\alpha_{ij}}(1/\wp^{\prime})$. Concretely, we have

$\displaystyle p_{2}:=p_{01}$
$\displaystyle=\frac{1}{\wp^{\prime}}+r_{1}\frac{1}{\wp^{\prime}}-r_{2}\frac{1}{\wp^{\prime}}-r_{1}r_{2}\frac{1}{\wp^{\prime}},$
(23) $\displaystyle p_{1}:=p_{10}$
$\displaystyle=\frac{1}{\wp^{\prime}}-r_{1}\frac{1}{\wp^{\prime}}+r_{2}\frac{1}{\wp^{\prime}}-r_{1}r_{2}\frac{1}{\wp^{\prime}},$
(24) $\displaystyle p_{0}:=p_{11}$
$\displaystyle=\frac{1}{\wp^{\prime}}-r_{1}\frac{1}{\wp^{\prime}}-r_{2}\frac{1}{\wp^{\prime}}+r_{1}r_{2}\frac{1}{\wp^{\prime}}.$
(25)

By construction, the functions $p_{ij}$ are elements of
$\mathcal{O}_{\mathbb{T}}^{\alpha_{ij}}$. Observe that they have at most order
1 poles in the set $\mathcal{S}$. Furthermore, they are odd functions since
$\wp^{\prime}$ is odd and $s$ (with $s(z)=-z$) and $r_{1},r_{2}$ commute. The
definition of the $p_{i}$ is motivated in Lemma 6.14.

######  Remark 6.11.

The $p_{j}$ can also be defined in terms of theta functions, as is done in
[3]. They define

$\lambda_{1}(z)=\frac{\vartheta_{00}(2z|\tau)}{\vartheta_{00}(0|\tau)\vartheta_{11}(2z|\tau)},\quad\lambda_{2}(z)=\frac{\vartheta_{10}(2z|\tau)}{\vartheta_{10}(0|\tau)\vartheta_{11}(2z|\tau)},\quad\lambda_{3}(z)=\frac{\vartheta_{01}(2z|\tau)}{\vartheta_{01}(0|\tau)\vartheta_{11}(2z|\tau)},$

where $\vartheta_{ab}(z|\tau)=\sum_{n\in{\mathbb{Z}}}\exp(2\pi
i[\tau(n+a/2)^{2}/2+(n+a/2)(z+b/2)])$, and where
$(a,b)=(0,0),(1,0),(0,1),(1,1)$. The relations between our projected
$1/\wp^{\prime}$ are, up to a scaling, as follows:

$p_{0}\leftrightarrow\lambda_{1},\quad p_{1}\leftrightarrow\lambda_{3},\quad
p_{2}\leftrightarrow\lambda_{2}.$

Written out explicitly, the group $C_{2}\times C_{2}$ acts on the $p_{i}$ as

$\displaystyle r_{1}\cdot p_{0}$ $\displaystyle=-p_{0},$ $\displaystyle
r_{2}\cdot p_{0}$ $\displaystyle=-p_{0},$ (26) $\displaystyle r_{1}\cdot
p_{1}$ $\displaystyle=-p_{1},$ $\displaystyle r_{2}\cdot p_{1}$
$\displaystyle=p_{1},$ (27) $\displaystyle r_{1}\cdot p_{2}$
$\displaystyle=p_{2},$ $\displaystyle r_{2}\cdot p_{2}$
$\displaystyle=-p_{2}.$ (28)

Let us obtain the series expansion of $\frac{1}{r\wp^{\prime}}$ about $z=0$,
for $r\in\langle r_{1},r_{2}\rangle\setminus 1$, which we will use to derive
the expansions of the $p_{i}^{2}$ about $z=0$ in Lemma 6.13. First of all, for
$r\in\langle r_{1},r_{2}\rangle\setminus 1$, we find for
$\frac{1}{r\wp^{\prime}}$ the following expansion about $z=0$:

$\displaystyle\frac{1}{r\wp^{\prime}(z)}=\frac{1}{6\wp(r\cdot
0)^{2}-\frac{1}{2}g_{2}}\frac{1}{z}-\frac{2\wp(r\cdot 0)}{6\wp(r\cdot
0)^{2}-\frac{1}{2}g_{2}}z+O(z^{2}).$ (29)

For example, for $r=r_{1}$, we can simplify this to

$\frac{1}{r_{1}\wp^{\prime}(z)}=\frac{1}{2(e_{1}-e_{3})(e_{1}-e_{2})}\frac{1}{z}-\frac{e_{1}}{(e_{1}-e_{3})(e_{1}-e_{2})}z+O(z^{2}),$

using that $6\wp(1/2)^{2}-g_{2}/2=2(e_{1}-e_{3})(e_{1}-e_{2})$. Notice that
the denominators of the coefficients are indeed nonzero, since for lattices
${\Lambda}$, the _discriminant_ $\Delta({\Lambda}):=g_{2}^{3}-27g_{3}^{2}$ is
nonzero and can be shown to be equal to
$16(e_{1}-e_{2})^{2}(e_{1}-e_{3})^{2}(e_{2}-e_{3})^{2}$.

Let us argue that the $p_{j}$ are not identically zero. To this end, consider
$p_{00}=4\pi_{\alpha_{00}}(1/\wp^{\prime})=\frac{1}{\wp^{\prime}}+r_{1}\frac{1}{\wp^{\prime}}+r_{2}\frac{1}{\wp^{\prime}}+r_{1}r_{2}\frac{1}{\wp^{\prime}}$.
This function is invariant under $C_{2}\times C_{2}$ and hence descends to a
function on $\tilde{T}:=T/\sigma(C_{2}\times C_{2})$. Now, $p_{00}$ has at
most order 1 poles and on $\tilde{T}$, there is therefore at most a single
pole. Thus $p_{00}$ must be constant. In fact, since $\wp^{\prime}$ is odd,
$p_{00}$ is odd as well. Hence $p_{00}=0.$ Knowing that $p_{00}$ is constant
allows us to draw conclusions about the $p_{i}$. By considering the divisor of
$\frac{1}{\wp^{\prime}}$ in (22), we see that the sets of poles of
$r_{1}\frac{1}{\wp^{\prime}},r_{2}\frac{1}{\wp^{\prime}}$ and
$r_{1}r_{2}\frac{1}{\wp^{\prime}}$ have pairwise precisely two poles in
common. We therefore see that the three functions

$\displaystyle
k_{1}:=r_{2}\frac{1}{\wp^{\prime}}+r_{1}r_{2}\frac{1}{\wp^{\prime}},\quad
k_{2}:=r_{1}\frac{1}{\wp^{\prime}}+r_{1}r_{2}\frac{1}{\wp^{\prime}},\quad
k_{3}:=r_{1}\frac{1}{\wp^{\prime}}+r_{2}\frac{1}{\wp^{\prime}}$

are all non-constant. As a consequence we get that
$p_{0}=p_{00}-2k_{3}=-2k_{3}$, $p_{1}=p_{00}-2k_{2}=-2k_{2}$ and
$p_{2}=p_{00}-2k_{1}=-2k_{1}$ are all nonzero.

The next lemma describes the algebra of $C_{2}\times C_{2}$-invariant subspace
of $\mathcal{O}_{\mathbb{T}}$.

######  Lemma 6.12.

We have $\mathcal{O}_{\mathbb{T}}^{\tilde{\sigma}(C_{2}\times
C_{2})}={\mathbb{C}}[\wp_{\frac{1}{2}{\Lambda}},\wp_{\frac{1}{2}{\Lambda}}^{\prime}]$.

###### Proof.

By definition, $\mathcal{O}_{\mathbb{T}}^{\tilde{\sigma}(C_{2}\times C_{2})}$
is the algebra of all $\frac{1}{2}{\Lambda}$-periodic meromorphic functions on
$T$ that are holomorphic on $T\setminus\frac{1}{2}{\Lambda}$, which
corresponds to
${\mathbb{C}}[\wp_{\frac{1}{2}{\Lambda}},\wp_{\frac{1}{2}{\Lambda}}^{\prime}]$
by Lemma 2.3. ∎

We now record the relations between the $p_{i}^{2}$ and
$\wp_{\frac{1}{2}{\Lambda}}$ in a lemma for later purposes.

######  Lemma 6.13.

We have

$\displaystyle p_{0}^{2}$
$\displaystyle=\frac{1}{(e_{1}-e_{3})^{2}(e_{2}-e_{3})^{2}}(\wp_{\frac{1}{2}{\Lambda}}-4e_{3})=\frac{16(e_{2}-e_{3})^{2}}{\Delta({\Lambda})}(\wp_{\frac{1}{2}{\Lambda}}-4e_{3}).$

Similarly, the relations of $p_{j}^{2}$ and $\wp_{\frac{1}{2}{\Lambda}}$ for
$j=1,2$ are obtained from $p_{0}^{2}$ by cyclic permutations of the $e_{i}$.
Consequently, there are the following relations:
$p_{1}^{2}=\alpha_{1}p_{2}^{2}+\alpha_{2}$,
$p_{0}^{2}=\beta_{1}p_{2}^{2}+\beta_{2}$, where

$\alpha_{1}=\left(\frac{e_{1}-e_{3}}{e_{2}-e_{3}}\right)^{2},\quad\alpha_{2}=\frac{4}{(e_{1}-e_{2})(e_{2}-e_{3})^{2}},$

and

$\beta_{1}=\left(\frac{e_{1}-e_{2}}{e_{2}-e_{3}}\right)^{2},\quad\beta_{2}=\frac{4}{(e_{1}-e_{3})(e_{2}-e_{3})^{2}}.$

Thus $\alpha_{1},\alpha_{2},\beta_{1},\beta_{2}\in{\mathbb{C}}^{*}$.

###### Proof.

Using the expansion in (29), we obtain for $r,\tilde{r}\in\langle
r_{1},r_{2}\rangle\setminus 1$:

$\displaystyle\left(\frac{1}{r\wp^{\prime}(z)}\right)\left(\frac{1}{\tilde{r}\wp^{\prime}(z)}\right)$
$\displaystyle=\frac{1}{(6\wp(r\cdot
0)^{2}-\frac{1}{2}g_{2})(6\wp(\tilde{r}\cdot
0)^{2}-\frac{1}{2}g_{2})}\frac{1}{z^{2}}-$
$\displaystyle\quad\frac{2\wp(r\cdot 0)+2\wp(\tilde{r}\cdot 0)}{(6\wp(r\cdot
0)^{2}-\frac{1}{2}g_{2})(6\wp(\tilde{r}\cdot
0)^{2}-\frac{1}{2}g_{2})}+O(z^{2}),$

where we use that $\wp^{\prime}$ is odd, and hence the above product is even.
Hence, for $r=r_{1}$ and $\tilde{r}=r_{2}$, we get

$\displaystyle p_{0}(z)^{2}=4k_{3}(z)^{2}$
$\displaystyle=4\left(\frac{1}{r_{1}\wp^{\prime}(z)}\right)^{2}+4\left(\frac{1}{r_{2}\wp^{\prime}(z)}\right)^{2}+8\left(\frac{1}{r_{1}\wp^{\prime}(z)}\right)\left(\frac{1}{r_{2}\wp^{\prime}(z)}\right)$
$\displaystyle=\frac{1}{(e_{1}-e_{3})^{2}(e_{2}-e_{3})^{2}}\frac{1}{z^{2}}-\frac{4e_{3}}{(e_{1}-e_{3})^{2}(e_{2}-e_{3})^{2}}+O(z^{2}).$

Similarly, one can show that $p_{j}^{2}$ $(j=1,2)$ is obtained from
$p_{0}^{2}$ by changing $e_{i}$ to $e_{i-j}$ in the expansions for $i=1,2,3$.

It is clear from the definition of $\wp$ as given in (3), that about $z=0$,
$\wp$ has the expansion $\wp(z)=\frac{1}{z^{2}}+O(z^{2})$. Since $p_{j}^{2}$
has order 2 poles and is invariant under $C_{2}\times C_{2}$, we know that
$p_{j}^{2}=c_{1}\wp_{\frac{1}{2}{\Lambda}}+c_{2}$ for some
$c_{1},c_{2}\in{\mathbb{C}}$. The last statement follows now directly. This
completes the proof. ∎

The symmetry groups for $\mathfrak{sl}_{2}$-based aLias having $C_{2}\times
C_{2}$ as a subgroup of translations, are limited to $A_{4}\cong
C_{3}\ltimes(C_{2}\times C_{2})$ (for hexagonal lattices) and $C_{2}\times
C_{2}$ itself. We are therefore interested in how $C_{3}$ acts on functions in
$\mathcal{O}_{\mathbb{T}}^{C_{2}\times C_{2}}$. This is the content of the
following lemma.

######  Lemma 6.14.

Suppose ${\Lambda}$ is homothetic to ${\Lambda}_{\omega_{6}}$ and $s$ acts on
${\mathbb{C}}/{\Lambda}$ as $s\cdot z=\omega_{3}z$. Then

$s\cdot p_{j}=p_{j+1},$

where the subscripts are taken modulo $3$. Furthermore,
$\alpha_{1}=\omega_{3},\beta_{1}=\omega_{3}^{2}$ if and only if ${\Lambda}$ is
homothetic to ${\Lambda}_{\omega_{6}}$.

###### Proof.

Assume that ${\Lambda}$ is homothetic to ${\Lambda}_{\omega_{6}}.$ Recall
$\langle s\rangle\ltimes\langle r_{1},r_{2}\rangle\cong
C_{3}\ltimes(C_{2}\times C_{2})\cong A_{4}$, where $s(z)=\omega_{3}z$ and
$r_{1}(z)=z+1/2$ and $r_{2}(z)=z+\omega_{3}/2$, cf. Lemma 4.4. The following
relations hold: $sr_{1}=r_{1}r_{2}s$ and $sr_{2}=r_{1}s.$ Using that
$s\cdot\wp^{\prime}=\wp^{\prime}$, we compute

$s\cdot
p_{2}=s\frac{1}{\wp^{\prime}}+sr_{1}\frac{1}{\wp^{\prime}}-sr_{2}\frac{1}{\wp^{\prime}}-sr_{1}r_{2}\frac{1}{\wp^{\prime}}=\frac{1}{\wp^{\prime}}+r_{1}r_{2}\frac{1}{\wp^{\prime}}-r_{1}\frac{1}{\wp^{\prime}}-r_{2}\frac{1}{\wp^{\prime}}=p_{0}.$

The other identities follow in the same way.

We now prove the second claim. We have

$p_{2}^{2}=s\cdot p_{1}^{2}=\alpha_{1}s\cdot
p_{2}^{2}+\alpha_{2}=\alpha_{1}p_{0}^{2}+\alpha_{2}=\alpha_{1}\beta_{1}p_{2}^{2}+\alpha_{1}\beta_{2}+\alpha_{2}$

and we thus see that $\alpha_{1}\beta_{1}=1$ and
$\alpha_{1}\beta_{2}=-\alpha_{2}$. By Lemma 6.13 we see
$p_{0}^{2}+p_{1}^{2}+p_{2}^{2}=(1+\alpha_{1}+\beta_{1})p_{2}^{2}+\alpha_{2}+\beta_{2}$.
Therefore the function $p_{0}^{2}+p_{1}^{2}+p_{2}^{2}$ is constant if and only
if $g_{2}=0$. Indeed, a computation shows that
$1+\alpha_{1}+\beta_{1}=\frac{g_{2}}{2(e_{2}-e_{3})^{2}}.$ We conclude that
$\alpha_{1}$ and $\beta_{1}$ are distinct third roots of unity if and only if
${\Lambda}$ is homothetic to a hexagonal lattice. Using that
$e_{1}^{2}+e_{1}e_{2}+e_{2}^{2}=0$ if and only if ${\Lambda}$ is homothetic to
${\Lambda}_{\omega_{6}}$, one can show that
$\alpha_{1}=(2e_{1}+e_{2})^{2}/(2e_{2}+e_{1})^{2}=e_{2}^{2}/e_{1}^{2}.$ Now,
$e_{1}=\omega_{3}e_{2}$ by (7). Thus, we get $\alpha_{1}=\omega_{3}$ and
$\beta_{1}=\omega_{3}^{2}$. ∎

Define

$\displaystyle
A_{1}=\left(\frac{e_{1}-e_{3}}{e_{2}-e_{3}}\right)^{\frac{3}{2}},\quad
B_{1}=\left(\frac{e_{1}-e_{2}}{e_{2}-e_{3}}\right)^{\frac{3}{2}}.$ (30)

We note that

$\frac{e_{1}-e_{2}}{e_{2}-e_{3}}=\frac{\vartheta_{01}^{4}(0|\tau)}{\vartheta_{00}^{4}(0|\tau)}=1-\lambda(\tau),$

where $\vartheta_{00},\vartheta_{01}$ are the Jacobi theta functions defined
in Remark 6.11 and $\lambda$ is the _modular lambda function_ , see for
example [29, Section 7.2].

Also, $\frac{e_{1}-e_{3}}{e_{2}-e_{3}}=1+\frac{e_{1}-e_{2}}{e_{2}-e_{3}},$ and
hence $\frac{e_{1}-e_{3}}{e_{2}-e_{3}}=2-\lambda(\tau).$ In particular, we can
express $\alpha_{1}$, $\beta_{1}$, $A_{1}$ and $B_{1}$ in terms of $\lambda$
as follows: $\alpha_{1}=(2-\lambda)^{2}$, $\beta_{1}=(1-\lambda)^{2}$,
$A_{1}=(2-\lambda)^{3/2}$ and $B_{1}=(1-\lambda)^{3/2}$.

For later reference, we will record the following identities.

######  Lemma 6.15.

The following identities hold:

$\displaystyle\left(\frac{A_{1}}{\alpha_{1}}p_{1}+\frac{B_{1}}{\beta_{1}}p_{0}\right)\left(\frac{A_{1}}{\alpha_{1}}p_{1}-\frac{B_{1}}{\beta_{1}}p_{0}\right)=p_{2}^{2},$
$\displaystyle\left(\frac{A_{1}}{\alpha_{1}}p_{1}\mp\frac{B_{1}}{\beta_{1}}p_{0}\right)(A_{1}p_{0}\pm
B_{1}p_{1})=p_{0}p_{1}\pm\sqrt{\alpha_{2}\beta_{2}}.$

Furthermore, if ${\Lambda}$ is homothetic to ${\Lambda}_{\omega_{6}}$, then
$A_{1}^{2}=-1$ and $B_{1}^{2}=1$.

###### Proof.

Using Lemma 6.14, we compute

$\left(\frac{A_{1}}{\alpha_{1}}p_{1}+\frac{B_{1}}{\beta_{1}}p_{0}\right)\left(\frac{A_{1}}{\alpha_{1}}p_{1}-\frac{B_{1}}{\beta_{1}}p_{0}\right)=\left(\frac{A_{1}^{2}}{\alpha_{1}}-\frac{B_{1}^{2}}{\beta_{1}}\right)p_{2}^{2}+\left(\frac{A_{1}^{2}\alpha_{2}}{\alpha_{1}^{2}}-\frac{B_{1}^{2}\beta_{2}}{\beta_{1}^{2}}\right).$

Another computation shows that

$\frac{A_{1}^{2}\alpha_{2}}{\alpha_{1}^{2}}=\frac{4}{(e_{1}-e_{3})(e_{2}-e_{3})(e_{1}-e_{2})}=\frac{B_{1}^{2}\beta_{2}}{\beta_{1}^{2}}$

and

$\frac{A_{1}^{2}}{\alpha_{1}}-\frac{B_{1}^{2}}{\beta_{1}}=\frac{e_{1}-e_{3}}{e_{2}-e_{3}}-\frac{e_{1}-e_{2}}{e_{2}-e_{3}}=1.$

This proves the first claim. The other identities follow by similar
calculations, where we note that the sign of $\sqrt{\alpha_{2}\beta_{2}}$
comes from the choice of the square roots in the definition of $A_{1}$ and
$B_{1}$. Notice that $\sqrt{\alpha_{2}\beta_{2}}\neq 0$ since
$\Delta({\Lambda})=16(e_{1}-e_{3})^{2}(e_{2}-e_{3})^{2}(e_{1}-e_{2})^{2}\neq
0$ for any lattice ${\Lambda}$. The final claim follows from the definition of
$A_{1}$ and $B_{1}$, cf. (30), and using that in this case,
$e_{1}=\omega_{3}e_{2}$. ∎

We will now define two functions which arise in the construction of an
intertwiner $\Psi$. The intertwiner will be used to construct a basis for the
$C_{2}\times C_{2}$-aLia with $g(T/C_{2}\times C_{2})=1$. Define

$\displaystyle\xi_{\pm}:=\sqrt{\frac{A_{1}}{\alpha_{1}}p_{1}\pm\frac{B_{1}}{\beta_{1}}p_{0}}=\sqrt{\sqrt{\frac{e_{2}-e_{3}}{e_{1}-e_{3}}}p_{1}\pm\sqrt{\frac{e_{2}-e_{3}}{e_{1}-e_{2}}}p_{0}},$
(31)

where we choose the branches in such way that $\xi_{-}\xi_{+}=p_{2}$ (having
Lemma 6.15 in mind). Notice that $\xi_{\pm}$ are not functions on any complex
torus.

We will now turn our attention to finding an intertwiner $\mathrm{Ad}(\Phi)$
as we have done for $\Gamma=C_{N}$, cf. Section 6.1. Recall that we
constructed an $SL_{2}({\mathbb{C}})$-valued map $\Phi$ such that $\Phi(r\cdot
z)=\delta(r)\Phi(z)$, where $r\in C_{N}$ and $\delta$ is a monomorphism
$C_{N}\rightarrow SL_{2}({\mathbb{C}})$. Composing $\Phi$ with $\mathrm{Ad}$
yields a map equivariant with respect to the action of $C_{N}$ on a torus $T$
and $\mathfrak{sl}_{2}$. Let us now argue that a similar construction, that
is, obtaining an intertwiner via conjugation of some $C_{N}$-equivariant
$SL_{2}({\mathbb{C}})$-valued map, cannot be carried out for
$\Gamma=C_{2}\times C_{2}$. A necessary condition for carrying out this
construction, is that $\delta:\Gamma\rightarrow SL_{2}({\mathbb{C}})$ should
be a monomorphism. However, by considering the character table of $C_{2}\times
C_{2}$, one concludes that there does not exist a faithful representation
$C_{2}\times C_{2}\rightarrow SL_{2}({\mathbb{C}})$, and hence there is no
monomorphism $\delta$ such that the above condition holds for nonzero $\Phi$.
However, there does exist a faithful representation $C_{2}\times
C_{2}\rightarrow\mathrm{Aut}(\mathfrak{sl}_{2})$, which is the context in
which we will look for an intertwiner below.

Define $\rho:C_{2}\times C_{2}\rightarrow\mathrm{Aut}(\mathfrak{sl}_{2})$ by

$\displaystyle\rho(r_{1})\begin{pmatrix}a&b\\\
c&-a\end{pmatrix}=\begin{pmatrix}a&-b\\\ -c&-a\end{pmatrix},$
$\displaystyle\quad\rho(r_{2})\begin{pmatrix}a&b\\\
c&-a\end{pmatrix}=\begin{pmatrix}-a&-c\\\ -b&a\end{pmatrix}.$ (32)

######  Remark 6.16.

We have used the quaternions $Q_{8}$ as a double cover of $C_{2}\times C_{2}$
in (32). The matrices $R_{1}=\begin{pmatrix}i&0\\\ 0&-i\end{pmatrix}$ and
$R_{2}=\begin{pmatrix}0&1\\\ -1&0\end{pmatrix}$ generate $Q_{8}$ inside
$SL_{2}({\mathbb{C}})$, which define a homomorphism $\zeta:Q_{8}\rightarrow
SL_{2}({\mathbb{C}})$. Then $\mathrm{Ad}(\zeta(Q_{8}))\cong C_{2}\times C_{2}$
inside $\mathrm{Aut}(\mathfrak{sl}_{2})$.

Introduce the following matrix, which will play an important role in the
construction of the intertwiner:

$\displaystyle\Omega=\begin{pmatrix}\sqrt{\frac{A_{1}}{\alpha_{1}}p_{1}-\frac{B_{1}}{\beta_{1}}p_{0}}&\frac{1}{2\sqrt{\alpha_{2}\beta_{2}}}(A_{1}p_{0}-B_{1}p_{1})\sqrt{\frac{A_{1}}{\alpha_{1}}p_{1}+\frac{B_{1}}{\beta_{1}}p_{0}}\\\
\sqrt{\frac{A_{1}}{\alpha_{1}}p_{1}+\frac{B_{1}}{\beta_{1}}p_{0}}&\frac{1}{2\sqrt{\alpha_{2}\beta_{2}}}(A_{1}p_{0}+B_{1}p_{1})\sqrt{\frac{A_{1}}{\alpha_{1}}p_{1}-\frac{B_{1}}{\beta_{1}}p_{0}}\end{pmatrix},$
(33)

where we recall that $\alpha_{j},\beta_{j}$ are defined in Lemma 6.14 and
$A_{1},B_{1}$ in (30).

The following two simple lemmata establish that $\mathrm{Ad}(\Omega)$ is a
$C_{2}\times C_{2}$-equivariant automorphism of
$\mathfrak{sl}_{2}\otimes_{{\mathbb{C}}}\mathcal{O}_{\mathbb{T}}$. First of
all, conjugation with $\Omega$ is an automorphism of
$\mathfrak{sl}_{2}\otimes_{{\mathbb{C}}}\mathcal{O}_{\mathbb{T}}$:

######  Lemma 6.17.

We have
$\Psi:=\mathrm{Ad}(\Omega)\in\mathrm{Aut}_{\mathcal{O}_{\mathbb{T}}}(\mathfrak{sl}_{2}\otimes_{{\mathbb{C}}}\mathcal{O}_{\mathbb{T}}).$
Explicitly, the matrix of $\Psi$ with respect to the basis
$\mathcal{B}=\\{h,e,f\\}$, is given by

$\displaystyle[\Psi]_{\mathcal{B}}=\begin{pmatrix}\frac{1}{\sqrt{\alpha_{2}\beta_{2}}}p_{0}p_{1}&-p_{2}&\frac{1}{4\alpha_{2}\beta_{2}}(A_{1}^{2}p_{0}^{2}-B_{1}^{2}p_{1}^{2})p_{2}\\\
\frac{1}{\sqrt{\alpha_{2}\beta_{2}}}(B_{1}p_{1}-A_{1}p_{0})p_{2}&\frac{A_{1}}{\alpha_{1}}p_{1}-\frac{B_{1}}{\beta_{1}}p_{0}&\frac{1}{4\alpha_{2}\beta_{2}}(B_{1}p_{1}-A_{1}p_{0})\tilde{p}_{-}\\\
\frac{1}{\sqrt{\alpha_{2}\beta_{2}}}(A_{1}p_{0}+B_{1}p_{1})p_{2}&-\frac{A_{1}}{\alpha_{1}}p_{1}-\frac{B_{1}}{\beta_{1}}p_{0}&\frac{1}{4\alpha_{2}\beta_{2}}(A_{1}p_{0}+B_{1}p_{1})\tilde{p}_{+}\end{pmatrix},$

where $\tilde{p}_{\pm}:=p_{0}p_{1}\pm\sqrt{\alpha_{2}\beta_{2}}$.

###### Proof.

We will first argue that $\det(\Omega)=1$. Recall
$\xi_{\pm}=\sqrt{\frac{A_{1}}{\alpha_{1}}p_{1}\pm\frac{B_{1}}{\beta_{1}}p_{0}}$.
Using Lemma 6.15, we compute

$\displaystyle\det(\Omega)$
$\displaystyle=\frac{1}{2\sqrt{\alpha_{2}\beta_{2}}}(\xi_{-}^{2}(A_{1}p_{0}+B_{1}p_{1})-\xi_{+}^{2}(A_{1}p_{0}-B_{1}p_{1}))$
$\displaystyle=\frac{1}{2\sqrt{\alpha_{2}\beta_{2}}}((p_{0}p_{1}+\sqrt{\alpha_{2}\beta_{2}})-(p_{0}p_{1}-\sqrt{\alpha_{2}\beta_{2}}))$
$\displaystyle=1.$

Let $m=\begin{pmatrix}a&b\\\ c&d\end{pmatrix}\in SL_{2}(R)$, where $R$ is some
unital ring. The matrix of $\mathrm{Ad}(m)$ with respect to $\mathcal{B}$ is
given by

$[\mathrm{Ad}(m)]_{\mathcal{B}}=\begin{pmatrix}bc+ad&-ac&bd\\\
-2ab&a^{2}&-b^{2}\\\ 2cd&-c^{2}&d^{2}\end{pmatrix}\in SL_{3}(R).$

Take $m=\Omega$ and after some work, using Lemma 6.15, one arrives at the
matrix in the claim. Since $\det(\Omega)=1$, the same holds for $\Psi$ and we
see that all the entries are in $\mathcal{O}_{\mathbb{T}}$. This shows that
$\Psi\in\mathrm{Aut}_{\mathcal{O}_{\mathbb{T}}}(\mathfrak{sl}_{2}\otimes_{{\mathbb{C}}}\mathcal{O}_{\mathbb{T}}).$
∎

######  Lemma 6.18.

The map $\Psi$ intertwines the actions of $C_{2}\times C_{2}$ on $T$ and
$\mathfrak{sl}_{2}$: $\Psi(r\cdot z)=\rho(r)\Psi(z)$ for all $r\in C_{2}\times
C_{2}$ and $z\in{\mathbb{T}}$.

###### Proof.

Using that

$\left[\mathrm{Ad}\begin{pmatrix}i&0\\\
0&-i\end{pmatrix}\right]_{\mathcal{B}}=\begin{pmatrix}1&0&0\\\ 0&-1&0\\\
0&0&-1\end{pmatrix},\quad\left[\mathrm{Ad}\begin{pmatrix}0&1\\\
-1&0\end{pmatrix}\right]_{\mathcal{B}}=\begin{pmatrix}-1&0&0\\\ 0&0&-1\\\
0&-1&0\end{pmatrix},$

it is straightforward to verify the claim for $r_{1}$ and $r_{2}$, and hence
for all $r\in C_{2}\times C_{2}$. ∎

######  Remark 6.19.

Notice that in the definition of $\Omega$ in (33), the choice of signs of the
constants $A_{1}$ and $B_{1}$ does not matter. Indeed, one could verify that
changing the signs corresponds to compose $\Psi=\mathrm{Ad}(\Omega)$ with
commuting (order 2) elements of $\mathrm{Aut}(\mathfrak{sl}_{2})$, which in
turn also commute with $\rho(r)$, for any $r\in C_{2}\times C_{2}$.

Define the following elements of
$\mathfrak{sl}_{2}\otimes_{{\mathbb{C}}}\mathcal{O}_{\mathbb{T}}$:

$\displaystyle h^{\prime}=\Psi h,\quad e^{\prime}=\Psi e,\quad f^{\prime}=\Psi
f.$ (34)

By the equivariance property of $\Psi$, cf. Lemma 6.18, the elements
$h^{\prime},e^{\prime}$ and $f^{\prime}$ are invariant with respect to the
action of $C_{2}\times C_{2}$. In fact, they will form a basis of the
$C_{2}\times C_{2}$-aLia as we will see in the following theorem.

######  Theorem 6.20.

Let $\rho:C_{2}\times C_{2}\rightarrow\mathrm{Aut}(\mathfrak{sl}_{2})$ and
$\sigma:C_{2}\times C_{2}\rightarrow\mathrm{Aut}(T)$ be monomorphisms and
assume that $g(T/\sigma(C_{2}\times C_{2}))=1$. Then

$(\mathfrak{sl}_{2}\otimes_{{\mathbb{C}}}\mathcal{O}_{\mathbb{T}})^{\rho\otimes\tilde{\sigma}(C_{2}\times
C_{2})}\cong\mathfrak{sl}_{2}\otimes_{{\mathbb{C}}}\mathcal{O}_{\mathbb{T}}^{\tilde{\sigma}(C_{2}\times
C_{2})}.$

A normal form is given by

$(\mathfrak{sl}_{2}\otimes_{{\mathbb{C}}}\mathcal{O}_{\mathbb{T}})^{\rho\otimes\tilde{\sigma}(C_{2}\times
C_{2})}={\mathbb{C}}\langle
h^{\prime},e^{\prime},f^{\prime}\rangle\otimes_{{\mathbb{C}}}{\mathbb{C}}[\wp_{\frac{1}{2}{\Lambda}},\wp_{\frac{1}{2}{\Lambda}}^{\prime}]$

with

$[h^{\prime},e^{\prime}]=2e^{\prime},\quad[h^{\prime},f^{\prime}]=-2f^{\prime},\quad[e^{\prime},f^{\prime}]=h^{\prime},$

where $h^{\prime},e^{\prime},f^{\prime}$ are defined in (34).

###### Proof.

By Lemma 6.12 , we know that
$\mathcal{O}_{\mathbb{T}}^{\tilde{\sigma}(C_{2}\times
C_{2})}={\mathbb{C}}[\wp_{\frac{1}{2}\Lambda},\wp_{\frac{1}{2}\Lambda}^{\prime}]$.
Now, by Lemma 6.18, we are in a position to apply Proposition 6.4 to
$\Psi=\mathrm{Ad}(\Omega)$ with $K=C_{2}\times C_{2}$. Hence

$(\mathfrak{sl}_{2}\otimes_{{\mathbb{C}}}\mathcal{O}_{\mathbb{T}})^{\rho\otimes\tilde{\sigma}(C_{2}\times
C_{2})}={\mathbb{C}}\langle
h^{\prime},e^{\prime},f^{\prime}\rangle\otimes_{{\mathbb{C}}}{\mathbb{C}}[\wp_{\frac{1}{2}{\Lambda}},\wp_{\frac{1}{2}{\Lambda}}^{\prime}].$

The Lie structure is the same as the Lie structure of ${\mathbb{C}}\langle
e,f,h\rangle$ and is therefore given by

$[h^{\prime},e^{\prime}]=2e^{\prime},\quad[h^{\prime},f^{\prime}]=-2f^{\prime},\quad[e^{\prime},f^{\prime}]=h^{\prime}.$

In particular, we see that
$(\mathfrak{sl}_{2}\otimes_{{\mathbb{C}}}\mathcal{O}_{\mathbb{T}})^{\rho\otimes\tilde{\sigma}(C_{2}\times
C_{2})}\cong\mathfrak{sl}_{2}\otimes_{{\mathbb{C}}}{\mathbb{C}}[\wp_{\frac{1}{2}{\Lambda}},\wp_{\frac{1}{2}{\Lambda}}^{\prime}]=\mathfrak{C}_{\frac{1}{2}{\Lambda}}.$
∎

Explicitly, the generators of the $C_{2}\times C_{2}$-aLia in normal form, are
given by

$\displaystyle h^{\prime}$
$\displaystyle=\frac{1}{\sqrt{\alpha_{2}\beta_{2}}}\begin{pmatrix}p_{0}p_{1}&-\left(\frac{e_{1}-e_{3}}{e_{2}-e_{3}}\right)^{\frac{3}{2}}p_{0}p_{2}+\left(\frac{e_{1}-e_{2}}{e_{2}-e_{3}}\right)^{\frac{3}{2}}p_{1}p_{2}\\\
\left(\frac{e_{1}-e_{3}}{e_{2}-e_{3}}\right)^{\frac{3}{2}}p_{0}p_{2}+\left(\frac{e_{1}-e_{2}}{e_{2}-e_{3}}\right)^{\frac{3}{2}}p_{1}p_{2}&-p_{0}p_{1}\end{pmatrix},$
$\displaystyle e^{\prime}$
$\displaystyle=\begin{pmatrix}-p_{2}&\sqrt{\frac{e_{2}-e_{3}}{e_{1}-e_{3}}}p_{1}-\sqrt{\frac{e_{2}-e_{3}}{e_{1}-e_{2}}}p_{0}\\\
-\sqrt{\frac{e_{2}-e_{3}}{e_{1}-e_{3}}}p_{1}-\sqrt{\frac{e_{2}-e_{3}}{e_{1}-e_{2}}}p_{0}&p_{2}\end{pmatrix},$
$\displaystyle f^{\prime}$
$\displaystyle=\frac{1}{4\alpha_{2}\beta_{2}}\begin{pmatrix}\left(\frac{\Delta({\Lambda})}{16(e_{2}-e_{3})^{4}}p_{2}^{2}+\frac{12e_{1}}{(e_{2}-e_{3})^{4}}\right)p_{2}&\left(\left(\frac{e_{1}-e_{2}}{e_{2}-e_{3}}\right)^{\frac{3}{2}}p_{1}-\left(\frac{e_{1}-e_{3}}{e_{2}-e_{3}}\right)^{\frac{3}{2}}p_{0}\right)\tilde{p}_{-}\\\
\left(\left(\frac{e_{1}-e_{3}}{e_{2}-e_{3}}\right)^{\frac{3}{2}}p_{0}+\left(\frac{e_{1}-e_{2}}{e_{2}-e_{3}}\right)^{\frac{3}{2}}p_{1}\right)\tilde{p}_{+}&-\left(\frac{\Delta({\Lambda})}{16(e_{2}-e_{3})^{4}}p_{2}^{2}+\frac{12e_{1}}{(e_{2}-e_{3})^{4}}\right)p_{2}\end{pmatrix}.$

where we recall that
$\tilde{p}_{\pm}=p_{0}p_{1}\pm\sqrt{\alpha_{2}\beta_{2}}$. We could
rewrite/simplify the term
$\frac{\Delta({\Lambda})}{16(e_{2}-e_{3})^{4}}p_{2}^{2}$ to
$\wp_{\frac{1}{2}{\Lambda}}-4e_{1}$ due to Corollary 6.13.

######  Remark 6.21.

The aLia
$\mathfrak{A}=(\mathfrak{sl}_{2}\otimes_{{\mathbb{C}}}\mathcal{O}_{\mathbb{T}})^{\rho\otimes\tilde{\sigma}(C_{2}\times
C_{2})}$ is the Lie algebra of holomorphic $K=C_{2}\times C_{2}$-equivariant
maps from a punctured torus to $\mathfrak{sl}_{2}$. In [3], the authors study
holomorphic $K$-equivariant maps from some $K$-invariant subset of a complex
torus ${\mathbb{C}}/{\mathbb{Z}}\oplus{\mathbb{Z}}\tau$ to
$SL_{2}({\mathbb{C}})$. They give expansions of the functions that appear in
these matrices in terms of $\lambda_{1},\lambda_{2}$ and $\lambda_{3}$, which
are related to our functions $p_{0},p_{1}$ and $p_{2}$ defined in (23)-(25).
With these functions, they construct $K$-equivariant maps with values in
$\mathfrak{sl}_{2}$ which they use in the context of integrable systems, in
particular in an algebraic description of the Landau-Lifshitz hierarchy. For