# Integrability of the Inozemtsev spin chain

Oleg Chalykh  School of Mathematics, University of Leeds, Leeds LS2 9JT, UK
<EMAIL_ADDRESS>
###### Abstract.

We show that the Inozemtsev spin chain is integrable. The conserved quantities
(commuting Hamiltonians) are constructed using elliptic Dunkl operators. We
also suggest a generalisation.

##  1\. Introduction

The Inozemtsev quantum spin chain [I1] is described by the Hamiltonian

$\mathscr{H}=\sum_{i<j}^{n}\wp\left(\frac{i-j}{n}\right)P_{ij}\,.$ (1.1)

Here $\wp(z)=\wp(z|1,\tau)$ is the Weierstrass $\wp$-function with periods
$1,\tau$ and $P_{ij}$ acts on the tensor power $U^{\otimes n}$ of
$U=\mathbb{C}^{m}$ by permuting the factors. This is viewed as a spin chain on
$n$ sites, with each copy of $U$ representing local spin states, and with
$\displaystyle\wp\left(\frac{i-j}{n}\right)$ expressing the strength of
interaction between the $i$th and $j$th sites. To show that it is integrable,
one would like to find elements of $\mathtt{End}_{\mathbb{C}}(U^{\otimes n})$
commuting with $\mathscr{H}$. This is an old problem, see [I2, DI1, DI2, SS,
FGL] for some of its history and related context. The trigonometric case –
with $\sin^{-2}(\pi z)$ in place of $\wp(z)$ – describes the celebrated
Haldane–Shastry chain [H, S] which is integrable and has the Yangian symmetry
[BGHP]. Nothing of that sort in known in the elliptic case so, as it stands,
(1.1) falls outside the class of problems solved by Quantum Inverse Scattering
or similar methods.

Nevertheless, as will be demonstrated below, one can construct commuting
Hamiltonians from those of the quantum elliptic spin Calogero–Moser system, by
“freezing”. This builds on (and elucidates) the original idea by Polychronakos
[P2], see also related works [FM, MP, TH]. Our main ingredients are the
elliptic Dunkl operators and Cherednik algebra. Along the way, we find a
generalisation of the Inozemtsev spin chain.

One would also like to calculate the eigenvalues and eigenvectors of
$\mathscr{H}$, see [KL] for the state of the art and references. Hopefully,
the link to the elliptic Dunkl operators can shed some light onto this. Our
methods apply, almost verbatim, to elliptic Calogero–Moser systems (and spin
chains) for arbitrary root systems. They can also be extended to
Calogero–Moser systems of $R$-matrix type and the corresponding spin chains
from [LOZ, SZ], which will be done separately [ChL].

Acknowledgement. I would like to thank Andrei Zotov for useful conversations.
I am also grateful to Jules Lamers and Rob Klabbers for stimulating
discussions and for waiting patiently for this paper to be written up.

##  2\. Rational Calogero–Moser system and Dunkl operators

Here we recall the well-known relationship between the rational Calogero–Moser
system, Dunkl operators and Cherednik algebra, see [E] and references therein.

### 2.1.

Let $V=\mathbb{C}^{n}$ be the $n$-dimensional complex Euclidean vector space
with the standard basis $(e_{i})$ and coordinates $(x_{i})$. The symmetric
group $S_{n}$ acts on $V$ by permuting the basis vectors. This induces an
$S_{n}$-action on the algebra $\mathbb{C}(V)$ of meromorphic functions in $n$
variables by $(w.f)(x)=f(w^{-1}.\,x)$ for $x\in V$, $w\in S_{n}$. The crossed
product $\mathbb{C}(V)*S_{n}$ is formed by taking the vector space
$\mathbb{C}(V)\otimes\mathbb{C}S_{n}$ with the multiplication $(f\otimes
w)(f^{\prime}\otimes w^{\prime})=f(w.f^{\prime})\otimes ww^{\prime}$. Let
$\mathscr{D}(V)$ denote the ring of differential operators on $V$ with
meromorphic coefficients. It is generated by the partial derivatives
$\partial_{i}=\frac{\partial}{\partial x_{i}}$ and operators of multiplication
by $g\in\mathbb{C}(V)$. We have a natural action of $S_{n}$ on
$\mathscr{D}(V)$, hence the crossed product $\mathscr{D}(V)*S_{n}$. As an
algebra, $\mathscr{D}(V)*S_{n}$ is generated by its two subalgebras,
$1\otimes\mathbb{C}S_{n}$ and $\mathscr{D}(V)\otimes 1$ which can be
identified with $\mathbb{C}W$ and $\mathscr{D}(V)$, respectively. Using these
identifications, we replace $a\otimes w$ by $aw$, so each element of
$\mathscr{D}(V)*S_{n}$ is written uniquely as $a=\sum_{w\in S_{n}}a_{w}w$ with
$a_{w}\in\mathscr{D}(V)$.

### 2.2.

The _quantum rational_ Calogero–Moser (CM) system is described by the
Hamiltonian

${H}^{(r)}=\sum_{i=1}^{n}\hbar^{2}\partial_{i}^{2}-\sum_{i<j}^{n}\frac{2\kappa(\kappa-\hbar)}{(x_{i}-x_{j})^{2}}\,.$
(2.1)

Here $\kappa\in\mathbb{C}$ is the coupling constant and $\hbar\neq 0$ is a
quantum parameter 111To follow the standard conventions of quantum mechanics,
$\hbar$ should be replaced $-\mathrm{i}\hbar$ and $\kappa$ should be purely
imaginary.. This is a completely integrable system, in the sense that it
admits $n$ commuting Hamiltonians. These are constructed using the _Dunkl
operators_ :

${y}^{(r)}_{i}=\hbar\partial_{i}-\kappa\sum_{j\neq
i}^{n}\frac{1}{x_{i}-x_{j}}s_{ij}\,,\qquad i=1,\dots,n\,.$ (2.2)

The key property of the these operators is their commutativity,
$[{y}^{(r)}_{i},{y}^{(r)}_{j}]=0$, and equivariance,

$s_{ij}{y}^{(r)}_{i}={y}^{(r)}_{j}s_{ij}\,,\qquad
s_{ij}{y}^{(r)}_{k}={y}^{(r)}_{k}s_{ij}\quad(k\neq i,j)\,.$

As a result, the assignment $\,e_{i}\mapsto{y}^{(r)}_{i}\,$ extends to a
$S_{n}$\- equivariant algebra map

$\mathbb{C}[V^{*}]\to\mathscr{D}(V)*S_{n}\,,\quad q\mapsto q({y}^{(r)})\ .$
(2.3)

Introduce the linear map

$\mathtt{Res}:\ \mathscr{D}(V)*S_{n}\,\to\,\mathscr{D}(V)\,,\qquad\sum_{w\in
S_{n}}a_{w}w\mapsto\sum_{w\in S_{n}}a_{w}\,.$ (2.4)

Combining this map with (2.3), define

${L}^{(r)}_{q}=\mathtt{Res}\,q({y}^{(r)})\,,\qquad
q\in\mathbb{C}[V^{*}]^{S_{n}}\,.$ (2.5)

The $S_{n}$-invariance of $q$ and the commutativity of the Dunkl operators
imply that (1) each ${L}^{(r)}_{q}$ is $S_{n}$-invariant, (2)
${L}^{(r)}_{q}{L}^{(r)}_{q^{\prime}}={L}^{(r)}_{qq^{\prime}}$ for any
$q,q^{\prime}\in\mathbb{C}[V^{*}]^{S_{n}}$, (3) the family
$\\{{L}^{(r)}_{q}\,,\ q\in\mathbb{C}[V^{*}]^{S_{n}}\\}$ is commutative.

Taking $q=e_{1}^{2}+\dots+e_{n}^{2}\in S^{2}V\subset\mathbb{C}[V^{*}]$, a
direct calculation shows that

$q({y}^{(r)})=({y}^{(r)}_{1})^{2}+\dots+({y}^{(r)}_{n})^{2}=\sum_{i=1}^{n}\hbar^{2}\partial_{i}^{2}-\sum_{i<j}^{n}\frac{2}{(x_{i}-x_{j})^{2}}\kappa(\kappa-\hbar
s_{\alpha})\,.$ (2.6)

Hence, ${L}^{(r)}_{q}=\mathtt{Res}\,q({y}^{(r)})$ in this case is precisely
the CM Hamiltonian (2.1). Other symmetric combinations of
${y}^{(r)}_{1},\dots,{y}^{(r)}_{n}$ produce higher commuting Hamiltonians.

### 2.3.

The classical limit corresponds to taking $\hbar\to 0$. More precisely, we
view the Dunkl operators as elements of the algebra

$A_{\hbar}=\mathbb{C}[[\hbar]]\otimes\mathbb{C}(V)[\hat{p}_{1},\dots,\hat{p}_{n}]\,,\qquad\hat{p}_{i}=\hbar\partial_{i}\,.$

The quantum momenta $\hat{p}_{i}$ satisfy the relations
$[\hat{p}_{i},f]=\hbar\,\partial_{i}f$ for $f\in\mathbb{C}(V)$. We have an
algebra isomorphism

$\eta_{0}:\ (A_{\hbar}*S_{n})/\hbar(A_{\hbar}*S_{n})\to A_{0}*S_{n}\,,\qquad
f\mapsto f\,,\ \ \hat{p}_{i}\mapsto p_{i}\,,\ \ w\mapsto w\,,$ (2.7)

where

$A_{0}=\mathbb{C}(V)\otimes\mathbb{C}[V^{*}]=\mathbb{C}(V)[p_{1},\dots,p_{n}]$

is the classical version of $A_{\hbar}$. Therefore, $A_{\hbar}$ and
$A_{\hbar}*S_{n}$ are formal deformations of $A_{0}$ and $A_{0}*S_{n}$,
respectively. Note that $A_{0}$ is commutative, with Poisson bracket defined
by $\\{\eta_{0}(a),\eta_{0}(b)\\}=\eta_{0}(\hbar^{-1}[a,b])$ for $a,b\in
A_{\hbar}$. For any $a\in A_{\hbar}*S_{n}$, we call $\eta_{0}(a)$ the
_classical limit_ of $a$. For instance, the classical limit of (2.2) are the
classical Dunkl operators

${y}^{(r)}_{i,c}=p_{i}-\kappa\sum_{j\neq
i}^{n}\frac{1}{x_{i}-x_{j}}s_{ij}\,,\qquad i=1,\dots,n\,.$ (2.8)

These are commuting, equivaraint elements of $A_{0}*S_{n}$, and we have a
classical variant of (2.3):

$\mathbb{C}[V^{*}]\to A_{0}*{S_{n}}\ ,\quad q\mapsto q({y}^{(r)}_{c})\,.$
(2.9)

### 2.4.

The classical limit of (2.1) gives the Hamiltonian

${H}^{(r)}_{c}=\sum_{i=1}^{n}p_{i}^{2}-\sum_{i<j}^{n}\frac{2\kappa^{2}}{(x_{i}-x_{j})^{2}}\,.$
(2.10)

Its complete integrability can be established similarly to the quantum case.
Namely, using the classical variant of the map (2.4), we set

${L}^{(r)}_{q,c}=\mathtt{Res}\,q({y}^{(r)}_{c})\,,\qquad
q\in\mathbb{C}[V^{*}]^{S_{n}}\,.$ (2.11)

By construction, ${L}^{(r)}_{q,c}$ is the classical limit of ${L}^{(r)}_{q}$.
Hence, the family $\\{{L}^{(r)}_{q,c}\,,\ q\in\mathbb{C}[V^{*}]^{S_{n}}\\}$ is
Poisson commutative, i.e. it defines a classical integrable system. Passing to
the classical limit in (2.6), we get

$({y}^{(r)}_{1,c})^{2}+\dots+({y}^{(r)}_{n,c})^{2}=\sum_{i=1}^{n}p_{i}^{2}-\sum_{i<j}^{n}\frac{2\kappa^{2}}{(x_{i}-x_{j})^{2}}\,,$
(2.12)

which is the Hamiltonian (2.10). Note that in this case there is no need to
apply $\mathtt{Res}$. This is true for the higher Hamiltonians as well:

######  Lemma 2.1 (Lemma 2.2 [EFMV]).

For any $q\in\mathbb{C}[V^{*}]^{S_{n}}$, we have $q({y}^{(r)}_{c})\in A_{0}$.
Hence, ${L}^{(r)}_{q,c}=q({y}^{(r)}_{c})$, i.e. the application of
$\mathtt{Res}$ is not necessary.

An analogous statement in the elliptic case will be important below.

### 2.5.

By definition, the rational Cherednik algebra $\mathtt{H}_{\hbar,\kappa}$ of
type $S_{n}$ is the subalgebra of $\mathscr{D}(V)*S_{n}$ generated by $w\in
S_{n}$, $x_{i}$ ($i=1,\dots,n$), and the Dunkl operators (2.2). The spherical
subalgebra of $\mathtt{H}_{\hbar,\kappa}$ is defined as
$e\,\mathtt{H}_{\hbar,\kappa}\,e\,$, where

$e=\frac{1}{n!}\sum_{w\in S_{n}}w\,.$ (2.13)

Restricting (2.4) onto the spherical subalgebra, one obtains an algebra map

$\mathtt{Res}:\,e\mathtt{H}_{\hbar,\kappa}e\hookrightarrow\mathscr{D}(V)^{S_{n}}\,.$
(2.14)

whose image is denoted as $\mathtt{B}_{\hbar,\kappa}$. Elements of
$B_{\hbar,\kappa}$ can be obtained by applying $\mathtt{Res}$ to symmetric
combinations of $x_{i}$ and ${y}^{(r)}_{i}$. Inside $\mathtt{B}_{0,\kappa}$ we
have the commutative algebra of quantum rational CM Hamiltonians
$\\{{L}^{(r)}_{q}\,,\,q\in\mathbb{C}[V^{*}]^{S_{n}}\\}$.

The classical limits are defined in a similar way. That is,
$\mathtt{H}_{0,\kappa}$ is the subalgebra of $A_{0}*S_{n}$ generated by $w\in
S_{n}$, $x_{i}$ ($i=1,\dots,n$), and the classical Dunkl operators (2.8). The
spherical subalgebra is defined as $e\,\mathtt{H}_{0,\kappa}\,e\,$, and we
have an algebra map

$\mathtt{Res}:\,e\mathtt{H}_{0,\kappa}e\hookrightarrow A_{0}^{S_{n}}\,.$
(2.15)

Its image, $\mathtt{B}_{0,\kappa}$, is a Poisson subalgebra of
$A_{0}^{S_{n}}$, containing a Poisson commutative algebra,
$\\{{L}^{(r)}_{q,c}\,,\,q\in\mathbb{C}[V^{*}]^{S_{n}}\\}$, of classical
rational CM Hamiltonians.

##  3\. Elliptic Dunkl operators and Calogero–Moser system

### 3.1.

The quantum elliptic CM system is described by the Hamiltonian

${H}=\hbar^{2}\sum_{i=1}^{n}\partial_{i}^{2}-2\kappa(\kappa-\hbar)\sum_{i<j}^{n}\wp(x_{i}-x_{j})\,.$
(3.1)

Here $\wp(z)=\wp(z|1,\tau)$ is the Weierstrass $\wp$-function. The classical
Hamiltonian is

$H_{c}=\sum_{i=1}^{n}p_{i}^{2}-2\kappa^{2}\sum_{i<j}^{n}\wp(x_{i}-x_{j})\,.$
(3.2)

### 3.2.

As in the rational case, both quantum and classical systems are completely
integrable [Ca, Pe, OP1, OP2, OS]. Following [EFMV], this can be shown using
_elliptic Dunkl operators_ [BFV] which, in the quantum case, are

$y_{i}=\hbar\partial_{i}-\kappa\sum_{j\neq
i}^{n}\phi(\lambda_{i}-\lambda_{j},x_{i}-x_{j})s_{ij}\,,\qquad\phi(\mu,z)=\frac{\sigma(z-\mu)}{\sigma(z)\sigma(-\mu)}\,.$
(3.3)

Here $\sigma(z)=\sigma(z|1,\tau)$ is the Weierstrass $\sigma$-function, and
$\lambda_{1},\dots,\lambda_{n}$ are auxiliary variables referred to as the
_spectral variables_. Hence, $y_{i}=y_{i}(\lambda)$ are $\lambda$-dependent
elements of $\mathscr{D}(V)*S_{n}$.

For $\xi\in V$, we write $y_{\xi}=\sum_{i=1}^{n}\xi_{i}y_{i}$ . Two main
properties of the operators $y_{\xi}$ are their commutativity and
equivariance: for all $\,\xi,\eta\in V\,$ and $w\in S_{n}$,

$y_{\xi}\,y_{\eta}=y_{\eta}\,y_{\xi}\,,\qquad\,w\,y_{\xi}(\lambda)=y_{w\xi}(w\lambda)\,w\,.$
(3.4)

Note that in the second relation the group action affects both $\xi$ and
$\lambda$. As before, the assignment $\,\xi\mapsto y_{\xi}\,$ extends to an
algebra map

$\mathbb{C}[V^{*}]\to\mathscr{D}(V)*S_{n}\ ,\quad q\mapsto q(y)\,.$ (3.5)

However, unlike in the rational case, this map is not $S_{n}$-equivariant and
constructing commuting quantum Hamiltonians requires certain regularization.

### 3.3.

To explain the regularization procedure [EFMV], we extend the algebra map
(3.5) by allowing polynomials with $\lambda$-dependent coefficients:

$\mathbb{C}(V)\otimes\mathbb{C}[V^{*}]\to\mathscr{D}(V)*S_{n}\ ,\quad f\mapsto
f(\lambda,y)\,.$ (3.6)

Next, recall the rational classical Hamiltonians
${L}^{(r)}_{q,c}=q({y}^{(r)}_{c})$, $q\in\mathbb{C}[V^{*}]^{S_{n}}$. These are
$S_{n}$-invariant elements of $A_{0}=\mathbb{C}(V)\otimes\mathbb{C}[V^{*}]$.

######  Theorem 3.1 ([EFMV]).

For $q\in\mathbb{C}[V^{*}]^{S_{n}}$, consider ${L}^{(r)}_{q,c}\in
A_{0}^{S_{n}}$. Identify $A_{0}$ with $\mathbb{C}(V)\otimes\mathbb{C}[V^{*}]$
in(3.6) and obtain ${L}^{(r)}_{q,c}(\lambda,y)\in\mathscr{D}(V)*W$ by applying
(3.6), i.e. by replacing classical momenta in ${L}^{(r)}_{q,c}$ with elliptic
Dunkl operators.

$(1)$ The elements ${L}^{(r)}_{q,c}(\lambda,y)$ are regular near $\lambda=0$
and so have a well-defined limit at $\lambda=0$.

$(2)$ Setting $L_{q}:=\mathtt{Res}\lim_{\lambda\to
0}\,{L}^{(r)}_{q,c}(\lambda,y)$ defines an algebra embedding
$\mathbb{C}[V^{*}]^{S_{n}}\to\mathscr{D}(V)^{S_{n}}$, $q\mapsto L_{q}$.

### 3.4.

To illustrate the theorem, let us follow a calculation in [BFV]. Choose
${L}^{(r)}_{q,c}$ to be the Hamiltonian ${H}^{(r)}_{c}$ (2.10). In this case,

${H}^{(r)}_{c}(\lambda,y)=y_{1}^{2}+\dots+y_{n}^{2}-\sum_{i<j}\frac{2\kappa^{2}}{(\lambda_{i}-\lambda_{j})^{2}}\,.$

By a direct calculation, with the help of the identity
$\phi(\mu,z)\phi(\mu,-z)=\wp(\mu)-\wp(z)$,

$y_{1}^{2}+\dots+y_{n}^{2}=\hbar^{2}\sum_{i=1}^{n}\partial_{i}^{2}-2\hbar\kappa\sum_{i<j}^{n}\phi^{\prime}(\lambda_{i}-\lambda_{j},x_{i}-x_{j})s_{ij}\\\
+2\kappa^{2}\sum_{i<j}^{n}\left(\wp(\lambda_{i}-\lambda_{j})-\wp(x_{i}-x_{j})\right)\,.$
(3.7)

Here $\phi^{\prime}(\mu,z)=\frac{d}{dz}\phi(\mu,z)$. Using that

$\wp(z)=\frac{1}{z^{2}}+o(z)\quad\text{as}\ z\to 0\,,\qquad\lim_{\mu\to
0}\phi^{\prime}(\mu,z)=-\wp(z)\,,$

one finds that

$\lim_{\lambda\to
0}{H}^{(r)}_{c}(\lambda,y)=\hbar^{2}\sum_{i=1}^{n}\partial_{i}^{2}-\sum_{i<j}^{n}2\kappa(\kappa-\hbar
s_{ij})\wp(x_{i}-x_{j})\,.$ (3.8)

Applying the map (2.4) gives the Hamiltonian (3.1):

$\mathtt{Res}\lim_{\lambda\to 0}{H}^{(r)}_{c}(\lambda,y)=H\,.$ (3.9)

######  Remark 3.2.

For the Calogero–Moser particles, their total momentum

$P=p_{1}+\dots+p_{n}$

is conserved. Replacing $p_{i}$ with the Dunkl operators $y_{i}$, one finds
that

$y_{1}+\dots+y_{n}=\hbar\partial_{1}+\dots+\hbar\partial_{n}\,,$ (3.10)

reflecting conservation of the total momentum in the quantum system.

### 3.5.

By passing to the classical limit in the above constructions, one obtains
Hamiltonians for the classical elliptic CM system. Namely, we think of the
operators (3.11) as elements of $A_{\hbar}*S_{n}$, so their classical limit
are the following commuting elements of $A_{0}*S_{n}$:

$y_{i,c}=p_{i}-\kappa\sum_{j\neq
i}^{n}\phi(\lambda_{i}-\lambda_{j},x_{i}-x_{j})s_{ij}\,.$ (3.11)

The classical limit of (3.6) is the map

$\mathbb{C}(V)\otimes\mathbb{C}[V^{*}]\to A_{0}*S_{n}\ ,\quad f\mapsto
f(\lambda,y_{c})\,.$ (3.12)

######  Theorem 3.3 ([EFMV]).

For $q\in\mathbb{C}[V^{*}]^{S_{n}}$, consider ${L}^{(r)}_{q,c}\in
A_{0}^{S_{n}}$. Identify $A_{0}$ with $\mathbb{C}(V)\otimes\mathbb{C}[V^{*}]$
in(3.12) and obtain ${L}^{(r)}_{q,c}(\lambda,y_{c})\in A_{0}*W$ by applying
(3.12), i.e. by replacing classical momenta in ${L}^{(r)}_{q,c}$ with
classical elliptic Dunkl operators.

$(1)$ The elements ${L}^{(r)}_{q,c}(\lambda,y_{c})$ are regular near
$\lambda=0$ and so have a well-defined limit at $\lambda=0$.

$(2)$ Setting $L_{q,c}:=\mathtt{Res}\lim_{\lambda\to
0}\,{L}^{(r)}_{q,c}(\lambda,y_{c})$ defines a Poisson-commutative subalgebra
$\\{L_{q,c},\,\,q\in\mathbb{C}[V^{*}]^{S_{n}}\\}$ in $A_{0}^{S_{n}}$.

$(3)$ In particular, similarly to (3.8),

$\mathtt{Res}\lim_{\lambda\to
0}{H}^{(r)}_{c}(\lambda,y_{c})=\sum_{i=1}^{n}p_{i}^{2}-2\kappa^{2}\sum_{i<j}^{n}\wp(x_{i}-x_{j})\,,$

which is the Hamiltonian (3.2).

### 3.6.

For later use, let us consider what happens when we substitute elliptic Dunkl
operators into _elliptic_ Hamiltonians (rather than rational ones).

######  Proposition 3.4 (Proposition 5.1 [Ch]).

For $q\in\mathbb{C}[V^{*}]^{S_{n}}$, let $L_{q,c}\in A_{0}^{S_{n}}$ be the
classcal Hamiltonian from Theorem 3.3. Then:

$(1)$ $L_{q,c}(\lambda,y)$ is regular near $\lambda=0$,

$(2)$ $L_{q,c}(\lambda,y_{c})\in A_{0}*S_{n}$ does not depend on $\lambda$.
Furthermore, expanding it into $\sum_{w\in S_{n}}a_{w}w$, we have $a_{w}=0$
for $w\neq\mathrm{id}$, i.e. $L_{q,c}(\lambda,y_{c})\in A_{0}\subset
A_{0}*S_{n}$.

Note that part $(2)$ of the proposition is an elliptic analogue of Lemma 2.1.
Now, viewing $L_{q,c}(\lambda,y)$ as elements of $A_{\hbar}*S_{n}$, denote

$\mathsf{H}_{q}:=\mathtt{Res}\lim_{\lambda\to 0}L_{q,c}(\lambda,y)\,,\qquad
q\in\mathbb{C}[V^{*}]^{S_{n}}.$ (3.13)

Clearly, $\mathsf{H}_{q}$ belongs to $A_{h}^{S_{n}}$ and commutes with all
higher CM Hamiltonians222In fact, one can show that
$\mathsf{H}_{q}=L_{q^{\prime}}$ for some
$q^{\prime}\in\mathbb{C}[V^{*}]^{S_{n}}$, but we will not need this..

######  Corollary 3.5.

For $q\in\mathbb{C}[V^{*}]^{S_{n}}$, we have

$\lim_{\lambda\to
0}L_{q,c}(\lambda,y)=\mathsf{H}_{q}+\hbar\mathsf{A}_{q}\,,\qquad\text{for
some}\ \mathsf{A}_{q}\in(A_{\hbar}*S_{n})^{S_{n}}\,.$ (3.14)

Indeed, the classical limit of the l.h.s. is $\lim_{\lambda\to
0}L_{q,c}(\lambda,y_{c})=\mathtt{Res}\lim_{\lambda\to
0}L_{q,c}(\lambda,y_{c})$ (the last equality is by part $(2)$ of Prop. 3.4).
Hence, $\lim_{\lambda\to 0}L_{q,c}(\lambda,y)$ and $\mathsf{H}_{q}$ have the
same classical limit. ∎

##  4\. Spin Calogero–Moser system

The Dunkl operators can be used to construct a spin generalisation of the
elliptic Calogero–Moser system. For the trigonometric case, see [P1, BGHP,
Che].

### 4.1.

Consider the Hamiltonian

$\widehat{H}=\sum_{i=1}^{n}\hbar^{2}\partial_{i}^{2}-2\sum_{i<j}^{n}\wp(x_{i}-x_{j})\kappa(\kappa-\hbar\widehat{s}_{ij})\,.$
(4.1)

This is viewed as an element of the algebra
$\mathscr{D}(V)\otimes\mathbb{C}S_{n}$, with the tensor-product sign omitted
and with the “hat” symbol used to distinguish this from the crossed product
$\mathscr{D}(V)*S_{n}$. Choosing an $S_{n}$-module $\mathscr{U}$ makes
$\widehat{H}$ a matrix-valued differential operator, i.e. an element of
$\mathscr{D}(V)\otimes\mathtt{End}_{\mathbb{C}}\mathscr{U}$, with
$\widehat{s}_{ij}$ acting on $\mathscr{U}$ (but not on $\mathbb{C}(V)$). For
example, we may choose $\mathscr{U}=U^{\otimes n}$, $U=\mathbb{C}^{m}$, with
$\widehat{s}_{ij}$ acting as the permutation $P_{ij}$ of the tensor factors.
Our goal is to construct a commutative subalgebra of
$\mathscr{D}(V)\otimes\mathbb{C}S_{n}$ containing $\widehat{H}$.

### 4.2.

We begin with considerations as in [P1, BGHP, Che], rephrased for convenience.
Let us enlarge the algebra $\mathscr{D}(V)*S_{n}$ by adding another copy of
the group algebra:

$\mathscr{A}:=(\mathscr{D}(V)*S_{n})\otimes\mathbb{C}S_{n}\,.$

Elements of $\mathscr{A}$ can be uniquely written as

$a=\sum_{w,w^{\prime}\in
S_{n}}a_{ww^{\prime}}w\otimes\widehat{w^{\prime}}\quad\text{with}\
a_{ww^{\prime}}\in\mathscr{D}(V)\,.$

Define a linear map

$\widehat{\mathtt{Res}}:\
\mathscr{A}\,\to\,\mathscr{D}(V)\otimes\mathbb{C}S_{n}\,,\qquad\sum_{w,w^{\prime}\in
S_{n}}a_{ww^{\prime}}w\otimes\widehat{w^{\prime}}\mapsto\sum_{w,w^{\prime}\in
S_{n}}a_{ww^{\prime}}\otimes\widehat{w^{\prime}w^{-1}}\,.$ (4.2)

Equivalently, $\widehat{\mathtt{Res}}(a)$ is the unique element
$L_{a}\in\mathscr{D}(V)\otimes\mathbb{C}S_{n}$ such that

$a\widehat{e}=L_{a}\widehat{e}\,,\qquad\widehat{e}:=\frac{1}{n!}\sum_{w\in
S_{n}}w\otimes\widehat{w}\,.$ (4.3)

We have an $S_{n}$-action on $\mathscr{A}$ and
$\mathscr{D}(V)\otimes\mathbb{C}S_{n}$ by conjugation,

$a\mapsto(w\otimes\widehat{w})a(w^{-1}\otimes\widehat{w^{-1}})\qquad\forall\
w\in S_{n}\,.$ (4.4)

It is easy to check that the map $\widehat{\mathtt{Res}}$ is
$S_{n}$-equivariant with respect to this action.

######  Lemma 4.1.

The restriction
$\widehat{\mathtt{Res}}\,:\,\mathscr{A}^{S_{n}}\to(\mathscr{D}(V)\otimes\mathbb{C}S_{n})^{S_{n}}$
of the map (4.2) is an algebra homomorphism.

Indeed, if $a\in\mathscr{A}^{S_{n}}$ and $L_{a}=\widehat{\mathtt{Res}}(a)$,
then $a\widehat{e}=\widehat{e}a$ and $L_{a}\widehat{e}=\widehat{e}L_{a}$.
Hence, for $a,b\in\mathscr{A}^{S_{n}}$ we have
$L_{ab}\widehat{e}=ab\widehat{e}=(a\widehat{e})(b\widehat{e})=(L_{a}\widehat{e})(L_{b}\widehat{e})=L_{a}L_{b}\widehat{e}$.
∎

### 4.3.

On $\mathscr{D}(V)*S_{n}\subset\mathscr{A}$, the action (4.4) is simply
$a\mapsto waw^{-1}$. Thus, restricting (4.2) further gives an algebra map

$\widehat{\mathtt{Res}}\,:\,(\mathscr{D}(V)*S_{n})^{S_{n}}\to(\mathscr{D}(V)\otimes\mathbb{C}S_{n})^{S_{n}}\,,\qquad\sum_{w\in
S_{n}}a_{w}w\mapsto\sum_{w\in S_{n}}a_{w}\otimes\widehat{w^{-1}}\,,$ (4.5)

converting reflection-differential operators to $\mathbb{C}S_{n}$-valued
differential operators. Recall that, by Theorem 3.1, we have well-defined
elements $\lim_{\lambda\to 0}{L}^{(r)}_{q,c}(\lambda,y)$, for every
$q\in\mathbb{C}[V^{*}]^{S_{n}}$. Since
$w{L}^{(r)}_{q,c}(\lambda,y)w^{-1}={L}^{(r)}_{q,c}(w\lambda,y)$, the limit at
$\lambda=0$ is $S_{n}$-invariant. Also, since these elements are constructed
from commuting Dunkl operators, they pairwise commute. Therefore, we have the
following result.

######  Proposition 4.2.

For $q\in\mathbb{C}[V^{*}]^{S_{n}}$, define
$\widehat{\mathsf{L}}_{q}=\widehat{\mathtt{Res}}\lim_{\lambda\to
0}{L}^{(r)}_{q,c}(\lambda,y)$. The family
$\\{\widehat{\mathsf{L}}_{q}\,,\,q\in\mathbb{C}[V^{*}]^{S_{n}}\\}$ forms a
commutative subalgebra in $(\mathscr{D}(V)\otimes\mathbb{C}S_{n})^{S_{n}}$.

Using our calculations in § 3.4, we find

$\displaystyle\widehat{\mathsf{L}}_{q}$
$\displaystyle=\hbar\partial_{1}+\dots+\hbar\partial_{n}\qquad$
$\displaystyle\text{for}\ q$ $\displaystyle=e_{1}+\dots+e_{n}\,,$ (4.6)
$\displaystyle\widehat{\mathsf{L}}_{q}$
$\displaystyle=\sum_{i=1}^{n}\hbar^{2}\partial_{i}^{2}-2\sum_{i<j}^{n}\wp(x_{i}-x_{j})\kappa(\kappa-\hbar\widehat{s}_{ij})\qquad$
$\displaystyle\text{for}\ q$ $\displaystyle=e_{1}^{2}+\dots+e_{n}^{2}\,.$
(4.7)

Hence, the operator (4.1) belongs to the constructed commutative family.

######  Remark 4.3.

The algebra $\mathscr{D}(V)*S_{n}$ has a nontrivial centre, spanned by the
central idempotents $c_{\pi}\in\mathbb{C}S_{n}$, $\pi\in\mathtt{Irr}S_{n}$.
The elements $\widehat{c_{\pi}}$ commute with any of the spin Hamiltonians.

### 4.4.

Due to extra degrees of freedom in the spin system, one expects more commuting
Hamiltonians. Recall the classical spherical subalgebra
$\mathtt{B}_{0,\kappa}$, a Poisson subalgebra of $A_{0}^{S_{n}}$, see § 2.5.
Also, recall the map $A_{0}\to\mathscr{D}(V)*S_{n}$, $f\mapsto f(\lambda,y)$,
see (3.6). We have the following result strengthening Proposition 4.2.

######  Theorem 4.4.

For any $f\in\mathtt{B}_{0,\kappa}$, the element
$f(\lambda,y)\in\mathscr{D}(V)*S_{n}$ is regular near $\lambda=0$. Define
$\widehat{\mathsf{L}}_{f}:=\widehat{\mathtt{Res}}\lim_{\lambda\to
0}f(\lambda,y)$. Then the family
$\\{\widehat{\mathsf{L}}_{f}\,,\,f\in\mathtt{B}_{0,\kappa}\\}$ forms a
commutative subalgebra in $(\mathscr{D}(V)\otimes\mathbb{C}S_{n})^{S_{n}}$.

The fact that $f(\lambda,y)$ are regular near $\lambda=0$ for any
$f\in\mathtt{B}_{0,\kappa}$ follows from the second proof of [EFMV, Theorem
3.1]. To be precise, in loc. cit. this is shown for $f={L}^{(r)}_{q,c}$, cf.
Theorem 3.1. However, the proof applies verbatim to any
$f\in\mathtt{B}_{0,\kappa}$, see [EFMV, Sec. 5.3] for the details. The
remaining statements are clear. ∎

### 4.5.

We can also substitute Dunkl operators into elliptic Hamiltonians (cf. Prop.
3.4) and obtain

$\widehat{\mathsf{H}}_{q}:=\widehat{\mathtt{Res}}\lim_{\lambda\to
0}L_{q,c}(\lambda,y)\,,\qquad q\in\mathbb{C}[V^{*}]^{S_{n}}\,.$ (4.8)

These obviously commute between themselves and with any of the Hamiltonians
$\widehat{\mathsf{L}}_{f}$, $f\in\mathtt{B}_{0,\kappa}$.333One can show that,
in fact, $\mathsf{H}_{q}=\widehat{\mathsf{L}}_{f}$ for some
$f\in\mathtt{B}_{0,\kappa}$, but we will not need this result. We will refer
to the elements $\widehat{\mathsf{L}}_{f}$, $f\in\mathtt{B}_{0,\kappa}$ as
higher spin CM Hamiltonians, and to $\widehat{\mathsf{H}}_{q}$,
$q\in\mathbb{C}[V^{*}]^{S_{n}}$ as _principal_ spin CM Hamiltonians.

##  5\. Inozemtsev quantum spin chain

We are now in a position to construct commuting Hamiltonians for the
Inozemtsev spin chain (1.1). They will be obtained from the spin CM
Hamiltonians by taking classical limits and evaluating at an equilibrium. In
particular, this will provide a full justification for the “freezing”
procedure [P2].

### 5.1.

Consider the classical Hamiltonian $H$ (3.2), assuming
$x_{i},p_{i}\in\mathbb{R}$. Suppose $\tau\in\mathrm{i}\mathbb{R}_{+}$, so
$\wp(z)$ is real for real $z$. The following fact is well known (it follows
easily from the convexity of $\wp(z)$ for $0<z<1$).

######  Lemma 5.1.

In the region $x_{1}<\dots<x_{n}<x_{1}+1$, all possible equilibria $(x,p)$ of
$H$ are of the form $x_{i}=c+\frac{i}{n}$ (with arbitrary $c\in\mathbb{R}$),
$p_{i}=0$ ($i=1,\dots,n$).

Write $(x^{*},p^{*})$ for an equlibrium of $H$ of the form
$x_{i}=\frac{i}{n}$, $p_{i}=0$.

######  Corollary 5.2.

If a function $F=F(x,p)$ satisfies $\\{F,H\\}=\\{F,x_{1}+\dots+x_{n}\\}=0$
then $(x^{*},p^{*})$ is an equilibrium for $F$.

Indeed, the Hamiltonian flow defined by $F$ should map an equilibrium for $H$
to an equilibrium, at the same time preserving $x_{1}+\dots+x_{n}$. ∎

######  Remark 5.3.

Assuming that $F$ depends analytically on $\tau$, the above corollary remains
valid for arbitrary, not necessarily real, $\tau$. This further implies that
for any primitive period $\omega\in\mathbb{Z}+\mathbb{Z}\tau$, the point
$(x,p)$ with $x_{i}=\frac{i}{n}\omega$, $p_{i}=0$ is an equilibrium for $F$.

### 5.2.

Recall CM Hamiltonians $\mathsf{H}_{q}$, see § 3.6. Let us restrict the choice
of $q$ to

$\mathbb{C}[V^{*}]_{0}^{S_{n}}:=\mathbb{C}[V^{*}]^{S_{n}}/\langle
e_{1}+\dots+e_{n}\rangle\,.$

In other words, we assume that $q$ is a symmetric polynomial of

$\xi_{i}=e_{i}-\frac{1}{n}(e_{1}+\dots+e_{n})\,,\qquad i=1,\dots,n\,.$

Let $y^{0}_{i}=y_{\xi_{i}}$ be the corresponding elliptic Dunkl operators. It
is easy to check that $[y^{0}_{i},x_{1}+\dots+x_{n}]=0$. Hence, the quantum CM
Hamiltonians $\mathsf{H}_{q}$ built from $y^{0}_{i}$ would commute with
$x_{1}+\dots+x_{n}$, and their classical limit, $\mathsf{H}_{q,c}$, would
Poisson-commute with $x_{1}+\dots+x_{n}$, so we can apply the Corollary 5.2.

######  Proposition 5.4.

The classical CM Hamiltonians $\mathsf{H}_{q,c}=\eta_{0}(\mathsf{H}_{q})$ with
$q\in\mathbb{C}[V^{*}]^{S_{n}}_{0}$ have a joint equilibrium at
$(x^{*},p^{*})$. Therefore,

$\\{\mathsf{H}_{q,c},G\\}=0\quad\text{at}\ (x,p)=(x^{*},p^{*})\qquad\forall\
G=G(x,p).$ (5.1)

∎

### 5.3.

Recall the commuting spin CM Hamiltonians $\widehat{\mathsf{L}}_{f}$,
$\widehat{\mathsf{H}}_{q}$, see § 4.4, § 4.5. Let us view them as elements of
$A_{\hbar}\otimes\mathbb{C}S_{n}$, so we can take classical limits using a
map, similar to (2.7):

$\eta_{0}\,:\,A_{\hbar}\otimes\mathbb{C}S_{n}\to
A_{0}\otimes\mathbb{C}S_{n}\,,\qquad f\mapsto f\,,\ \ \hat{p}_{i}\mapsto
p_{i}\,,\ \ w\mapsto w\,.$

By applying $\widehat{\mathtt{Res}}$ to (3.14), we have

$\widehat{\mathsf{H}}_{q}=\mathsf{H}_{q}+\hbar\widehat{\mathsf{A}}_{q}\qquad\text{with}\
\widehat{\mathsf{A}}_{q}\in A_{\hbar}\otimes\mathbb{C}S_{n}\,.$ (5.2)

We define _principal Inozemtsev Hamiltonians_ by taking the classical limit of
the spin part $\widehat{\mathsf{A}}_{q}$ and evaluating it at equilibrium ,
i.e.

$\mathscr{H}_{q}:=\eta_{0}(\widehat{\mathsf{A}}_{q})|_{(x,p)=(x^{*},p^{*})}\,,\quad
q\in\mathbb{C}[V^{*}]^{S_{n}}\,.$ (5.3)

For example, for $q=e_{1}^{2}+\dots+e_{n}^{2}$ one finds from (4.7):

$\displaystyle\mathscr{H}_{q}$
$\displaystyle=2\kappa\sum_{i<j}^{n}\wp\left(\frac{i-j}{n}\right)(\widehat{s}_{ij}-\widehat{\mathrm{id}})\,.$

The multiple of the identity can be removed so this is essentially the same as
(1.1).

######  Theorem 5.5.

The elements $\mathscr{H}_{q}\in\mathbb{C}S_{n}$ pairwise commute.

###### Proof.

First, note that for $q=e_{1}+\dots+e_{n}$, $\mathscr{H}_{q}=0$. Hence, it is
sufficient to prove commutativity for $q\in\mathbb{C}[V^{*}]^{S_{n}}_{0}$. We
need the following elementary lemma.

######  Lemma 5.6.

Let $B=B_{0}+\hbar B_{1}$, $C=C_{0}+\hbar C_{1}$ be elements of
$A_{\hbar}\otimes\mathbb{C}S_{n}$ with $B_{0},C_{0}\in A_{\hbar}\subset
A_{\hbar}\otimes\mathbb{C}S_{n}$. Denote by $b_{0},b_{1},c_{0},c_{1}$ the
classical limits of $B_{0},B_{1},C_{0},C_{1}$. Then, assuming
$[B_{0},C_{0}]=0$, one has $[B,C]\in\hbar^{2}A_{\hbar}\otimes\mathbb{C}S_{n}$,
and

$\eta_{0}(\hbar^{-2}[B,C])=\\{b_{0},c_{1}\\}-\\{c_{0},b_{1}\\}+[b_{1},c_{1}]\,.$

Here the Poisson bracket in the r.h.s. is understood as
$\\{a,b\otimes\widehat{w}\\}:=\\{a,b\\}\otimes\widehat{w}$ for $a,b\in A_{0}$.

Now, given $q,q^{\prime}\in\mathbb{C}[V^{*}]^{S_{n}}_{0}$, we apply the lemma
to $B=\widehat{\mathsf{H}}_{q}$, $C=\widehat{\mathsf{H}}_{q^{\prime}}$ and use
the decomposition (5.2). Since $[B,C]=0$ in this case, we get
$0=\\{b_{0},c_{1}\\}-\\{c_{0},b_{1}\\}+[b_{1},c_{1}]$. The first two terms in
the r.h.s. vanish at $(x,p)=(x^{*},p^{*})$, by Prop. 5.4. As a result,
$[b_{1},c_{1}]$ vanishes at $(x^{*},p^{*})$, i.e.
$[\mathscr{H}_{q},\mathscr{H}_{q^{\prime}}]=0$.444When finalising this paper,
we learned about the work [LRS] where a similar idea was used for the
Haldane–Shastry chain. ∎

### 5.4.

We proceed to define _higher Inozemtsev Hamiltonians_ by

$\mathscr{H}_{f}:=\eta_{0}(\widehat{\mathsf{L}}_{f})|_{(x,p)=(x^{*},p^{*})}\,,\quad
f\in\mathtt{B}_{0,\kappa}\,.$ (5.4)

######  Theorem 5.7.

The elements $\mathscr{H}_{f}$ pairwise commute, as well as commuting with the
principal Hamiltonians $\mathscr{H}_{q}$.

The fact that $[\mathscr{H}_{f},\mathscr{H}_{f^{\prime}}]=0$ for
$f,f^{\prime}\in\mathtt{B}_{0,\kappa}$ is immediate from
$[\widehat{\mathsf{L}}_{f},\widehat{\mathsf{L}}_{f^{\prime}}]=0$ (evaluation
at $(x^{*},p^{*})$ plays no role here). The fact that
$[\mathscr{H}_{q},\mathscr{H}_{f}]=0$ follows from the following lemma
(obtained by setting $C_{0}=0$ in Lemma 5.6).

######  Lemma 5.8.

Let $B=B_{0}+\hbar B_{1}$, $C$ be elements of
$A_{\hbar}\otimes\mathbb{C}S_{n}$ with $B_{0}\in A_{\hbar}\subset
A_{\hbar}\otimes\mathbb{C}S_{n}$. Denote by $b_{0},b_{1},c$ the classical
limits of $B_{0},B_{1},C$. Then one has $[B,C]\in\hbar
A_{\hbar}\otimes\mathbb{C}S_{n}$, and

$\eta_{0}(\hbar^{-1}[B,C])=\\{b_{0},c\\}+[b_{1},c]\,.$

∎

######  Remark 5.9.

The above results and constructions remain valid for elliptic Calogero–Moser
systems associated for any root system (including the $BC_{n}$ version) - all
the necessary ingredients can already be found in the cited sources. In
general, equilibria for these systems will not have such a simple form as
above, which makes the corresponding spin chains less natural. In some cases,
however, simple equilibria configurations are possible, see, e.g., [IS, BPS].

##  6\. Explicit examples

Here we calculate two of the Hamiltonians explicitly, as an illustration and
to compare with known results.

### 6.1.

We use the following shorthand notation:
$\phi_{ij}^{ab}=\phi(\lambda_{a}-\lambda_{b},x_{i}-x_{j})$,
$\phi_{ij}:=\phi_{ij}^{ij}$,
$h_{ij}^{ab}=-\phi^{\prime}(\lambda_{a}-\lambda_{b},x_{i}-x_{j})$,
$h_{ij}:=h_{ij}^{ij}$, $u_{ij}=\wp(x_{i}-x_{j})$,
$u^{ab}=\wp(\lambda_{a}-\lambda_{b})$, $\theta_{i}=\sum_{j\neq
i}\phi_{ij}s_{ij}$. With these notations, the Dunkl operators are
$y_{i}={\hat{p}}_{i}-\kappa\theta_{i}$. Let us substitute them into the cubic
classical Hamiltonian $L=L_{q}$ with $q=\sum_{i<j<k}e_{i}e_{j}e_{k}$:

$L=\sum_{i<j<k}\left(p_{i}p_{j}p_{k}+\kappa^{2}u^{ij}p_{k}+\kappa^{2}u^{ik}p_{j}+\kappa^{2}u^{jk}p_{i}\right)\,.$

Thus, we need to evaluate

$L(\lambda,y)=\frac{1}{6}\sum_{i\neq j\neq
k}y_{i}y_{j}y_{k}+\kappa^{2}\sum_{i\neq j\neq k}u^{ij}y_{k}\,.$ (6.1)

First, consider

$y_{i}y_{j}y_{k}={\hat{p}}_{i}{\hat{p}}_{j}{\hat{p}}_{k}-\kappa({\hat{p}}_{i}{\hat{p}}_{j}\theta_{k}+{\hat{p}}_{i}\theta_{j}{\hat{p}}_{k}+\theta_{i}{\hat{p}}_{j}{\hat{p}}_{k})+\kappa^{2}(\theta_{i}\theta_{j}{\hat{p}}_{k}+\theta_{i}{\hat{p}}_{j}\theta_{k}+{\hat{p}}_{i}\theta_{j}\theta_{k})-\kappa^{3}\theta_{i}\theta_{j}\theta_{k}\,.$

When expanding each $\theta_{a}$ in terms of $\phi_{ab}s_{ab}$ and taking the
sum of these over all $i\neq j\neq k$, some terms cancel due to
$\phi_{ia}=-\phi_{ai}$. For example, the term
${\hat{p}}_{i}{\hat{p}}_{j}\phi_{ka}s_{ka}$ (with $a\neq i,j,k$) coming from
$y_{i}y_{j}y_{k}$ cancels with a similar term
${\hat{p}}_{i}{\hat{p}}_{j}\phi_{ak}s_{ak}$ coming from $y_{i}y_{j}y_{a}$. In
addition, some terms linear in momenta cancel due to Fay’s identity,

$\phi_{ij}^{ik}\phi_{jk}=\phi_{ik}^{jk}\phi_{ij}+\phi_{jk}^{ji}\phi_{ik}\qquad(i\neq
j\neq k)\,,$

which implies that for $a\neq i,j$

$\phi_{ia}s_{ia}\phi_{ja}s_{ja}+\ \text{(cyclic permutations of $i,j,a$)}\
=0\,.$

Similarly, when expanding $\theta_{i}\theta_{j}\theta_{k}$, we may neglect
terms with indices outside $\\{i,j,k\\}$ because of the previous identity and
its higher analogue, valid for $a\neq i,j,k$:

$\phi_{ia}s_{ia}\phi_{ja}s_{ja}\phi_{ka}s_{ka}+\ \text{(cyclic permutations of
$i,j,k,a$)}\ =0\,.$

As a result, when calculating $\sum_{i\neq j\neq k}y_{i}y_{j}y_{k}$, it is
sufficient to keep in each $y_{i}y_{j}y_{k}$ only the terms with indices from
$\\{i,j,k\\}$ set (i.e. calculating $y_{i}y_{j}y_{k}$ _as if_ it was just for
three particles at $x_{i},x_{j},x_{k}$.) By the same token, in the second sum
in (6.1), we have a cancellation of, e.g., $u^{ij}\phi_{ka}s_{ka}$ with
$u^{ij}\phi_{ak}s_{ak}$ (if $a\neq i,j,k$). As a result, we may replace

$u^{ij}y_{k}\mapsto
u^{ij}({\hat{p}}_{k}-\kappa\phi_{ki}s_{ki}-\kappa\phi_{kj}s_{kj})\,.$

Altogether, (6.1) can be replaced with $\sum_{i<j<k}H_{ijk}$ where

$H_{ijk}=({\hat{p}}_{i}-\kappa\phi_{ij}s_{ij}-\kappa\phi_{ik}s_{ik})({\hat{p}}_{j}-\kappa\phi_{ji}s_{ji}-\kappa\phi_{jk}s_{jk})({\hat{p}}_{k}-\kappa\phi_{ki}s_{ki}-\kappa\phi_{kj}s_{kj})\\\
+\kappa^{2}\left\\{u^{ij}({\hat{p}}_{k}-\kappa\phi_{ki}s_{ki}-\kappa\phi_{kj}s_{kj})+\
\text{(cyclic permutations of $i,j,k$)}\right\\}.$

With the help of Fay’s identity and
$\phi_{ij}^{ij}\phi_{ij}^{ji}=u_{ij}-u^{ij}$, this becomes

$\displaystyle H_{ijk}=$
$\displaystyle{\hat{p}}_{i}{\hat{p}}_{j}{\hat{p}}_{k}+\kappa^{2}u_{ij}{\hat{p}}_{k}+\kappa^{2}u_{ik}{\hat{p}}_{j}+\kappa^{2}u_{jk}{\hat{p}}_{i}$
$\displaystyle-\hbar\kappa(h_{ij}{\hat{p}}_{k}s_{ij}+h_{jk}{\hat{p}}_{i}s_{jk}+h_{ki}{\hat{p}}_{j}s_{ki})$
$\displaystyle+\hbar\kappa^{2}\left\\{(h_{ij}\phi_{ki}^{kj}+h_{ij}^{ik}\phi_{jk})+\
\text{(cyclic permutations of $i,j,k$)}\right\\}(ijk)$
$\displaystyle+\hbar\kappa^{2}\left\\{(h_{ji}\phi_{kj}^{ki}+h_{ji}^{jk}\phi_{ik})+\
\text{(cyclic permutations of $i,j,k$)}\right\\}(kji)\,.$

Taking $\lambda\to 0$ gives

$\displaystyle H_{ijk}=$
$\displaystyle{\hat{p}}_{i}{\hat{p}}_{j}{\hat{p}}_{k}+\kappa^{2}u_{ij}{\hat{p}}_{k}+\kappa^{2}u_{ik}{\hat{p}}_{j}+\kappa^{2}u_{jk}{\hat{p}}_{i}$
$\displaystyle-\hbar\kappa(u_{ij}{\hat{p}}_{k}s_{ij}+u_{jk}{\hat{p}}_{i}s_{jk}+u_{ki}{\hat{p}}_{j}s_{ki})$
$\displaystyle+\hbar\kappa^{2}\left\\{(u_{ij}+u_{jk}+u_{ki})(\zeta_{ij}+\zeta_{jk}+\zeta_{ki})+\frac{1}{2}(u^{\prime}_{ij}+u^{\prime}_{jk}+u^{\prime}_{ki})\right\\}{(ijk)}$
$\displaystyle+\hbar\kappa^{2}\left\\{(u_{kj}+u_{ji}+u_{ik})(\zeta_{kj}+\zeta_{ji}+\zeta_{ik})+\frac{1}{2}(u^{\prime}_{kj}+u^{\prime}_{ji}+u^{\prime}_{ik})\right\\}{(kji)}\,.$

Applying $\widehat{\mathtt{Res}}$ gives a spin Hamiltonian that matches the
expression for $I_{2}$ in [DI2]. Furthermore, picking terms of order $\hbar$
and evaluating at an equilibrium in the classical limit, we get a principal
Inozemtsev Hamiltonian

$\displaystyle\mathscr{H}_{3}=$
$\displaystyle\sum_{i<j<k}\left\\{(\wp_{ij}+\wp_{jk}+\wp_{ki})(\zeta_{ij}+\zeta_{jk}+\zeta_{ki})+\frac{1}{2}(\wp^{\prime}_{ij}+\wp^{\prime}_{jk}+\wp^{\prime}_{ki})\right\\}(\widehat{(ijk)}-\widehat{(kji)})\,.$

Here $\wp_{ij}=\wp(\frac{i-j}{n})$, $\zeta_{ij}=\zeta(\frac{i-j}{n})$,
$\wp^{\prime}_{ij}=\wp^{\prime}(\frac{i-j}{n})$. Note that by the addition
formula for $\wp(z)$,
$\wp_{ij}+\wp_{jk}+\wp_{ki}=(\zeta_{ij}+\zeta_{jk}+\zeta_{ki})^{2}$, so
$\mathscr{H}_{3}$ matches $\hat{I}_{2}$ from [I1].

### 6.2.

Let us also consider higher Hamiltonians obtained from
$f\in\mathtt{B}_{0,\kappa}$. First, let us try

$f=\mathtt{Res}\sum_{i=1}^{n}x_{i}{y}^{(r)}_{i,c}=\sum_{i=1}^{n}x_{i}p_{i}\,.$

Replacing the classical momenta with elliptic Dunkl operators, we get

$f(\lambda,y)=\sum_{i=1}^{n}\lambda_{i}y_{i}=\sum_{i=1}^{n}\lambda_{i}({\hat{p}}_{i}+\kappa\sum_{j\neq
i}\phi_{ij}s_{ij})\,.$

We have

$\phi_{ij}=-\frac{1}{\lambda_{i}-\lambda_{j}}+\zeta(x_{i}-x_{j})+o(1)\quad\text{near}\
\lambda=0\,.$ (6.2)

Using this in the previous formula, we find that

$\lim_{\lambda\to 0}f(\lambda,y)=-\sum_{i<j}s_{ij}\,,$

which commutes with other spin Hamiltonians for trivial reasons. A more
interesting choice is

$f=\sum_{i\neq j\neq
k}x_{i}\left(p_{j}p_{k}+\frac{\kappa^{2}}{x_{jk}^{2}}\right)\,.$

Such $f$ can be obtained by applying $\mathtt{Res}$ to $\sum_{i\neq j\neq
k}x_{i}{y}^{(r)}_{j,c}{y}^{(r)}_{k,c}$ (and subtracting a multiple of
$p_{1}+\dots+p_{n}$). We then need to consider

$f(\lambda,y)=\sum_{i\neq j\neq
k}\lambda_{i}\left(y_{j}y_{k}+\frac{\kappa^{2}}{\lambda_{jk}^{2}}\right)\,.$

Since we are interested in the $\lambda\to 0$ limit, we may expand
$y_{j}y_{k}$ using (6.2) and keep track only of the terms singular in
$\lambda$. After a straightforward rearrangement, the limiting form of
$f(\lambda,y)$ is found as

$-\kappa\sum_{i\neq j\neq
k}{\hat{p}}_{i}s_{jk}-\kappa^{2}\sum_{i<j<k}\left\\{\zeta(x_{i}-x_{j})+\zeta(x_{j}-x_{k})+\zeta(x_{k}-x_{i})\right\\}((ijk)-(kji))\,.$

Applying $\widehat{\mathtt{Res}}$ gives an additional spin CM Hamiltonian

$\widehat{\mathsf{H}}_{f}=-\kappa\sum_{i\neq j\neq
k}{\hat{p}}_{i}\widehat{s_{jk}}+\kappa^{2}\sum_{i<j<k}\left\\{\zeta(x_{i}-x_{j})+\zeta(x_{j}-x_{k})+\zeta(x_{k}-x_{i})\right\\}(\widehat{(ijk)}-\widehat{(kji)})\,.$

This matches the quantum spin Hamiltonian $I_{1}$ found in [DI2]. Taking
classical limit and evaluating at equilibrium, we obtain (setting $\kappa=1$)

$\mathscr{H}_{f}=\sum_{i<j<k}\left\\{\zeta_{ij}+\zeta_{jk}+\zeta_{ki}\right\\}(\widehat{(ijk)}-\widehat{(kji)})\,,\qquad\zeta_{ij}=\zeta\left(\frac{i-j}{n}\right)\,.$

This coincides with $\hat{I}_{1}$ from [I1]. Note that it is surprisingly
difficult to check that $[\hat{I}_{1},\hat{I}_{2}]=0$ by a direct calculation,
see [DI1].

######  Remark 6.1.

For calculating $\mathscr{H}_{f}$ with $f(x,p)\in B_{0,\kappa}$, we may take
classical limit and evaluate at equilibrium immediately, i.e. substitute
$x_{i}\mapsto\lambda$, $p_{i}\mapsto\theta_{i}=\sum_{j\neq i}\phi_{ij}s_{ij}$.
However, finding the limiting value at $\lambda\to 0$ seems a lengthy process
in general.

##  7\. A generalisation

### 7.1.

Consider the following modification of the Hamiltonian (1.1):

$\mathscr{H}^{\vee}=\sum_{i<j}^{n}\phi^{\prime}(\lambda_{i}-\lambda_{j},\frac{i-j}{n})P_{ij}\,.$
(7.1)

Here $\phi^{\prime}(\mu,z)=\frac{d}{dz}\phi(\mu,z)$ is considered as a formal
series in $\mu$,

$\phi^{\prime}(\mu,z)=\sum_{i=0}^{\infty}c_{i}\mu^{i}\,,\qquad
c_{0}=-\wp(z)\,,\ c_{1}=\wp^{\prime}(z)+2\zeta(z)\wp(z)\,,\dots\,,$

making $\phi^{\prime}(\lambda_{i}-\lambda_{j},\frac{i-j}{n})$ a formal series
in $\lambda_{i},\lambda_{j}$. The operator (7.1) is viewed as acting on the
tensor product $U^{\otimes n}$ of $U=\mathbb{C}^{m}\otimes\mathbb{C}[[s]]$,
with $P_{ij}$ permuting the factors, and with the $i$th spectral variable
$\lambda_{i}$ acting as $s^{(i)}$, i.e. as multiplication by $s$ on the $i$th
factor and identity on the others. Note that the action of $\mathscr{H}$
naturally descends onto $(U/s^{l+1}U)^{\otimes n}$, for any $l\geq 0$. In
particular, taking $l=0$ recovers the original Hamiltonian (1.1).

### 7.2.

To demonstrate the integrability of the Hamiltonian (7.1), we first construct
a dynamical spin model, modifying the approach in § 4 to incorporate
dependence on the spectral variables $\lambda$. It will be convenient to
introduce an auxiliary space $V^{\vee}\cong V$ with $\lambda\in V^{\vee}$, and
denote by $S_{n}^{\vee}$ a copy of the symmetric group acting on $V^{\vee}$.
Consider the algebra $\mathscr{D}^{\vee}(V)$ of differential operators on $V$
whose coefficients also depend on $\lambda$; it is generated by (operators of
multiplication by) functions $f\in\mathbb{C}(V\times V^{\vee})$ and the
derivations $\partial_{i}=\partial/\partial x_{i}$. The group $S_{n}\times
S_{n}^{\vee}$ acts on $V\times V^{\vee}$ and on $\mathscr{D}^{\vee}(V)$, so we
can form a crossed product

$\mathscr{A}^{\vee}:=\mathscr{D}^{\vee}(V)*(S_{n}\times S_{n}^{\vee})\,.$

Elements of $\mathscr{A}^{\vee}$ can be uniquely written as

$a=\sum_{w_{1},w_{2}\in S_{n}}a_{w_{1}w_{2}}w_{1}\otimes
w_{2}^{\vee}\quad\text{with}\ a_{w_{1}w_{2}}\in\mathscr{D}^{\vee}(V)\,.$

Define a linear map

$\displaystyle{\mathtt{Res}}^{\vee}$ $\displaystyle:\
\mathscr{A}^{\vee}\,\to\,\mathscr{D}^{\vee}(V)*S_{n}^{\vee}\,,$
$\displaystyle\sum_{w_{1},w_{2}\in S_{n}}a_{w_{1}w_{2}}w_{1}\otimes
w_{2}^{\vee}\ \mapsto\sum_{w_{1},w_{2}\in
S_{n}}a_{w_{1}w_{2}}(w_{2}w_{1}^{-1})^{\vee}\,.$ (7.2)

Equivalently, ${\mathtt{Res}}^{\vee}(a)$ is the unique element
$L_{a}\in\mathscr{D}^{\vee}(V)*S_{n}^{\vee}$ such that

$ae^{\vee}=L_{a}e^{\vee}\,,\qquad e^{\vee}:=\frac{1}{n!}\sum_{w\in
S_{n}}w\otimes w^{\vee}\,.$ (7.3)

We have an $S_{n}$-action on $\mathscr{A}^{\vee}$ and
$\mathscr{D}^{\vee}(V)*S_{n}^{\vee}$ by conjugation,

$a\mapsto(w\otimes w^{\vee})a(w\otimes w^{\vee})^{-1}\qquad\forall\ w\in
S_{n}\,.$ (7.4)

It is easy to check that the map ${\mathtt{Res}}^{\vee}$ is
$S_{n}$-equivariant with respect to this action.

######  Lemma 7.1.

The restriction
${\mathtt{Res}}^{\vee}\,:\,(\mathscr{A}^{\vee})^{S_{n}}\to(\mathscr{D}^{\vee}(V)*S_{n}^{\vee})^{S_{n}}$
of the map (4.2) is an algebra homomorphism.

Proof is identical to that of Lemma 4.1. ∎

### 7.3.

Restricting (7.2) further gives an algebra map

$\displaystyle\mathtt{Res}^{\vee}\,:$
$\displaystyle\,(\mathscr{D}^{\vee}(V)*S_{n})^{S_{n}}\to(\mathscr{D}^{\vee}(V)*S_{n}^{\vee})^{S_{n}}\,,$
$\displaystyle\sum_{w\in S_{n}}a_{w}w\mapsto\sum_{w\in
S_{n}}a_{w}(w^{-1})^{\vee}\,.$

Comparing with a similar map in § 4, it takes elements invariant under
“diagonal” $S_{n}$-action (on both $x,\lambda$). Another difference is the
crossed-product structure in the resulting operator so it acts on the spectral
variables.

As for examples of elements in $(\mathscr{D}^{\vee}(V)*S_{n})^{S_{n}}$, we can
substitute elliptic Dunkl operators into any $S_{n}$-invariant function on
$V^{\vee}\otimes V^{*}$. However, we would like the result to be regular near
$\lambda=0$ so we can expand the coefficients into a formal series on
$\lambda$ as we did for (7.1). This motivates the following definitions:

$\displaystyle\mathsf{L}^{\vee}_{f}:=\mathtt{Res}^{\vee}f(\lambda,y)\,,\quad\mathsf{H}^{\vee}_{q}:=\mathtt{Res}^{\vee}L_{q,c}(\lambda,y)\,,\qquad\text{with}\
f\in\mathtt{B}_{0,\kappa}\,,q\in\mathbb{C}[V^{*}]^{S_{n}}\,.$ (7.5)

These are similar to the definitions in § 4, but one important difference is
that we do not take limit $\lambda\to 0$.

######  Theorem 7.2.

The elements $\mathsf{L}^{\vee}_{f}$ and $\mathsf{H}^{\vee}_{q}$ are pairwise
commuting, $S_{n}$-equivariant elements of
$\mathscr{D}^{\vee}(V)*S_{n}^{\vee}$, regular near $\lambda=0$.

Proofs are the same as in § 4. ∎

As an example, using the calculations in § 3.4, we find that for
$q=e_{1}^{2}+\dots+e_{n}^{2}$,

$\mathsf{H}^{\vee}_{q}=\sum_{i=1}^{n}\hbar^{2}\partial_{i}^{2}-2\kappa^{2}\sum_{i<j}^{n}\wp(x_{i}-x_{j})-2\hbar\kappa\sum_{i<j}^{n}\phi^{\prime}(\lambda_{i}-\lambda_{j},x_{i}-x_{j})s^{\vee}_{ij}\,.$
(7.6)

This is a generalisation of the spin Calogero–Moser Hamiltonian (4.1).
Choosing any $\mathbb{C}[[V^{\vee}]]*S_{n}^{\vee}$-module $\mathscr{U}$, and
expanding $\mathsf{H}^{\vee}_{q}$ into a formal series in $\lambda$ makes it a
differential operator acting on $\mathscr{U}$-valued functions
$\varphi:V\to\mathscr{U}$. For example, we can choose $\mathscr{U}=U^{\otimes
n}$ with $U=\mathbb{C}^{m}\otimes\mathbb{C}[[s]]$ and with $\lambda_{i}$
acting by $s^{(i)}$, in the same way as it was done for the Hamiltonian (7.1).

### 7.4.

Now the commuting Hamiltonians for the spin chain (7.1) can be constructed in
the same way as before. We omit the proofs as they are identical to those in §
5 and use evaluation at an equilibrium $(x^{*},p^{*})$.

Let us view the Hamiltonians (7.5) as elements of $A_{\hbar}*S_{n}^{\vee}$, so
we can take classical limits. We have a counterpart of (5.2):

$\mathsf{H}^{\vee}_{q}=\mathsf{H}_{q}+\hbar\mathsf{A}^{\vee}_{q}\qquad\text{with}\
\mathsf{A}^{\vee}_{q}\in A_{\hbar}*S_{n}^{\vee}\,.$ (7.7)

We define _principal Hamiltonians_ by

$\mathscr{H}_{q}^{\vee}:=\eta_{0}(\mathsf{A}^{\vee}_{q})|_{(x,p)=(x^{*},p^{*})}\,,\quad
q\in\mathbb{C}[V^{*}]^{S_{n}}\,.$ (7.8)

For example, for $q=e_{1}^{2}+\dots+e_{n}^{2}$ one finds from (7.6):

$\mathscr{H}_{q}^{\vee}=-2\kappa\sum_{i<j}^{n}\phi^{\prime}\left(\lambda_{i}-\lambda_{j},\frac{i-j}{n}\right){s}_{ij}^{\vee}-2\kappa\sum_{i<j}^{n}\wp\left(\frac{i-j}{n}\right)\mathrm{id}^{\vee}\,.$
(7.9)

The multiple of the identity can be removed, giving

$\mathscr{H}^{\vee}=\sum_{i<j}^{n}\phi^{\prime}\left(\lambda_{i}-\lambda_{j},\frac{i-j}{n}\right){s}_{ij}^{\vee}\,.$
(7.10)

We also define _higher Hamiltonians_ by

$\mathscr{H}^{\vee}_{f}:=\eta_{0}(\mathsf{L}^{\vee}_{f})|_{(x,p)=(x^{*},p^{*})}\,,\quad
f\in\mathtt{B}_{0,\kappa}\,.$ (7.11)

######  Theorem 7.3.

The elements $\mathscr{H}_{q}^{\vee}$, $\mathscr{H}_{f}^{\vee}$ with
$q\in\mathbb{C}[V^{*}]^{S_{n}}$, $f\in\mathtt{B}_{0,\kappa}$ form a
commutative family in $\mathbb{C}[[V^{\vee}]]*S_{n}^{\vee}$.

Choosing $\mathscr{U}=U^{\otimes n}$ with
$U=\mathbb{C}^{m}\otimes\mathbb{C}[[s]]$ as a representation of
$\mathbb{C}[[V^{\vee}]]*S_{n}^{\vee}$ identifies (7.10) with the Hamiltonian
(7.1), hence, it is integrable.

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