Garnier integrable system

Let ${\displaystyle {\mathfrak {g}}}$ be a quadratic Lie algebra, that is, a Lie algebra with a non-degenerate invariant bilinear form ${\displaystyle \kappa }$. If ${\displaystyle {\mathfrak {g}}}$ is complex and simple, this can be taken to be the Killing form.

The dual, denoted ${\displaystyle {\mathfrak {g}}^{*}}$, can be made into a linear Poisson structure by the Kirillov–Kostant bracket.

The phase space ${\displaystyle M}$ of the classical Gaudin model is then the Cartesian product of ${\displaystyle N}$ copies of ${\displaystyle {\mathfrak {g}}^{*}}$ for ${\displaystyle N}$ a positive integer.

Associated to each of these copies is a point in ${\displaystyle \mathbb {C} }$, denoted ${\displaystyle \lambda _{1},\cdots ,\lambda _{N}}$, and referred to as sites.

Fixing a basis of the Lie algebra ${\displaystyle \{I^{a}\}}$ with structure constants ${\displaystyle f_{c}^{ab}}$, there are functions ${\displaystyle X_{(r)}^{a}}$ with ${\displaystyle r=1,\cdots ,N}$ on the phase space satisfying the Poisson bracket

$${\displaystyle \{X_{(r)}^{a},X_{(s)}^{b}\}=\delta _{rs}f_{c}^{ab}X_{(r)}^{c}.}$$

These in turn are used to define ${\displaystyle {\mathfrak {g}}}$-valued functions

$${\displaystyle X^{(r)}=\kappa _{ab}I^{a}\otimes X_{(r)}^{b}}$$

with implicit summation.

Next, these are used to define the Lax matrix which is also a ${\displaystyle {\mathfrak {g}}}$ valued function on the phase space which in addition depends meromorphically on a spectral parameter ${\displaystyle \lambda }$,

$${\displaystyle {\mathcal {L}}(\lambda )=\sum _{r=1}^{N}{\frac {X^{(r)}}{\lambda -\lambda _{r}}}+\Omega ,}$$

and ${\displaystyle \Omega }$ is a constant element in ${\displaystyle {\mathfrak {g}}}$, in the sense that it Poisson commutes (has vanishing Poisson bracket) with all functions.

The (quadratic) Hamiltonian is

$${\displaystyle {\mathcal {H}}(\lambda )={\frac {1}{2}}\kappa ({\mathcal {L}}(\lambda ),{\mathcal {L}}(\lambda ))}$$

which is indeed a function on the phase space, which is additionally dependent on a spectral parameter ${\displaystyle \lambda }$. This can be written as

$${\displaystyle {\mathcal {H}}(\lambda )=\Delta _{\infty }+\sum _{r=1}^{N}\left({\frac {\Delta _{r}}{(\lambda -\lambda _{r})^{2}}}+{\frac {{\mathcal {H}}_{r}}{\lambda -\lambda _{r}}}\right),}$$

with

$${\displaystyle \Delta _{r}={\frac {1}{2}}\kappa (X^{(r)},X^{(r)}),\Delta _{\infty }={\frac {1}{2}}\kappa (\Omega ,\Omega )}$$

and

$${\displaystyle {\mathcal {H}}_{r}=\sum _{s\neq r}{\frac {\kappa (X^{(r)},X^{(s)})}{\lambda _{r}-\lambda _{s}}}+\kappa (X^{(r)},\Omega ).}$$

From the Poisson bracket relation

$${\displaystyle \{{\mathcal {H}}(\lambda ),{\mathcal {H}}(\mu )\}=0,\forall \lambda ,\mu \in \mathbb {C} ,}$$

by varying ${\displaystyle \lambda }$ and ${\displaystyle \mu }$ it must be true that the ${\displaystyle {\mathcal {H}}_{r}}$'s, the ${\displaystyle \Delta _{r}}$'s and ${\displaystyle \Delta _{\infty }}$ are all in involution. It can be shown that the ${\displaystyle \Delta _{r}}$'s and ${\displaystyle \Delta _{\infty }}$ Poisson commute with all functions on the phase space, but the ${\displaystyle {\mathcal {H}}_{r}}$'s do not in general. These are the conserved charges in involution for the purposes of Arnol'd Liouville integrability.

One can show

$${\displaystyle \{{\mathcal {H}}_{r},{\mathcal {L}}(\lambda )\}=\left[{\frac {X^{(r)}}{\lambda -\lambda _{r}}},{\mathcal {L}}(\lambda )\right],}$$

so the Lax matrix satisfies the Lax equation when time evolution is given by any of the Hamiltonians ${\displaystyle {\mathcal {H}}_{r}}$, as well as any linear combination of them.

The quadratic Casimir gives corresponds to a quadratic Weyl invariant polynomial for the Lie algebra ${\displaystyle {\mathfrak {g}}}$, but in fact many more commuting conserved charges can be generated using ${\displaystyle {\mathfrak {g}}}$-invariant polynomials. These invariant polynomials can be found using the Harish-Chandra isomorphism in the case ${\displaystyle {\mathfrak {g}}}$ is complex, simple and finite.