Feigin, B. (ed.) et al., Topics in quantum groups and finite-type invariants. Mathematics at the Independent University of Moscow. Providence, RI: American Mathematical Society. Transl. Math. Monogr. 185(38), 65-84 (1998).

Let

$\mathcal{E}$ be an elliptic curve,

$\tau \in \mathcal{E}$,

$0<k<n$ integers with

$gcd(k,n)=1$. To this datum, the authors attached [in Preprint Inst. Theor. Phys., Kiev (1989); see also Funct. Anal. Appl. 23, No. 3, 207-214 (1989); translation from Funkts. Anal. Prilozh. 23, No. 3, 45-54 (1989;

Zbl 0687.17001)] an associative algebra

${Q}_{n,k}(E,\tau )$ generalizing previous work of Sklyanin. In the classical limit “

$\tau \to 0$” the algebra

${Q}_{n,k}(E,\tau )$ becomes abelian and determines a Hamiltonian structure on

$\u2102{\mathbb{P}}^{n-1}$. One of the main results of the present paper is the determination of the symplectic leaves of this structure in terms of moduli spaces of bundles on

$\mathcal{E}$. Given an indecomposable bundle

${\xi}_{n,k}$ of rank

$n$ and degree

$k>0$, the moduli space of vector bundles

$Y$ with a sub-bundle

$(\nu ,\rho )\simeq {\xi}_{0,1}$ and quotient

$Y/(\nu ,\rho )\simeq {\xi}_{n,k}$ is isomorphic to

$\mathbb{P}\left(\text{Ext}({\xi}_{0,1},{\xi}_{n,k})\right)$. The decomposition of this moduli space as a union of strata, where each stratum corresponds to a type of

$k+1$-dimensional bundles, coincides with the decomposition into the union of symplectic leaves. The authors consider also the more general situation of moduli spaces of

$P$-bundles on

$\mathcal{E}$, where

$P$ is a parabolic subgroup of a Kac-Moody group

$G$, with Hamiltonian structure coming form the standard Lie bialgebra structure on

$\text{Lie}\phantom{\rule{4.pt}{0ex}}G$. They address the question of the combinatorial structure of the stratification of

$\mathbb{P}\left(\text{Ext}\right(A,B\left)\right)$, where

$A$ and

$B$ are bundles on

$\mathcal{E}$. Then some associative algebras are introduced, generalizing

${Q}_{n,k}(E,\tau )$; they allow to quantize the above mentioned Hamiltonian structures in the case when

$P$ is a Borel subgroup of

$G$.