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Vector bundles on an elliptic curve and Sklyanin algebras. (English) Zbl 0916.16014
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 $\Cal E$ be an elliptic curve, $\tau\in{\Cal E}$, $0<k<n$ 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 $\bbfC\bbfP^{n-1}$. 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 $\Cal E$. 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 $\bbfP(\text{Ext}(\xi_{0,1},\xi_{n,k}))$. 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 $\Cal E$, where $P$ is a parabolic subgroup of a Kac-Moody group $G$, with Hamiltonian structure coming form the standard Lie bialgebra structure on $\text{Lie }G$. They address the question of the combinatorial structure of the stratification of $\bbfP(\text{Ext}(A,B))$, where $A$ and $B$ are bundles on $\Cal E$. 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 $G$. For the entire collection see [Zbl 0892.00020].

##### MSC:
 16S80 Deformation theory of associative ring and algebras 14H52 Elliptic curves 14H60 Vector bundles on curves and their moduli 16W30 Hopf algebras (associative rings and algebras) (MSC2000) 17B37 Quantum groups and related deformations 17A45 Quadratic algebras (but not quadratic Jordan algebras)