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Zbl 0916.16014
Feigin, B.L.; Odesskij, A.V.
Vector bundles on an elliptic curve and Sklyanin algebras.
(English)
[A] 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). ISBN 0-8218-1084-7/hbk

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 $G$.
[N.Andruskiewitsch (Córdoba)]
MSC 2000:
*16S80 Deformation theory of associative ring and algebras
14H52 Elliptic curves
14H60 Vector bundles on curves
16W30 Hopf algebras (assoc. rings and algebras)
17B37 Quantum groups and related deformations

Keywords: Sklyanin algebras; elliptic curves; symplectic leaves; moduli spaces of bundles; Kac-Moody groups

Citations: Zbl 0687.17001

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