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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 be an elliptic curve, τ, 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,τ) generalizing previous work of Sklyanin. In the classical limit “τ0” the algebra Q n,k (E,τ) becomes abelian and determines a Hamiltonian structure on 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 . Given an indecomposable bundle ξ n,k of rank n and degree k>0, the moduli space of vector bundles Y with a sub-bundle (ν,ρ)ξ 0,1 and quotient Y/(ν,ρ)ξ n,k is isomorphic to (Ext(ξ 0,1 ,ξ 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 , where P is a parabolic subgroup of a Kac-Moody group G, with Hamiltonian structure coming form the standard Lie bialgebra structure on LieG. They address the question of the combinatorial structure of the stratification of (Ext(A,B)), where A and B are bundles on . Then some associative algebras are introduced, generalizing Q n,k (E,τ); they allow to quantize the above mentioned Hamiltonian structures in the case when P is a Borel subgroup of G.
MSC:
16S80Deformation theory of associative ring and algebras
14H52Elliptic curves
14H60Vector bundles on curves and their moduli
16W30Hopf algebras (assoc. rings and algebras) (MSC2000)
17B37Quantum groups and related deformations
17A45Quadratic algebras (but not quadratic Jordan algebras)