We construct quadratic finite-dimensional Poisson algebras corresponding to a rank-N degree-one vector bundle over an elliptic curve with n marked points and also construct the quantum version of the algebras. The algebras are parameterized by the moduli of curves. For N = 2 and n = 1, they coincide with Sklyanin algebras. We prove that the Poisson structure is compatible with the Lie-Poisson structure defined on the direct sum of n copies of sl(N). The origin of the algebras is related to the Poisson reduction of canonical brackets on an affine space over the bundle cotangent to automorphism groups of vector bundles.
机构:
Chinese Acad Sci, Acad Math & Syst Sci, Morningside Ctr Math, Beijing 100190, Peoples R ChinaUniv Cambridge Emmanuel Coll, Cambridge CB2 3AP, England
Li, Yongxiong
Tian, Ye
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Chinese Acad Sci, Acad Math & Syst Sci, Morningside Ctr Math, Beijing 100190, Peoples R ChinaUniv Cambridge Emmanuel Coll, Cambridge CB2 3AP, England
Tian, Ye
Zhai, Shuai
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Shandong Univ, Sch Math, Jinan 250100, Shandong, Peoples R China
Univ Cambridge, Dept Pure Math & Math Stat, Cambridge CB3 0WB, EnglandUniv Cambridge Emmanuel Coll, Cambridge CB2 3AP, England