2017/12/05 by Zheng Hua, Hua, Zheng, Alexander Polishchuk +1
Mathematics · #14Dxx #20Gxx #53D17 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Quantum Algebra (math.QA) #Symplectic Geometry (math.SG)
paper · pdf · doi:10.48550/arxiv.1712.01659
openalex publication_date 2017/12/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper we classify symplectic leaves of the regular part of the projectivization of the space of meromorphic endomorphisms of a stable vector bundle on an elliptic curve, using the study of shifted Poisson structures on the moduli of complexes from our previous work \citeHP17. This Poisson ind-scheme is closely related to the ind Poisson-Lie group associated to Belavin's elliptic r-matrix, studied by Sklyanin, Cherednik and Reyman and Semenov-Tian-Shansky. Our result leads to a classification of symplectic leaves on the regular part of meromorphic matrix algebras over an elliptic curve, which can be viewed as the Lie algebra of the above-mentioned ind Poisson-Lie group. We also describe the decomposition of the product of leaves under the multiplication morphism and show the invariance of Poisson structures under autoequivalences of the derived category of coherent sheaves on an elliptic curve.